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+A Quantum Monte Carlo study of the structural, energetic, and magnetic
+properties of two-dimensional (2D) H and T phase VSe2
+Daniel Wines,1, a) Juha Tiihonen,2 Kayahan Saritas,3 Jaron Krogel,3 and Can Ataca4, b)
+1)Materials Science and Engineering Division, National Institute of Standards and Technology (NIST), Gaithersburg,
+MD 20899
+2)Department of Physics, Nanoscience Center, University of Jyväskylä, P.O. Box 35, Finland
+3)Material Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge,
+Tennessee 37831
+4)Department of Physics, University of Maryland Baltimore County, Baltimore MD 21250
+(Dated: 30 January 2023)
+Previous works have controversially claimed near-room temperature ferromagnetism in two-dimensional (2D) VSe2,
+with conflicting results throughout the literature. These discrepancies in magnetic properties between both phases (T
+and H phase) of 2D VSe2 are most likely due to the structural parameters being coupled to the magnetic properties.
+Specifically, both phases have a close lattice match and similar total energies, which makes it difficult to determine
+which phase is being observed experimentally. In this study, we used a combination of density functional theory
+(DFT), highly accurate diffusion Monte Carlo (DMC) and a surrogate Hessian line-search optimization technique to
+resolve the previously reported discrepancy in structural parameters and relative phase stability. With DMC accuracy,
+we determined the freestanding geometry of both phases and constructed a phase diagram. Our findings demonstrate
+the successes of the DMC method coupled with the surrogate Hessian structural optimization technique when applied
+to a 2D magnetic system.
+I.
+INTRODUCTION
+One of the most promising two-dimensional (2D) mag-
+netic materials that has been extensively studied experimen-
+tally and theoretically is 2D VSe2. Similar to other 2D tran-
+sition metal dichalcogenides (such as MoS2)1, VSe2 exists
+in two phases, the T (octahedral phase (1T)-centered honey-
+combs) phase which is metallic and the H (the trigonal pris-
+matic phase (2H)-hexagonal honeycombs, see Fig. 1) phase
+which is semiconducting. Several experimental and theoret-
+ical studies have controversially claimed near-room tempera-
+ture ferromagnetism in VSe2, with conflicting results through-
+out the literature. Density functional theory (DFT) along with
+classical Monte Carlo simulations have been used to obtain
+a)
+b)
+1T-VSe2
+2H-VSe2
+FIG. 1. Top and side view of the atomic structure of monolayer VSe2
+in the a) 1T and b) 2H phase.
+a)Electronic mail: [email protected]
+b)Electronic mail: [email protected]
+an estimate of the Curie temperature of H-VSe2 (291 K)2, but
+the model Ising Hamiltonian used did not take into account the
+magnetic anisotropy energies, which are essential for an accu-
+rate estimation of the Curie temperature of a 2D lattice. The
+Curie temperature of multilayered 2D H-VSe2 has been ex-
+perimentally measured to be 425 K, with the ferromagnetism
+softening as the thickness of the sample increases3. Addi-
+tionally, the experimental Curie temperature for monolayer T-
+VSe2 has ranged from 300 K to 470 K4,5 depending on which
+substrate is used (MoS2, graphite, SiO2-coated silicon). The
+experimental magnetization of T-VSe2 has also been met with
+controversy, with values of 15 µB and 5 µB (per formula unit)
+being reported from two separate studies4,6. Insight has also
+been reported with regards to how the ferromagnetism is en-
+hanced with defects, molecular adsorption and the choice of
+substrate for VSe24,5,7. A wide range of values have also been
+reported for the charge density wave (CDW) transition tem-
+perature for T-VSe2, ranging from 120 K to 350 K3,6,8–10.
+These discrepancies in the electronic and magnetic proper-
+ties of either phase of 2D VSe2 arise from the structural pa-
+rameters of each phase being coupled closely to the magnetic
+and electronic properties and the external factors (substrates,
+defects) of the individual samples. One example of this has
+been a reported discrepancy on which phase (T or H) is en-
+ergetically more favorable. Both the T and H phases have a
+close lattice match and similar total energies, which makes it
+difficult to determine which phase is being observed experi-
+mentally. Recently, it has been reported experimentally that
+the T phase is favored for bulk VSe2, but with dimension-
+ality decrease, the H phase is favored3,11. It has also been
+reported that a T-to-H phase transition can be realized by ther-
+mal annealing11. This same structural phase transition has
+even been reported by applying a biaxial strain of ≈ 3 % (from
+calculated results)7,11,12. Researchers have proposed that this
+lattice strain can be induced by the mismatch that occurs from
+arXiv:2301.11404v1  [cond-mat.str-el]  26 Jan 2023
+
+b
+C
+aC
+a
+bb2
+putting 2D VSe2 on a substrate7,12.
+From a computational perspective, results for VSe2 depend
+heavily on which methodology is employed. In most cases,
+DFT with an empirical Hubbard correction (+U) for corre-
+lated electrons is used13. For example, if the U correction is
+applied for T and H-VSe2, the T phase is more energetically
+favorable, while if no U correction is applied, the H phase
+is more favorable14. In addition to the discrepancies in re-
+sults calculated with DFT+U, results between van der Waals
+(vdW) corrected functionals and hybrid functionals are also
+inconclusive14 in terms of predicting the relative phase stabil-
+ity. In order to alleviate the uncertainty in DFT methods, more
+sophisticated methods can be used such as Diffusion Monte
+Carlo (DMC)15. DMC is a correlated, many-body electronic
+structure method that has demonstrated success for the elec-
+tronic and magnetic properties of a variety of bulk and 2D
+systems16–24. This method has a weaker dependence on the
+starting density functional and U parameter and can success-
+fully achieve results with an accuracy beyond the DFT+U15.
+Due to the fact that T and H-VSe2 have structural parame-
+ters that are coupled to their electronic and magnetic proper-
+ties, it makes it difficult to produce conclusive results that rely
+solely on DFT or DFT+U. For this reason, we employed our
+recently developed energy-based surrogate Hessian method
+for structural optimization with stochastic electronic structure
+theories (such as DMC)22 to obtain the geometry of T and
+H-VSe2 with DMC accuracy, resulting in high-accuracy bond
+lengths that resolve previous functional dependent structural
+discrepancies. After obtaining an accurate geometry for both
+structures, we constructed a phase diagram between T and H-
+VSe2 using DMC calculated energies and obtained accurate
+magnetic properties of each structure. The accurate estimates
+for lattice geometry, relative phase energy and the DMC phase
+diagram assist in clarifying previously inconclusive theoreti-
+cal and experimental results regarding T and H phase VSe2.
+For full details of the computational methods used, see the
+Supporting Information (SI).
+As an initial starting point for our study, we performed
+benchmarking DFT and DFT+U calculations using a variety
+of density functionals (local density approximation (LDA)25,
+Perdew-Burke-Ernzerhof (PBE)26, and strongly constrained
+and appropriately normed (SCAN)27 meta-GGA functionals,
+see SI for more details) and the Vienna Ab initio Simulation
+Package (VASP) code for monolayer T-VSe2 and H-VSe2.
+The goal of these simulations were to assess how sensitive
+the relative energy between the T and H phase is with re-
+spect to functional and material geometry. Another goal of
+these simulations was to benchmark the structural parameters
+of each material with respect to several density functionals. It
+is advantageous to perform these reference calculations with
+VASP and PAW pseudopotentials as a precursor to the more
+expensive DMC calculations due to the fact that they require
+a much smaller cutoff energy and are more cost effective for
+a large number of simulations. It is important to note that
+for all DFT and DMC simulations, we assumed a ferromag-
+netic ground state for both T and H-VSe2. Although recent
+reports have suggested that T-VSe2 could be experimentally
+paramagnetic3, we infer that this paramagnetism can be in-
+duced by magnetic anisotropy. In addition, the modeling of
+paramagnetism with computational methods imposes a great
+challenge, which is why we focus on the freestanding ferro-
+magnetic ground states of both phases. A more robust treat-
+ment of the magnetic structure can be explored in future work,
+but is beyond the scope of this work which primarily focuses
+on determining the geometric structure and phase stability of
+2D T and H-VSe2.
+In Fig. 2 we present a comprehensive look at the difference
+in total energy between T-VSe2 and H-VSe2, using several
+DFT functionals under different geometric constraints. We
+performed these calculations for a variety of U values in three
+different ways: fully relaxing the structure at each value of U
+(Fig. 2 a) ), fixing the lattice and atomic positions to the U
+= 0 eV relaxed geometry of that particular functional and cal-
+culating the static energy at each value of U (Fig 2 b)), fixing
+the lattice to the U = 0 eV relaxed geometry of that particular
+functional and relaxing just the atomic positions at each value
+of U (Fig. 2 c)). The results in Fig. 2 indicate that there is
+a significant disagreement between DFT functionals, U value
+used, and material geometries, with all three factors playing
+a significant role in the energy difference between T and H
+phase. Specifically, regardless of relaxation method, all bare
+(no U correction) SCAN, PBE, and PBEsol functionals pre-
+dict H favorable, while bare LDA predicts T favorable. For
+all functionals, there is a critical value of U that reverses the
+relative phase stability, which is dependent on functional and
+relaxation method. The SCAN functional with a U correction
+predicts T phase favorable, with larger energy differences. As
+seen in Fig. 2, the trends in the relative phase stability be-
+tween Fig. 2 b) and c) are nearly identical, but significantly
+vary from Fig. a). This implies that the density functional is
+strongly coupled to material geometry, but the lattice constant
+change has more of an effect on phase stability than atomic
+positions and bond distances. This is most prevalent for higher
+U values (> 2 eV), where the relaxed geometry changes more
+drastically with U. The interrelated nature of the material’s
+geometry, density functional, and value of U are reasons to
+seek out higher levels of theory beyond DFT/DFT+U such as
+DMC to accurately determine the optimal geometry and rela-
+tive energy between the phases of 2D VSe2.
+The relaxed lattice constants, V-Se distances, and T - H en-
+ergies from Fig. 2 a) are presented in Table I and Fig. 3,
+along with additional VASP reference calculations performed
+with the vdW corrected functionals (PBE-D228, PBE-D329,
+SCAN+rvv1030). The DMC computed parameters are also
+given for comparison in Table I and Fig. 3 (more discussion
+to follow). We observe a ≈ 7 % variability in lattice constant
+across the different methods for T-VSe2 and a ≈ 4 % variabil-
+ity in lattice constant across the different methods for H-VSe2.
+Between both phases, we observe a ≈ 3 % variability in V-Se
+distance (dV−Se). Most strikingly, the energy difference be-
+tween the T and H phases (ET−H) drastically varies depend-
+ing on the material geometry and computational methodology,
+ranging from -0.2 eV/f.u. to 0.06 eV/f.u.. Due to the fact
+that a strain-induced phase transition has been reported be-
+tween T- and H-VSe27,11,12, we decided to perform additional
+VASP benchmarking calculations that involved the applica-
+
+3
+0
+1
+2
+3
+4
+U (eV)
+0
+1
+2
+3
+4
+U (eV)
+0
+1
+2
+3
+4
+U (eV)
+-0.6
+-0.5
+-0.4
+-0.3
+-0.2
+-0.1
+0
+0.1
+T - H Energy (eV/f.u.)
+-0.3
+-0.2
+-0.1
+0
+0.1
+-0.05
+ 0.05
+-0.15
+-0.25
+-0.35
+T - H Energy (eV/f.u.)
+-0.3
+-0.2
+-0.1
+0
+0.1
+-0.05
+ 0.05
+-0.15
+-0.25
+-0.35
+T - H Energy (eV/f.u.)
+PBE
+LDA
+SCAN
+PBESOL
+Full relaxation
+Fixed lattice/relaxed positions
+Fixed lattice/positions
+a)
+b)
+c)
+FIG. 2. Relative (T - H) energy between T and H phase 2D VSe2 as a function of U parameter for several density functionals and methods of
+atomic relaxation: a) fully relaxing the structure, b) fixing the lattice and atomic positions to the U = 0 eV relaxed geometry of that particular
+functional and calculating the static energy, c) fixing the lattice to the U = 0 eV relaxed geometry of that particular functional and relaxing just
+the atomic positions. The dotted line indicates 0 eV.
+TABLE I. Tabulated results for lattice constant, V-Se distance, and relative energy (T - H) for both T and H phase 2D VSe2 for several
+computational methods. DMC error bars (standard error about the mean) are included in parenthesis.
+T-VSe2
+H-VSe2
+Method
+a (Å)
+dV−Se (Å) a (Å)
+dV−Se (Å) ET−H (eV/f.u.)
+PBE
+3.336
+2.489
+3.333
+2.502
+0.045
+PBE+U=2
+3.435
+2.526
+3.364
+2.520
+-0.008
+LDA
+3.228
+2.438
+3.229
+2.445
+-0.026
+LDA+U=2
+3.277
+2.455
+3.266
+2.464
+0.045
+SCAN
+3.387
+2.486
+3.329
+2.486
+0.045
+SCAN+U=2
+3.462
+2.524
+3.353
+2.502
+-0.202
+PBEsol
+3.262
+2.458
+3.272
+2.471
+0.013
+PBEsol+U=2 3.323
+2.483
+3.301
+2.487
+0.025
+PBE-D2
+3.323
+2.484
+3.318
+2.496
+0.010
+PBE-D3
+3.315
+2.485
+3.319
+2.497
+0.042
+SCAN+rvv10 3.379
+2.481
+3.319
+2.482
+0.051
+DMC
+3.414(12) 2.505(7)
+3.335(8) 2.503(5)
+0.06(2)
+tion of tensile and compressive strain for each monolayer. We
+performed these calculations for PBE, SCAN, and LDA (with
+U = 0 eV and U = 2 eV), starting from the U = 0 eV geom-
+etry for each functional. The resulting equations of state are
+depicted in Fig. S3. As seen in the figure, the equation of
+state and resulting strain-induced phase transition is entirely
+dependent on the functional and U value, with no consistent
+trend.
+The strong sensitivity of each monolayer with respect to
+geometry and functional are grounds for using a higher-order
+method such as DMC to obtain a statistically accurate estimate
+of the lattice parameters and relative energy between phases.
+Prior to performing the DMC/line-search calculations, we op-
+timized our nodal surface (orbitals selected for DFT wave-
+function generation). Since DMC has the zero-variance prop-
+erty, it means that as the trial wave function approaches the
+
+4
+ET-H - ET-H
+(DMC)(eV/f.u.)
+dV-Se - dV-Se
+(DMC) (Å)
+dV-Se - dV-Se
+(DMC) (Å)
+a - aDMC (Å)
+a - aDMC (Å)
+T-VSe2
+H-VSe2
+a)
+b)
+c)
+FIG. 3. A summary of the deviation of the geometric properties rel-
+ative to the DMC calculated geometric properties for a) T-VSe2 and
+b) H-VSe2 and c) the the deviation of T - H energy relative to the
+DMC calculated T - H energy for a variety of DFT functionals (U =
+2 eV), where the DMC error bar (standard error about the mean) is
+represented by the red bars.
+exact ground state, the statistical fluctuations in the energy
+reduce to zero15. Although there have been instances where
+various sophisticated methods have been used to optimize the
+nodal surface31–34, we employed the PBE+U approach, where
+the Hubbard (U) value was used as a variational parameter
+to optimize the nodal surface using DMC (similar to other
+successful DMC studies of magnetic materials16,20,21,24,35–37).
+We performed these calculations for both T and H-VSe2 (24
+atom supercells), where we tuned the U value from (1 to 4) eV
+while creating the trial wavefunction and computed the DMC
+energy. The results of these calculations are depicted in Fig.
+S4, where we observe that U = 2 eV yields the lowest energy
+for both phases. It is important to note that for the H phase,
+the DMC energies for U = 1 and U = 2 eV are statistically
+identical. Based on this, we created the trial wavefunction us-
+ing PBE+U (U = 2 eV) for all subsequent DMC calculations
+within the surrogate Hessian line-search for both phases (all
+52 DMC energy evaluations). Since we obtained an optimal
+U value of 2 eV for both materials, we focused our DFT+U
+benchmarking efforts more on U = 2 eV (Fig. 3, Fig 5, Table
+I, Fig. 2, Fig. S3).
+Based on the DMC line-search results, we determined ac-
+curate bounds on the lattice parameter (a) and off-plane dis-
+placement of Se (z), within an error tolerance of 0.018 Å or
+lower for both parameters. This translates to within ≈ 0.5%
+accuracy in a parameter set of a and dV−Se with 95% con-
+fidence. Convergence (absence of significant displacements
+2.45
+2.50
+2.55
+2.60
+V-Se distance (˚A)
+T-phase
+H-phase
+Fit eqm. (T)
+Fit eqm. (H)
+LS eqm. (T)
+LS eqm. (H)
+3.2
+3.3
+3.4
+3.5
+3.6
+3.7
+3.8
+Lattice constant (˚A)
+−2459.5
+−2459.0
+−2458.5
+Energy/f.u. (eV)
+PES (T)
+PES (H)
+FIG. 4. (Top) The phase diagram of 2D VSe2 in terms of a and
+dV−Se. The phase boundary (solid line, black) is estimated from
+bicubic fits. To assure quality of the fits, the estimated ±0.01 eV
+error contours (dotted line) and the minima from the fits (’x’) and the
+line-search (’o’) are all well separated. (Bottom) Slices of the PES at
+dV−Se = 2.505 Å.
+outside of the error tolerance) was achieved after two parallel
+line-search iterations for both phases. This convergence is il-
+lustrated in Fig. S5, where the convergence of the parameter
+offsets of a and z and the convergence of the total energy per
+f.u. are depicted for both T and H phase 2D VSe2 for the ini-
+tial DFT relaxed structure (1) and both subsequent iterations
+of DMC (2 - 3). In addition, the final energy of both of the
+fitted structures (square points) are given.
+The final geometric parameters and relative phase energies
+determined with DMC are given in Table I and Fig. 3. For
+T-VSe2, we determined a lattice constant of 3.414(12) Å and
+a V-Se distance of 2.505(7) Å . For H-VSe2, we determined a
+lattice constant of 3.335(8) Å and a V-Se distance of 2.503(5)
+Å . The DMC finite-size extrapolated energy difference (T
+- H) between the two phases was determined to be 0.06(2)
+eV/f.u., indicating that in freestanding form at the equilibrium
+geometry, H-VSe2 is favored over T-VSe2. When comparing
+these DMC results to the other DFT functionals in Table I and
+Fig. 3, it is clear that very few DFT functionals can repro-
+duce the DMC results for lattice constant, V-Se distance and
+relative energy difference. The SCAN functional comes the
+closest to reproducing all three simultaneous DMC values, but
+still falls slightly short for the V-Se distances of both phases
+and the lattice constant of T-VSe2. The fact that SCAN+U
+successfully predicts the structural properties (for H-VSe2)
+and the fact that SCAN+rvv10 produces an energy difference
+closest to the average DMC energy difference for both phases
+loosely implies that a simultaneous description of correlated
+magnetism and vdW interactions are both needed to correctly
+represent the physics of VSe2. Experimental measurements of
+
+5
+the lattice constant and V-Se distance of freestanding mono-
+layer VSe2 are scarce and often times dependent on external
+factors such as the substrate (more discussion to follow) and
+sample preparation technique4,5,38,39. However, Chen et al.38
+have recently reported a lattice constant of 3.4 Å for thin films
+of T-VSe2 and Liu et al.39 have recently reported a lattice
+constant of 3.3 Å for epitaxially grown monolayer H-VSe2.
+Both of these measured values are in excellent agreement with
+our DMC computed lattice constants. Additionally, we deter-
+mined the near-equilibrium PES of both T and H 2D VSe2
+with DMC accuracy, which are both depicted in Fig. S6.
+The phase diagram presented in Fig. 4 is based on similar
+fits to data, where the z displacement has been remapped to
+dV−Se. This DMC phase diagram can directly be compared to
+the energy vs. strain DFT benchmarking calculations in Fig.
+S3, which emphasizes the need for an accurate representation
+of the phase boundary between the two phases. The freestand-
+ing geometries of both T and H lie in the energetic H phase,
+but a slice of the phase diagram along dV−Se = 2.505 Å in-
+dicates that the T phase becomes favorable over H at biaxial
+strain of a ≳ 3.5 Å. This implies that in freestanding form,
+once T-VSe2 is positively strained at least ≈ 2.5 %, T phase is
+favored over H. Alternatively, if freestanding H-VSe2 is pos-
+itively strained at least ≈ 5 %, T phase is also favored over
+H This strain can easily be accomplished by placing mono-
+layer VSe2 on a substrate with significant lattice mismatch. In
+fact, this type of mismatch has been reported to alter the mate-
+rial properties4,5,40,41, significantly contributing to the contro-
+versies of T and H-VSe2 (for energetic favorability, magnetic
+properties). Whether or not the changes in energetic favorabil-
+ity or magnetic properties with respect to the substrate are due
+to lattice mismatch or more complicated interactions between
+the substrate and the monolayer remains to be answered and
+is beyond the scope of this work, which has focused solely on
+the freestanding forms of T and H-VSe2. However, such cal-
+culations can be employed for future work using higher order
+methods such as DMC. The proximity of the phase boundary
+between T and H phase (Fig. 4) is emphasized by the small en-
+ergy difference between the two phases (0.06(2) eV/f.u., at the
+equilibrium geometry) between the two curves. Since this en-
+ergy difference is so close to room temperature (≈ 0.024 eV),
+this implies that a process such as thermal annealing can eas-
+ily induce a phase transition. In fact, recently it was demon-
+strated that a structural phase transition of multilayer VSe2
+from T to H occurs through annealing at 650 K, along with a
+metal-insulator transition11.
+To gain a deeper understanding of the magnetic properties
+of 2D T and H-VSe2, we extracted the spin densities (using a
+trial wavefunction at U = 2 eV and 24 atom supercell at the
+final equilibrium geometry predicted by DMC/line-search).
+The spin density isosurfaces of each phase (ρup - ρdown) are
+depicted in the insets of Fig. 5 a) and c) for T-VSe2 and H-
+VSe2 respectively. For both phases, we observe the V atoms
+are highly spin-polarized, while the Se atoms are slightly an-
+tiparallel with respect to the V atoms. For more calculation
+details regarding spin density, see SI.
+We went on to plot the radial averaged spin densities as a
+function of distance, separately for V and Se for T and H-VSe2
+(depicted in Fig. 5 a) - d)). This allows us to view the spa-
+tial variations in spin density. Additionally, we benchmarked
+these V and Se radially averaged densities with PBE+U (U
+= 2 eV) using NC pseudopotentials at the equilibrium geom-
+etry (the calculation required to create the trial WF for the
+subsequent DMC runs). As seen in Fig. 5 a) and c), there is
+a substantial difference in the V spin density between DMC
+and PBE+U (U = 2 eV) for both T and H phase. This same
+substantial difference between DMC and PBE+U also occurs
+for the total charge density. This discrepancy is most preva-
+lent near the radial density peak (peak of d orbital) and can
+be attributed to the fact that DFT functionals (even with the
+added Hubbard correction) tend to delocalize and unsuccess-
+fully capture 3d orbitals. This large discrepancy in the spin
+densities highlights the need for more accurate, many-body
+computational methodologies for correlated materials such as
+VSe2, where DFT fails. In contrast, there is closer agreement
+between the DMC and PBE+U spin densities for Se in T and
+H-VSe2 (see Fig. 5 b) and d).
+Finally, we estimated the site-averaged atomic magnetic
+moments per V and Se for both T and H phase by integrating
+the DMC and PBE+U spin densities depicted in Fig. 5. At the
+DMC level, we estimated a magnetic moment of 1.06(2) µB
+for V and -0.09(2) µB for Se in T-VSe2 and a magnetic mo-
+ment of 1.02(1) µB for V and -0.14(1) µB for Se in H-VSe2.
+At the PBE+U (U = 2 eV) level, we estimated a magnetic mo-
+ment of 1.30 µB for V and -0.12 µB for Se in T-VSe2 and a
+magnetic moment of 1.40 µB for V and -0.15 µB for Se in H-
+VSe2. Consistent with the radial spin density results in Fig.
+5, we find that the DMC and PBE+U magnetic moments for
+Se are in much closer agreement than for V (for both T and
+H phase). By analyzing the spin densities and obtaining the
+on-site magnetic moments, we obtain a clear picture of how
+the magnetization of each ion depends on the computational
+method used, serving as a benchmark for the magnetic prop-
+erties of 2D VSe2.
+In this work, we used a combination of DFT, DMC and
+a recently developed surrogate Hessian line-search optimiza-
+tion technique to resolve the previously reported discrepancy
+in structural parameters and relative phase stability of mono-
+layer T-VSe2 and H-VSe2. Using these methods, we deter-
+mined the lattice constant and V-Se distance (with DMC ac-
+curacy) to be 3.414(12) Å and 2.505(7) Å respectively for T-
+VSe2 and 3.335(8) Å and 2.503(5) respectively for H-VSe2.
+In addition, we find the relative energy between the phases (T
+- H) to be 0.06(2) eV/f.u. at the DMC level, indicating that
+in freestanding form, H-VSe2 is more energetically favorable
+than T-VSe2. We went on to obtain a phase diagram between
+T and H phase from the PES and determined that a phase tran-
+sition can be induced by strain or mechanisms such as ther-
+mal annealing. Additionally, we benchmarked the magnetic
+properties such as spin density and on-site magnetic moment
+for both phases and find substantial differences between DMC
+and DFT. The results of this study demonstrate the successes
+of the DMC method coupled with the surrogate Hessian line-
+search structural optimization technique when applied to a 2D
+magnetic system.
+The estimates for lattice constant, bond
+distance, relative phase energy and the extracted structural-
+
+6
+a)
+b)
+c)
+d)
+4̟r2[ρup - ρdown ] (Ne/Å)
+4̟r2[ρup - ρdown ] (Ne/Å)
+4̟r2[ρup - ρdown ] (Ne/Å)
+4̟r2[ρup - ρdown ] (Ne/Å)
+T-VSe2 (V)
+T-VSe2 (Se)
+H-VSe2 (V)
+H-VSe2 (Se)
+r (Å)
+r (Å)
+r (Å)
+r (Å)
+MV= 1.30 µB
+MV= 1.06(2) µB
+MV= 1.40 µB
+MV= 1.02(1) µB
+MSe= -0.12 µB
+MSe= -0.09(2) µB
+MSe= -0.15 µB
+MSe= -0.14(1) µB
+FIG. 5. The radially averaged spin density (ρup - ρdown) as a function of distance, calculated with DMC and PBE+U (U = 2 eV) of a) V and
+b) Se for 2D T-VSe2 and c) V and d) Se for 2D H-VSe2. The inset of a) and c) depicts the spin isosurface density of T-VSe2 and H-VSe2
+respectively, where the isosurface value was set to 6 x 10−3 e/Å3. The standard error about the mean for DMC is indicated by error bars in
+blue.
+dependent phase diagram assist in clarifying previously incon-
+clusive theoretical and experimental results regarding T and H
+phase VSe2.
+II.
+CODE AVAILABILITY STATEMENT
+Software packages mentioned in the article can be found at
+https://github.com/usnistgov/jarvis. Please note that the use of
+commercial software (VASP) does not imply recommendation
+by the National Institute of Standards and Technology.
+III.
+COMPETING INTERESTS
+The authors declare no competing interests.
+IV.
+ACKNOWLEDGMENTS
+The authors thank the National Institute of Standards
+and Technology for funding,
+computational,
+and data-
+management resources.
+The authors thank Dr.
+Kamal
+Choudhary and Dr.
+Francesca Tavazza for fruitful discus-
+sions.
+We acknowledge grants of computer capacity from
+the Finnish Grid and Cloud Infrastructure (persistent identi-
+fier urn:nbn:fi:research-infras-2016072533).
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+
+Supporting Information: A Quantum Monte
+Carlo study of the structural, energetic, and
+magnetic properties of two-dimensional (2D) H
+and T phase VSe2
+Daniel Wines,∗,† Juha Tiihonen,‡ Kayahan Saritas,¶ Jaron Krogel,§ and Can
+Ataca∗,∥
+†Materials Science and Engineering Division, National Institute of Standards and
+Technology (NIST), Gaithersburg, MD 20899
+‡Department of Physics, Nanoscience Center, University of Jyv¨askyl¨a, P.O. Box 35,
+Finland
+¶ Department of Applied Physics, Yale University, New Haven CT 06520
+§ Material Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge,
+Tennessee 37831
+∥Department of Physics, University of Maryland Baltimore County, Baltimore MD 21250
+E-mail: [email protected]; [email protected]
+Computational Methods
+Density functional theory (DFT) benchmarks for the T and H phase of 2D VSe2 were per-
+formed using the Vienna Ab initio Simulation Package (VASP) code with projector aug-
+mented wave (PAW) pseudopotentials.1,2 For these calculations, the local density approxi-
+S1
+arXiv:2301.11404v1  [cond-mat.str-el]  26 Jan 2023
+
+mation (LDA),3 Perdew-Burke-Ernzerhof (PBE),4 and strongly constrained and appropri-
+ately normed (SCAN)5 meta-GGA functionals were used with the added Hubbard correction
+(U)6 to treat the on-site Coulomb interaction of the 3d orbitals of the V atoms. At least 20
+˚A of vacuum was given between periodic layers of VSe2 in the c-direction. In addition, we
+used a reciprocal grid of 24x24x1 and a kinetic energy cutoff of 400 eV.
+Our Quantum Monte Carlo (QMC) simulations used DFT-PBE to generate the trial
+wavefunction for fixed-node diffusion Monte Carlo (DMC) calculations.
+The Quantum
+Espresso (QE)7 code was used for our DFT calculations to create the trial wavefunction.
+This trial wavefunction was created for the ferromagnetic configuration of 2D VSe2 using
+different U values with the goal of variationally determining the optimal nodal surface (U
+value that yields the lowest total energy). For V, we used norm-conserving (NC) RRKJ
+(OPT) pseudopotentials8 and for Se, we used NC Burkatzki-Fillipi-Dolg (BFD) pseudopo-
+tentials.9 After testing at the DFT level, a kinetic energy cutoff of 4,080 eV (300 Ry) and
+a k-grid of 6x6x1 was used (see Fig. S1 and S2) to generate trial wavefunctions for DMC.
+To accelerate the line-search method convergence for the metallic T phase, we increased the
+k-grid to 12x12x1.
+After the trial wavefunction was generated with DFT, Variational Monte Carlo (VMC)
+and DMC10,11 calculations were performed using the QMCPACK12,13 code. The single de-
+terminant DFT wavefunction is converted into a many-body wavefunction by use of the Jas-
+trow parameters,14,15 which assist in modeling electron correlation with the goal of reducing
+the statistical uncertainty in DMC calculations.16,17 Up to two-body Jastrow18 correlation
+functions were included, where the linear method19 was used to minimize the variance and
+energy of the VMC energies. The cost function of the variance optimization is 100 % vari-
+ance minimization and the cost function of the energy optimization is split as 95 % energy
+minimization and 5 % variance minimization, which has been proven to reduce the uncer-
+tainty of DMC calculated results.16 The Nexus20 software suite was used to automate the
+DFT-VMC-DMC workflow. The locality approximation17 was used to evaluate the nonlocal
+S2
+
+part of the pseudopotentials in DMC and an optimal timestep of 0.01 Ha−1 was determined
+for DMC simulations due to the fact that it yielded an acceptance ratio greater than 99 %
+(see Table S1). A full summary of the VMC and DMC methods can be found in reference.10
+The total charge density and spin density was extracted from our DMC calculations.
+The spin density is defined as the difference between the spin-up contribution to the total
+charge density and the spin-down contribution to the total charge density (ρup − ρdown). We
+used an extrapolation scheme on the DMC charge densities with the goal of eliminating the
+bias that occurs from using a mixed estimator. Since the charge density estimator does not
+commute with the fixed-node Hamiltonian, the DMC charge density was obtained from a
+mixed estimator between the pure fixed-node DMC and VMC densities. The extrapolation
+formula takes the form:10
+ρ1 = 2ρDMC − ρVMC + O[(Φ − ΨT)2]
+(1)
+where ρDMC and ρVMC are the DMC and VMC charge densities respectively. Φ is the trial
+wavefunction from the DMC Hamiltonian and ΨT is the trial wavefunction from VMC.
+In addition, we integrated the DFT+U and DMC spin densities up to a cutoff radius
+rcut (which we define as 1.34 ˚A , due to the fact that it is approximately half of the V-Se
+bond distance in 2D T and H-VSe2) in order to estimate the site-averaged atomic magnetic
+moment per V and Se. To obtain these magnetic moments per atom (MA), we sum over the
+spherically interpolated spin densities:
+MA = 4π
+� rcut
+0
+r2ρs(r)dr ≈ 4π
+rcut/∆r
+�
+i=0
+r2
+i ρs(ri)∆r
+(2)
+where ri is the distance from the center of the atom to a given point on the grid and ∆r is
+the radial grid size.
+To optimize the structural parameters of both T and H-VSe2 according to the DMC po-
+tential energy surface (PES), we use a surrogate Hessian accelerated optimization method.21
+S3
+
+In the method, we consider the PES around equilibrium as the second-order expansion in
+Wyckoff parameter space, p:
+E(p) = E0 + 1
+2(p − p0)THp(p − p0),
+(3)
+where Hp is the Hessian, or the force-constant matrix, E0 is the energy minimum and p0
+the energy-minimizing parameters. Diagonalizing the parameter Hessian, i.e., Hp = U TΛU,
+forms an optimal basis for a conjugate line-search in the parameter space, namely the eigen-
+vectors U. The line-searches along U can be conducted in parallel, and ideally, they locate
+the minimum in just one parallel iteration within the quadratic region. Here, we conduct
+the line-search according to a set of 2 parameters: the lattice constant a and the Wyckoff
+parameter z, which is the unsigned displacement of the Se atoms along the z axis (see Fig.
+1). For reporting purposes, the line-search parameters a and z are remapped to a and d,
+where d is the V-Se distance.
+In the surrogate Hessian scheme, we obtain a cheap but relatively accurate Hessian from
+DFT, and use it to the inform line-search on the DMC PES, in particular by providing the
+search directions. We also resample the DFT PES to predict fitting errors. Thus, we may
+minimize the computational cost of the DMC runs, while maintaining an error tolerance.
+The surrogate DFT PES was based on QE with a 4,080 eV (300 Ry) cutoff using PBE with
+no DFT+U correction. The DMC PES was based on DFT-PBE with U = 2 eV orbitals
+and finite-size extrapolation through supercell sizes of 9 and 24 atoms. Each line-search was
+based on a 3rd order polynomial fit and set to contain 7 points, or displaced geometries,
+totaling 13 energy evaluations per phase, per iteration. However, alternative techniques,
+including (bi)polynomial fitting, were used in some parts to incorporate auxiliary DMC
+data and ensure convergence to the quadratic region. Effectively, two parallel line-search
+iterations for both phases were carried out, and the convergence was claimed in the absence
+of significant displacements.
+S4
+
+a)
+b)
+Figure S1: The total energy per atom of the unit cell (3 atoms) of 2D a) T-VSe2 and b)
+H-VSe2 as a function of plane wave cutoff energy for the norm-conserving pseudopotentials
+calculated with DFT using the PBE functional at a k-point grid of 6x6x1. The results show
+a converged cutoff energy of 4,080 eV (300 Ry) for both phases.
+a)
+b)
+Figure S2: The total energy per atom of the unit cell (3 atoms) of 2D a) T-VSe2 and b)
+H-VSe2 as a function of K-point grid for the norm-conserving pseudopotentials calculated
+with DFT (PBE) at the converged cutoff energy (see Fig. S1). The results show a converged
+k-point grid of 6x6x1 (36) for both monolayers. The number of K-points was scaled appro-
+priately to obtain the converged grid depending on the supercell size and shape for all DFT
+and DMC calculations.
+S5
+
+PBE (U = 0)
+PBE (U = 2)
+SCAN (U = 0)
+SCAN (U = 2)
+3.15
+3.25
+3.35
+3.45
+3.55
+3.15
+3.25
+3.35
+3.45
+3.55
+3.15
+3.25
+3.35
+3.45
+3.55
+3.15
+3.25
+3.35
+3.45
+3.55
+-18.04
+-18.00
+-17.96
+-17.92
+-17.88
+-17.84
+-16.00
+-15.95
+-15.90
+-15.85
+-15.80
+-15.65
+-15.75
+-15.70
+-59.90
+-59.85
+-59.80
+-59.75
+-58.70
+-59.65
+-58.10
+-58.00
+-57.90
+-57.80
+-57.70
+-57.60
+Total Energy (eV)
+Total Energy (eV)
+Total Energy (eV)
+Total Energy (eV)
+Lattice Constant (Å)
+Lattice Constant (Å)
+Lattice Constant (Å)
+Lattice Constant (Å)
+3.05
+3.15
+3.25
+3.35
+-18.04
+-18.00
+-17.90
+-17.80
+Total Energy (eV)
+Lattice Constant (Å)
+LDA (U = 0)
+3.05
+3.15
+3.25
+3.35
+Lattice Constant (Å)
+LDA (U = 2)
+-20.30
+-20.20
+-20.15
+-20.10
+Total Energy (eV)
+-20.25
+-20.05
+T
+H
+Figure S3: Total energy as a function of lattice strain for T (blue) and H (red) phase 2D
+VSe2, calculated with various functionals and U values. Density functionals include LDA,
+PBE, and SCAN.
+S6
+
+Table S1: Tabulated results for the DMC timestep convergence of a 12 atom cell of 2D
+T-VSe2 and H-VSe2. The acceptance ratio of 0.99 indicates that 0.01 Ha−1 is an appropriate
+timestep to use for all subsequent DMC simulations.
+T-VSe2
+Timestep (Ha−1)
+DMC Total Energy (Ha)
+Error (Ha)
+Acceptance Ratio
+0.02
+-361.730
+0.001
+0.985
+0.01
+-361.709
+0.002
+0.994
+0.005
+-361.709
+0.003
+0.997
+0.002
+-361.702
+0.002
+0.999
+H-VSe2
+Timestep (Ha−1)
+DMC Total Energy (Ha)
+Error (Ha)
+Acceptance Ratio
+0.02
+-361.673
+0.001
+0.985
+0.01
+-361.657
+0.002
+0.994
+0.005
+-361.654
+0.002
+0.998
+0.002
+-361.657
+0.003
+0.999
+1
+2
+3
+4
+U (eV)
+-2460.30
+-2460.25
+-2460.20
+-2460.15
+-2460.10
+-2460.05
+-2460.00
+-2459.95
+Total Energy (eV/f.u.)
+T
+H
+Figure S4: DMC calculated total energies of a 24-atom supercell (normalized per formula
+unit (f.u.))
+of 2D T (blue) and H (red) phase VSe2 calculated as a function of the U
+parameter used to variationally determine the optimal trial wave function. The DMC error
+bars represent the standard error about the mean.
+S7
+
+1.0
+1.5
+2.0
+2.5
+3.0
+−0.02
+−0.01
+0.00
+0.01
+a (˚A )
+1.0
+1.5
+2.0
+2.5
+3.0
+−0.010
+−0.005
+0.000
+0.005
+0.010
+z (˚A )
+1
+2
+3
+Iteration
+-2459.55
+-2459.6
+-2459.65
+-2459.7
+E/f.u. (eV )
+T
+H
+Figure S5: The convergence of the a and z parameters and DMC energies per f.u. for both
+T (blue) and H (red) phase of 2D VSe2 based on parallel line-search iterations along the
+DMC PES. The starting parameters (iteration 1) are from DFT, the zero offset is the mean
+over iterations 2 and 3, and dotted lines indicate the error tolerances for each case (95 %
+confidence). The DMC energies from respective equilibrium geometries are plotted with
+1SEM (one standard error of the mean) uncertainties, with extra squares marking energies
+from the predicted minimum geometry.
+S8
+
+3.00
+3.25
+3.50
+3.75
+4.00
+Lattice constant (˚A )
+1.0
+1.2
+1.4
+1.6
+1.8
+2.0
+Z-offset (˚A )
+LS eqm .
+Fit eqm .
+LS # 0
+LS # 1
+3.00
+3.25
+3.50
+3.75
+4.00
+Lattice constant (˚A )
+0.00
+0.02
+0.04
+0.06
+0.08
+0.10
+Figure S6: Contour reconstructions of the DMC PESs (eV) of T (left) and H (right) phases of
+2D VSe2 with respect to a and z parameters. The contours are based on bicubic fits to sparse
+data, and thus, subject to biases and statistical uncertainties not indicated in the figures.
+The markers (’x’ and ’+’) indicate data points from two parallel line-search iterations.
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+

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+AI based approach to Trailer Generation for Online
+Educational Courses
+1st Prakhar Mishra
+IIIT
+Bangalore, India
+[email protected]
+2nd Chaitali Diwan
+IIIT
+Bangalore, India
+[email protected]
+3rd Srinath Srinivasa
+IIIT
+Bangalore, India
+[email protected]
+4th G. Srinivasaraghavan
+IIIT
+Bangalore, India
+[email protected]
+Abstract—In this paper, we propose an AI based approach
+to Trailer Generation in the form of short videos for online
+educational courses. Trailers give an overview of the course to
+the learners and help them make an informed choice about the
+courses they want to learn. It also helps to generate curiosity
+and interest among the learners and encourages them to pursue
+a course. While it is possible to manually generate the trailers, it
+requires extensive human efforts and skills over a broad spectrum
+of design, span selection, video editing, domain knowledge, etc.,
+thus making it time-consuming and expensive, especially in an
+academic setting. The framework we propose in this work is a
+template based method for video trailer generation, where most of
+the textual content of the trailer is auto-generated and the trailer
+video is automatically generated, by leveraging Machine Learning
+and Natural Language Processing techniques. The proposed
+trailer is in the form of a timeline consisting of various frag-
+ments created by selecting, para-phrasing or generating content
+using various proposed techniques. The fragments are further
+enhanced by adding voice-over text, subtitles, animations, etc., to
+create a holistic experience. Finally, we perform user evaluation
+with 63 human evaluators for evaluating the trailers generated
+by our system and the results obtained were encouraging.
+Index Terms—Video Trailer Generation, Machine Learning,
+Natural Language Processing
+I. INTRODUCTION
+The growth of the internet has significantly increased the
+amount of free instructional content. These resources are
+offered not only by big institutions but also by individual
+content creators over various platforms such as Coursera,
+Udemy, YouTube, etc. This increase in content production rate
+has resulted in the creation of redundant courses and tutoring
+videos for many topics over time. In spite of advantages
+like on-demand accessibility, the abundance of options has
+increased confusion and made it more challenging to select
+a course that might be in line with learner’s interests. And
+often, enrolling to a course that doesn’t meet the learner’s
+expectations for a course’s curriculum and other aspects such
+as expected level of commitment, the availability of support,
+etc., causes the learner to lose motivation and eventually drop
+the course. [1], [2].
+This problem can be tackled to a certain extent by presenting
+a video trailer to the learners before the start of the course
+(learning pathway) to help them quickly glance through the
+pathway and get an overall idea of the course content and its
+format [3]–[5].
+The idea of Trailers is not brand-new, and the film industry
+has been using them extensively for a while. Trailers, in
+context of movies are mostly about advertising. They notify
+viewers about an upcoming movie while generating interest
+among them. Often the effectiveness of a trailer affects the
+perception of the movie, even before it is released publicly.
+The course trailers serve a greater purpose in the educational
+context than simple course promotion. Before beginning the
+learning journey, they aid in helping learners set realistic
+expectations for their learning outcomes and competency
+mastery.
+Concept of trailers might resemble with that of summariza-
+tion [6]–[8], but apart from incorporating a few elements of
+summarization like shortening and abstracting out information
+from substantial sized input source, trailers are different in
+terms of their motivation, purpose and the impact they cre-
+ate on the end users. Unlike summaries, trailers need not
+be complete in their coverage. Also, they are designed to
+give glimpses of a few interesting segments of the narrative
+without revealing the main plot or climax of the underlying
+narrative [9]. Although there is no clear demarcation of what
+a climax is in academic narratives, based on our analysis of
+many academic course trailers in popular MOOCs (Massive
+Open Online Courses) such as Udemy1 and Coursera2, we
+see prevalence of a common pattern in trailer timelines. The
+timeline starts with an introduction about the course and the
+instructor and ends with a call-to-action (CTA) which offers
+opportunity to the learners to take action or start the course.
+In between, there are several elements and factoids about the
+course and its contents, that aim to arouse viewer interest.
+The current approach of generating trailers is manual,
+cumbersome and time-consuming, it requires someone with
+relevant skills like designing, video editing, and a subject
+matter expert to help in curating the trailer content. Although,
+there are software products like Apple iMovie3, Windows
+Movie Maker4 and others that people can use for generating
+trailers by performing basic editing like cuts, merging frames,
+1https://www.udemy.com
+2https://www.coursera.org
+3https://www.apple.com/in/imovie
+4https://www.microsoft.com/en-us/p/movie-maker-video-editor/
+9mvfq4lmz6c9
+arXiv:2301.03957v1  [cs.CL]  10 Jan 2023
+
+Fig. 1. Trailer Structure
+etc. Yet the content to be placed in the trailer has to be curated
+entirely by a human expert.
+In our work, we propose a semi-automatic template based
+framework for generating video trailers for learning pathways,
+which are a sequence of related educational documents of
+various forms [10]–[12]. Here, most of the content that is
+placed in the trailer is auto-generated with a scope for taking
+inputs from the creator. The framework for trailer generation
+consists of various essential trailer fragments arranged as
+a timeline of the trailer. Each fragment is composed of a
+sequence of frames that are coherent within themselves in
+terms of the topical information they present. And inherently,
+each frame is composed of various types of elements and their
+properties like font size, text styling, image size, etc. Fig. 1
+shows the illustration for the same.
+Once all the elements are generated and placed at their
+respective positions within a frame of a trailer fragment,
+a template is applied to it. The template consists of the
+multi-modal experiences such as voice-over, subtitles, sounds,
+animations, etc. It also determines the elements of the trailer
+design such as number and ordering of fragments, frames and
+elements. Fig. 2 shows the visual view of some of the frames
+for one of the templates with it’s corresponding elements and
+their positioning in the frames.
+II. RELATED WORK
+There are studies that discuss the idea, use and motivation
+of having trailers for academic courses [3]–[5]. Also, there are
+online educational platforms like Coursera and Udemy which
+have course trailers. However, we could not find literature on
+approaches to generating trailers for academic courses. Hence,
+in the following paragraphs we discuss some of the pioneering
+works of trailer generation in-general across other domains.
+Trailer generation can also be seen as special case of larger
+research interest of adding an element of surprise to the engage
+receiver’s attention in midst of information overload [13], [14].
+Authors in [15]–[18] present an approach for automatic
+trailer generation from movies as input. Hermes et al. [16]
+create trailers for action movies by analyzing audio and
+video signals present in movies and automatically detecting
+features like faces, scene cuts, sound-volume, etc and use
+ontology of the corresponding domain for producing trailers.
+Irie et al. [17] propose a movie trailer generation method
+which extracts symbols like title logo, main theme music and
+selects impressive shot or speech segments based on clustering
+methods and EM algorithm. Brachmann et al. [15] propose an
+approach of generating action movie trailers using the concept
+of trailer grammar, knowledge base and various ML techniques
+for analyzing audio and images present in the movie. Smith
+et al. [18] propose a system that understands and encodes
+the patterns and emotions present in horror movies using
+Convolution Neural Networks(CNN).
+All the above methods use visual and audio cues to derive
+the trailer frames, whereas we use raw text data and build
+the necessary discriminative and generative Neural Network
+models to create frames and its elements to be placed in the
+trailer.
+Hesham et al. in [19] explore the idea of creating movie
+trailers from their subtitles. They first classify the movie by
+genre, identify important keywords and then rank important
+subtitles. The trailer is then generated by stacking the movie
+time-frames corresponding to the important subtitles. Gaikwad
+et al. in [20] propose a technique to create previews of movies
+by utilizing subtitles and finding the most representative scenes
+by matching them with the plot summaries. Chi et al. [21]
+propose an approach to automatically create marketing-style
+short videos for a given product page url by extracting
+elements and their styles present in the product html page
+under specified tags.
+Unlike the aforementioned works which primarily focus
+on generating trailers based on an extractive strategies, in
+our work we develop various modules that comprehend in-
+put document and generate content for the trailer either by
+paraphrasing or by using Natural Language Generator based
+model.
+As far as we know, automatic/semi-automatic generation
+of video trailers for learning pathways is unexplored. Our
+proposed approach of video trailer generation using Machine
+Learning, Natural Language Processing and Generation tech-
+niques is also unique.
+III. PROPOSED SYSTEM
+We propose a framework for trailer generation consisting of
+different trailer fragments that form a trailer timeline, genera-
+tion of the trailer fragments and finally applying templates that
+determine the look and feel of the trailer. Based on our analysis
+of multiple trailers presented for various online courses offered
+on various educational platforms like Coursera and Udemy, we
+designed and structured our trailer elements, fragments and
+overall flow of the trailer.
+We propose a trailer timeline consisting of 7 trailer frag-
+ments namely, Splash, Trailer Title, Author Details, Outline,
+Meta-Information, Social Proof and finally the Call-to-Action.
+Figure 3 shows the timeline of all the above-mentioned frag-
+ments in the trailer. Each of these fragments define a specific
+part of the trailer, their purpose and their importance in the
+trailer. We define the fragments in detail in further part of
+this section. As discussed earlier, fragments are composed of
+
+Trailer
+ Fragment 1
+Fragment 2
+Fragment t 
+Frame? 1
+Frame? 2
+Frame?2 f
+Element 1  Element 1
+Element eFig. 2. Illustration of Frames
+Fig. 3. Trailer Timeline
+a sequence of frames and each frame is composed of various
+types of elements and their properties.
+The overall approach for trailer generation is illustrated in
+Fig. 4. All the resources mapped to a learning pathway form
+the input to our Fragment Data Generator (FDG) module.
+Template constraints that define the elements, fragments and
+frames also form the input to FDG. The FDG along with other
+sources like creator’s input, any images or information from
+the web or knowledge bases, etc., can be incorporated into
+the frames or the fragments. Once the elements for all the
+frames across all the fragments are generated, we pass it to
+the composition module for adding in other important aspects
+of the trailer like voice-over, subtitles, sounds, etc., to add to
+its multi-modal experience.
+A. Fragment Data Generation
+Following are the proposed trailer fragments arranged in the
+order of their appearance in the trailer timeline-
+Splash Fragment: The idea of splash fragment is to
+display any introductory information related to the trailer such
+as credits, software logo, etc., mostly obtained from creator’s
+input. This optional fragment could also be the last fragment
+in the trailer depending on the creator’s preference.
+Trailer Title Fragment: In this fragment we generate a
+short yet representative title for the entire trailer, hence giving
+a quick idea about the topic that summarizes the underlying
+pathway or the set of resources. We apply Hierarchical Title
+Generation model [22] over the resources mapped to the
+learning pathway to get the list of trailer titles. We select a title
+among them based on their Term Frequency. In case, none of
+the titles are above a threshold, we fall back on the fact that the
+first resource in the pathway is the proxy to the introductory
+resource, and we generate the trailer title for it by applying
+Single Document Title Generator [23], [24]. Figure 5 shows
+the trailer title fragment generation flow.
+Author Details Fragment: A quick introduction about
+the author or the instructor of the learning pathway could
+help the learners build an implicit connect and trust. Majority
+of the elements in the Author Details Fragment like author
+names, affiliations and author’s image are expected from the
+creator while creating the trailer. Template constraints such
+as addressing multiple authors with different frame elements,
+handling and getting relevant images to be put in this fragment
+etc are also obtained from trailer creator. These inputs and
+template constraints are plugged in the automation system
+to fill the overall author frame. Additionally, we crawl the
+web to get relevant images, for example: we crawl the web
+and get relevant affiliation images and place it in the desired
+coordinates as defined by the template. Also for the templates
+that allow for having only the frontal face of author, we make
+use of an open-sourced face recognition model5 to crop the
+face from the uploaded author image. In case no author image
+is provided to the system by the creator, we place a dummy
+caricatured relevant sized image. Similarly, we have defined
+defaults for the features, frames and templates in case there is
+no input from the trailer creator. For example, when multiple
+authors exists, we display information w.r.t to the the first
+author entered by the creator and treat him/her as the primary
+instructor and rest all the authors are abstracted by placing
+them under the “and others” category.
+Outline Fragment: This fragment gives an idea about
+the specific topics that would be covered in the learning
+pathway. This could help in setting learners’ expectation in
+terms of the topics covered and in deciding whether the content
+aligns to his/her end goals. For this we use Single Document
+5https://docs.opencv.org/3.4/db/d28/tutorial cascade classifier.html
+
+AddTextHere
+Add Text Here
+What you will learn ..
+AddTextHere
+Add Text Here
+Add Text Here
+1
+Add TextHere
+②
+3
+4
+Add Text
+Add Text
+Add Text
+Here
+Here
+Here
+Frame 1
+Frame 2
+Frame 3Meta-
+Splash
+Title
+Author
+Outline
+Information
+Social Proof
+CTA
+Introduction
+Introduction
+Overview of
+Course
+Building
+Credits/Logo
+Defining
+to the Course
+about the
+topics covered
+Structure and
+Validation
+Next Steps
+Instructor
+other details
+and TrustFig. 4. Trailer Generation Flow
+Fig. 5. Trailer Title Fragment Generation Flow
+Title Generator [23], [24] model to generate titles for all the
+resources in the learning pathway which represents the outline
+of the learning pathway.
+Every template under the outline fragment limits the number
+of text elements to be listed on the screen with the aim to
+balance aesthetics and information at the same time. To adhere
+to this prior constraint, we design a multi-step process to select
+diverse, yet impactful set of elements from a relatively larger
+list of outlines generated in the previous step. Fig. 6 shows
+the entire pipeline of Outline Text Selection.
+Let K be the number of text elements that the frame requires
+and N be the total number of resources we have as input
+and let K < N. We start with all the resources (N) given
+by the user and remove any instance of assessments and
+short documents under the assumption that such documents
+won’t hold much informational content. After this we remove
+any occurrence of exact duplicates and near duplicates in the
+remaining set and pass the remaining resource list to the title
+generator system to generate title for every resource.
+Post this, we fix the first and the last position of the outline
+with the first and last resource title. We specifically do this
+action because of the inherent ordering present in the input
+resource as a part of learning pathway. Also intuitively, picking
+first and last sets a bound over the topic space to be covered
+under a particular course.
+Finally on this reduced set, we divide the space into bins of
+equal size from which we randomly sample one outline ele-
+ment from each bin to remaining K−2 positions in the outline
+list. We use threshold based Jaccard and cosine similarity for
+filtering syntactic and semantic duplicates respectively. The
+Jaccard similarity between any two documents is calculated as
+an intersection over union of word sets for both documents. It
+helps us get sense of syntactic similarity between documents.
+For calculating cosine similarity, we vectorise our inputs using
+
+Learning Pathway
+R1
+RR3
+R4
+R5
+Ra
+R7
+Rs
+Template
+Constraints
+Fragment Data Generator
+OtherSources
+Creator
+Input
+Fragment Data
+Splash
+Trailer Title
+Outline
+Meta-
+Social Proof
+Call-to-
+Knowledge
+AuthorDetails
+Information
+Action
+Base
+Fragment Data
+Web
+Composition
+Voice-over
+Text-to-
+Frame
+Subtitle
+ixal
+Speech
+Generation
+Generation
+Duration
+Generation
+Trailer
+Music
+ArchiveTask: Generate
+No
+Hierarchical Title
+Titles List
+/Winning
+No
+KUserInput
+Pick 1st Resource
+Trailer Title
+Generation
+Title ?
+from Input
+Yes
+Yes
+ Single Document Title Generator
+Trailer Title
+4pre-trained Sentence Transformers [25] and then measure the
+semantic closeness between them using cosine similarity.
+Algorithm 1 Duplicates Filter
+1: resources = Array(1, 2, . . . , N − 1, N)
+2: remaining resources = Array(1, N)
+3: for i ← 2 to N − 1 do
+4:
+scores = Array()
+5:
+for r ← remaining resources do
+6:
+scores ← calculate similarity(i, r)
+7:
+end for
+8:
+if max(scores) < threshold then
+9:
+remaining resources ← i
+10:
+end if
+11: end for
+12: return remaining resources
+Since every pathway is composed of different resources of
+various properties like length, style, etc., having one threshold
+that fits all does not work. Hence, our threshold is adaptable in
+a way that guarantees at-least K items are selected post any of
+the syntactic or semantic pruning steps. The threshold search
+space is between 0 to 1 where for efficiency and tractability we
+quantize it at 0.1. Then for each threshold we get remaining
+resources as defined in Algorithm 1. Finally the threshold that
+guarantees at-least K items and possibly reduces the input set
+by maximum is chosen as the final threshold.
+Meta-Information Fragment: The idea of having Meta-
+Information Fragment is to inform learners about other impor-
+tant aspects of the course like course structure, total reading
+time, total number of resources, etc. We believe this would
+help learners understand more about the learning pathway or
+resources apart from just knowing the topics that would be
+covered. Also, such information can be used by learners in
+charting out their learning hours and estimating the efforts
+it would take for the successful completion of the course.
+Some of the elements that we generate automatically as part
+of this fragment are: generating topical word clouds 6 bases on
+word frequencies after pre-processing like stop-word removal,
+estimating total reading time based on average reading speed
+statistics and other pathway level derived statistics like total
+resources, availability of discussion forum, etc.
+Social Proof Fragment: Social Proof is one of the most
+prominent ways of social influence and is based on the
+heuristic that the users follow others similar to them when
+uncertain [26]. We collect these statistics from the deployed
+learning environments. This information is added to the video
+trailer over time when different learners take this course and
+the analytical data is available.
+Call-to-Action Fragment: CTA is a marketing term which
+is designed to push the audience in taking the desired actions.
+It is an important aspect of any trailer because all of the
+enthusiasm that is built in a learner while watching the trailer is
+of no use if the learner is not clear on the next actionable [27],
+6https://pypi.org/project/wordcloud/
+[28] item. In our system, we randomly select phrases from a
+set of pre-defined list of potential key-phrases to be placed on
+the screen at a pre-defined location under this fragment. Some
+of the phrases we use are ‘Start your learning today’, ‘Let’s
+get started’, ‘Are you ready?’, etc., along with the action that
+will take the learner on the learning pathway.
+B. Additional Elements
+In this subsection, we discuss two other interesting elements
+that we propose to be added to the trailers, namely, Definition
+Extractor and Paraphraser. These are shown as suggestions
+to the trailer creator and it’s up to the creator to include them
+and decide their placement in the trailer.
+Definition Extractor: Definitions are descriptive elements
+that we believe can help in introduction of concepts. To
+select the definition from the learning resource, we propose a
+discriminative model that classifies a given piece of text into
+Definition or Non-Definition class. For building the classifier
+model, we use a dataset7 that contains positive and negative
+definition candidates extracted from Wikipedia for various
+topics. Our best performing model is a fine-tuned DistilBERT-
+base-uncased8 model with a Definition class F1-score of 0.96
+and Non-Definition class F1-score of 0.97 on the test set.
+Paraphraser: We believe that this is an useful utility that
+can be used in the Outline and Trailer title fragments. This
+gives the creator an ability to re-write concisely any substan-
+tially larger textual content present in any frame. We use a
+publicly available pre-trained model9 for this task which fine-
+tunes a large sized T5 (Text-to-Text Transfer Transformer) [7]
+model on a parallel corpus of sentence and it’s corresponding
+paraphrase.
+C. Video Composition
+Video Composition module is responsible for stitching
+together all the elements that need to be part of the trailer, such
+as the Frame data, Voice-over text, Text-to-Speech (TTS), etc.,
+into a trailer video. Fig. 4 pictorially shows the overall flow
+of the various components that are a part of the video compo-
+sition. We use Python’s MoviePy library10 as our choice for
+video editing and composition of the templates as it provides
+us with all the necessary editing functions like inserting text,
+concatenations and cuts, which we use to draft our templates.
+After the frame-level data elements are in-place, the next
+step is to generate voice-over text for each of the frames.
+Voice-over text is defined as the spoken-text that the narrator
+speaks while a frame is displayed on the screen. For this,
+we select grammar from a pre-defined set of slot based
+text grammars which we define per frame. The slots in the
+grammar are nothing but the screen’s text elements. Finally,
+once the Voice-over Text is generated for every frame, we
+pass them through the IBM Watson’s Text-to-speech (TTS)
+7http://nlp.uniroma1.it/wcl/
+8https://huggingface.co/distilbert-base-uncased
+9https://github.com/ramsrigouthamg/Questgen.ai
+10 https://zulko.github.io/moviepy
+
+Fig. 6. Outline Text Selection
+API11 with relevant parameters such as voice-type, gender,
+etc., by choosing from a list of speaker profiles to get the
+audio files for every frame. Fig. 7 illustrates the flow from
+grammar selection to voice generation for the Trailer Title
+Fragment. We then derive the frame duration accordingly to
+make sure that the visual and audio aspects of the frames are
+in sync and minimize any kind of lag on either ends. Finally,
+along with all the above details, we input template constraints
+like positioning of elements, and styles, user preferences, and
+some basic animations like fade-in and fade-out settings to
+come up with the final trailer.
+IV. EXPERIMENTS
+In this section, we describe the dataset, evaluation strategy
+and results obtained for the trailers generated by our proposed
+system.
+Dataset: Apart from the datasets which we have used for
+training and evaluating specific modules that are responsible
+for generating fragment relevant data. We created three dif-
+ferent learning pathways for our experiments and evaluation
+of the generated trailers. Each learning pathway differs with
+each other in the number of resources and stylometry. Two of
+the pathways are based on text book chapters with difference
+in number of resources mapped, and one pathway is video
+lectures. We tried to take different pathways to evaluate our
+model’s flexibility on different types of learning pathways.
+First one was created by sampling some chapters sequentially
+from a freely available Machine Learning textbook [29]. For
+second, we chose the speech-to-text transcription of a week’s
+video lectures from an academic course on NLP. Our third
+learning pathway is the entire ML textbook [29]12. All the
+three corpus are analogous to learning pathways as they are all
+semantically coherent, progressive and share the same global
+topic.
+Evaluation and Results: Trailers can be seen as gen-
+erative tasks with an inherent notion of creativity. Here the
+objective evaluation is not straight-forward because the ef-
+fectiveness of a trailer is highly subjective and relies on the
+human perception. However, we think that human evaluation
+11https://cloud.ibm.com/catalog/services/speech-to-text
+12Datasets can be found at: https://bit.ly/3ro3JLO
+1
+The first trailer looked more catchy compared to the second
+one. Being generated by an AI agent, both seems to be good.
+2
+Looks amazing. Great work!
+3
+You guys have truly done a remarkable work!
+4
+Good job, keep it up!
+5
+Great!
+TABLE I
+POSITIVE COMMENTS
+1
+Maybe I just felt that he was conveying info too fast
+2
+As of now, it sounds a bit robotic. Some improvements w.r.t
+the TTS can help make it better.
+3
+Slowing the video when the information that is being conveyed
+is relatively dense would be helpful. For example, when going
+through the list of topics, speaking slowly helps. When giving
+instructor names, one can be fast.
+4
+Also, if there’s some way to bring viewer’s attention to the
+part of the slide that’s being mentioned, that would be better
+where the content is not sequential.
+5
+Remove the date from the frame. Add something about what
+can they do once they learn the course(what type of problems
+can they solve)
+TABLE II
+IMPROVEMENTS SUGGESTED BY USERS
+on various trailers generated can give us a good perspective
+on the quality of the trailers. We had 63 human evaluators
+consisting of Engineering graduates, Post-graduates and PhD
+students well versed in the technical domain that represent our
+dataset.
+We evaluate 6 trailers13 in total that were generated from 3
+different learning pathways as discussed above, i.e., 2 trailer
+per learning pathway. These two trailers are based on two
+templates T1, T2 created by us. Both the templates differ in
+aesthetics and level-of-detail(LOD). The evaluation for each
+trailer was done on a set of 8 questions on Likert-scale from
+1 to 5, where 1 would mean very poor and 5 would mean very
+good.
+There were three separate groups of evaluators. Each group
+was provided with 2 trailers based on 2 templates for the
+same pathway. We thoughtfully perform this diversification to
+simulate the cluster sampling procedure, since showing all 6
+trailers to the same evaluators would have created boredom,
+resulting in not so accurate evaluation.
+13Sample Trailers: https://bit.ly/3Hscie9
+
+Filtering Less-informative Documents
+Syntactic Filters over Document Text
+All Input
+Filter Assessments
+Filter Short
+Filter Exact
+Filter Near
+Generate Title for
+Resources
+Documents
+Duplicates
+Duplicates
+every Resource
+R = [1, 2, ... N-1, N]
+Select 1st and pth
+Randomly Select 1
+resource and add in
+Outline Elements
+resource from each
+Outline then Divide
+Filter Semantic
+Filter Near
+Filter Exact
+bin
+P-2 resources into K-
+Duplicates
+Duplicates
+Duplicates
+O = [1, 2, ... K-1, K]
+2 equal spaced bins
+Semantic Filter over Titles
+R = [1, 2, .. P-1, P]
+Syntactic Filters over Titles
+where, K<=P<=NFig. 7. Flow of Grammar selection to Voice-over generation
+We also encouraged the evaluators to give free comments
+for the trailers they evaluated, as this would help us improve
+our system in future iterations. Table. I and II lists down some
+of the positive comments and improvements suggested by the
+users. Fig. 8 shows some of the trailer fragments generated by
+our proposed system14.
+Following is the list of 8 questions that were asked to the
+evaluator during the evaluation. The text in italics highlights
+the broader aspect of the evaluation feature.
+Q1. Did you find the trailer to be self-contained?
+Q2. How were the fonts and styles used in the trailer in
+terms of readability?
+Q3. How did you find the length and pace of the trailer?
+Q4. As a user, how impressed are you with this trailer
+overall?
+Q5. Could this trailer evoke interest in someone taking
+this course? (Ignoring any prior inclination to the topic)
+Q6. How was the average duration of each frame?
+Q7. Based on the trailer you just saw, do you think you
+have a good impression of the course now?
+Q8. How did you find the sync between the audio and
+visuals you saw?
+As can be seen in Fig. 9, the scores obtained for each of the
+survey questions are good and far above the average(score of
+3) for almost all the trailers generated by our approach. Also,
+in our study, we found both the templates performed equally
+good. However, for Q5, the average scores is relatively lower
+compared to other questions. On digging deeper we found
+some of the comments of total 24 comments we received
+mentioned about the difficulty of the course for not getting
+interested in the course. This could mean that this question
+(Q5) is more subjective.
+14Detailed
+demo
+walk-through:
+https://www.youtube.com/watch?v=
+06VVuAlFhTk
+V. CONCLUSIONS AND FUTURE WORK
+In this paper, we presented a novel framework for au-
+tomatically generating video trailers for a learning pathway
+using ML and NLP techniques. We validated our trailers on
+multiple corpus of varied granularity with human evaluation
+and the results obtained were encouraging. This approach can
+be adapted to different domains given enough data to train
+the models involved in the entire process. We believe that
+this approach can lay foundation to building more advanced
+versions of trailer.
+In future, we plan to improve the existing system by
+incorporating suggestions obtained in the user evaluation and
+adding more interesting themes like automatically detecting
+learning outcomes given the resources. We also intend to create
+an interactive dashboard to take inputs from the creator and
+allow the creator to make edits to the auto-generated content.
+ACKNOWLEDGMENT
+We thank the Center of Excellence on Cognitive Com-
+puting, funded by Mphasis F1 Foundation for funding this
+research. We also thank Dr. Prasad Ram and Gooru team
+(https://gooru.org) for the topical discussions and encourage-
+ment.
+REFERENCES
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+
+Title Generator
+Sample Title: Graph Neural Networks
+Hello, Welcome to this course on _TITLE_
+Hi, Welcome to this course on _TITLE
+Grammar
+Random Selection
+ Hi, Welcome to this course on -_TITLE
+Hello, Welcome to _TITLE_ course.
+Hi, Welcome to the course_TITLE_
+Trailer Title Fragment
+Hi, Welcome to this course on Graph Neural Networks
+Text-to-speech engineFig. 8. Trailer Fragments
+Fig. 9. Average scores per Survey Question for all 3 pathways and trailers. Here P1, P2, P3 represent 3 Pathways and T1, T2 represent Templates
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+and
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+IEEE/CVF International Conference on Computer Vision, 2021, pp.
+3205–3214.
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+title generation for learning resources and pathways with pre-trained
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+[23] ——, “Automatic title generation for text with pre-trained transformer
+language model,” in 2021 IEEE 15th International Conference on
+Semantic Computing (ICSC).
+IEEE, 2021, pp. 17–24.
+[24] J. Tan, X. Wan, and J. Xiao, “From neural sentence summarization to
+headline generation: A coarse-to-fine approach.” in IJCAI, vol. 17, 2017,
+pp. 4109–4115.
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+siamese bert-networks,” arXiv preprint arXiv:1908.10084, 2019.
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+Pearson
+education Boston, MA, 2009, vol. 4.
+[27] “Call-to-action (cta),” https://bit.ly/3DDUBp4, accessed: 2021-12-08.
+[28] “3 reasons a call to action is important,” https://bit.ly/33c7WbO, ac-
+cessed: 2021-12-08.
+[29] J. Gareth, W. Daniela, H. Trevor, and T. Robert, An introduction to
+statistical learning: with applications in R.
+Spinger, 2013.
+
+What you will learn ...
+October 6, 20:
+October 6, 2021
+Readtimo
+~8hr
+model
+Resources
+oneet al
+Let's get started!
+Regularization in
+Convolutional Neural
+function
+Deep Learning
+Networks
+1
+2
+3
+4
+6
+Text
+layergradient
+Regular
+Resources
+neuralnetwork
+Assessments
+Are you
+rward Networks
+Optimization Techniques
+Recurrent Noural
+ready?
+Networks
+Training
+Active
+Discussion
+Forum
+Ihis course, In this specaly curated course
+ol the curriculum you wll go througn
+startyourjourney
+Outline Frame
+Meta-Information Frame
+CTA Frame5
+4
+Likert Value
+3
+2
+1
+0
+Q1
+Q2
+Q3
+Q4
+Q5
+Q6
+Q7
+Q8
+SurveyQuestion
+P1-T1
+P1-T2
+P2-T1
+P2-T2
+P3-T1
+P3-T2

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+MNRAS 000, 1–18 (222)
+Preprint 6 January 2023
+Compiled using MNRAS LATEX style file v3.0
+A Gaian Habitable Zone
+Rudy Arthur,1★ Arwen Nicholson,2†
+1University of Exeter, Department of Computer Science
+2University of Exeter, Department of Physics and Astronomy
+Accepted XXX. Received YYY; in original form ZZZ
+ABSTRACT
+When searching for inhabited exoplanets, understanding the boundaries of the habitable zone around the parent star is key. If
+life can strongly influence its global environment, then we would expect the boundaries of the habitable zone to be influenced
+by the presence of life. Here using a simple abstract model of ‘tangled-ecology’ where life can influence a global parameter,
+labelled as temperature, we investigate the boundaries of the habitable zone of our model system. As with other models of
+life-climate interactions, the species act to regulate the temperature. However, the system can also experience ‘punctuations’,
+where the system’s state jumps between different equilibria. Despite this, an ensemble of systems still tends to sustain or even
+improve conditions for life on average, a feature we call Entropic Gaia. The mechanism behind this is sequential selection with
+memory which is discussed in detail. With this modelling framework we investigate questions about how Gaia can affect and
+ultimately extend the habitable zone to what we call the Gaian habitable zone. This generates concrete predictions for the size
+of the habitable zone around stars, suggests directions for future work on the simulation of exoplanets and provides insight into
+the Gaian bottleneck hypothesis and the habitability/inhabitance paradox.
+Key words: Gaia – Habitable Zone – Biosignatures
+1 INTRODUCTION
+The Gaia hypothesis is that life influences the Earth’s feedback mech-
+anisms to form a self-regulating system, and therefore life can help
+maintain habitable conditions on its host planet Lovelock & Mar-
+gulis (1974). Distinct from the biosphere Huggett (1999), Gaia is
+the whole life-earth system, considered as a single entity. The impor-
+tance of life’s interactions with the non-living environment are now
+common sense, and the discipline of Earth System Science Lenton
+& Watson (2013) studies the various feedback loops that constitute
+‘Gaia’s body’ Volk (2012). Gaia theory itself takes a very broad
+perspective, aiming to describe life at a planetary scale. Gaia theory
+asks questions like: Is Gaia inevitable on a planet that hosts life, or
+is it due to chance? What mechanisms can create a long lived Gaian
+system? How will we detect other ‘Gaias’ beyond our solar system,
+where direct planetary exploration is not an option? The astrophysical
+point of view was crucial in the early development of Gaia, with the
+search for life on Mars providing the initial inspiration for the Gaia
+hypothesis Lovelock (1965). When looking at Earth from afar, Gaia
+is what we see and the search for habitable or inhabited exoplanets is
+the search for other Gaias.
+Methods for exoplanet detection have developed considerably
+since Gaia was first proposed. New telescopes, such as the James
+Webb Space telescope and the Extremely Large Telescope (currently
+under construction), and future missions, such as the Large Ultravi-
+olet Optical Infrared Surveyor, mean that searching for signs of alien
+★ E-mail: [email protected]
+† E-mail: [email protected]
+life will be possible within the coming decades Snellen et al. (2021);
+Quanz et al. (2021). While robotic missions to potentially habitable
+exoplanets remain unfeasible, evidence for alien life will only be ob-
+servable for exoplanets that have been dramatically shaped by their
+biospheres. Exoplanets with newly emerging life, or those with the
+remnants of a once-thriving biosphere that has since collapsed, will
+be unlikely to produce a remotely observable signature. Answering
+the key questions of Gaia theory not only informs how we think about
+the history of life on Earth, but can form the theoretical foundation
+for the study of life in the universe.
+Catling et. al. Catling et al. (2018) proposed a framework for as-
+sessing potential biosignatures using a probabilistic approach that
+combines observations of the candidate planet and host star with
+models of the possible abiotic and biotic planetary processes to de-
+termine the probability of the planet being inhabited. With a great
+diversity of exoplanets being found, any potential biosignature must
+be considered within the context of its host planet Seager (2013);
+Claudi. (2017); Kiang et al. (2018); Schwieterman et al. (2018);
+Krissansen-Totton et al. (2022). Detailed abiotic models of exoplan-
+ets are being developed for a wide range of detected planets, see e.g.
+Amundsen et al. (2016); Boutle et al. (2017); Collins (2021); Fauchez
+et al. (2021), and sophisticated models of biogeochemistry exist for
+different points in Earth’s history, e.g. Kharecha et al. (2005); Daines
+et al. (2017); Lenton et al. (2018b); Zakem et al. (2020).
+Detailed and realistic modelling of life on other planets is im-
+portant, however this paper will take a broader view that aims to
+understand the generic mechanisms that lead to Gaia. We build on
+recent work Ford Doolittle (2014); Lenton et al. (2018a); Arthur
+& Nicholson (2022) on Gaian selection principles. We argued in
+© 222 The Authors
+arXiv:2301.02150v1  [astro-ph.EP]  5 Jan 2023
+
+2
+Arthur & Nicholson
+Arthur & Nicholson (2022) that some approaches to Gaian selec-
+tion Ford Doolittle (2014); Lenton et al. (2018a) lead to anthropic
+reasoning - we see Gaia because if we didn’t we wouldn’t exist. An-
+thropic reasoning is controversial, with its opponents arguing that it
+unfalsifiable with limited (if any) predictive power Smolin (2007).
+The coming era of exoplanet astronomy gives new context and pur-
+pose to these discussions. If our aim is for Gaia theory to inform our
+search for life in the universe, then anthropic arguments are clearly
+inadequate.
+In Arthur & Nicholson (2022) we argue for ‘Entropic Gaia’ - that
+the emergence of Gaia is a statistical tendency for planets that host
+life. This means that life history on a single planet can be chaotic
+and have periods of stability and collapse, however there is a trend
+towards increasing biomass, stability, habitability and other Gaian
+features. Any single planetary history for a life-bearing planet, such
+as Earth, is likely to follow a (bumpy) trajectory towards Gaia. The
+micro-mechanism leading to this behaviour was argued to be ‘Se-
+quential Selection with Memory’ or an ‘entropic ratchet’. In brief, this
+mechanism starts from the observation that coupled life-environment
+systems move between regulation and disregulation. By definition,
+disregulating systems quickly destroy themselves while regulating
+systems persist, this is sequential selection Lenton et al. (2018a). In
+models of ecology and Gaia (e.g. Becker & Sibani (2014); Hard-
+ing (1999)) increasing diversity and complexity is associated with
+increasing stability 1. More diverse ecosystems can generate more
+novel species through mutation. Thus after every ecosystem collapse
+(caused by internal or external factors) if a new ecosystem arises
+it is likely to be more diverse, having started with a greater ‘pool’
+of species, and therefore also more stable. Sequential selection with
+memory describes a sequence of distinct stable states that tends to
+get ‘more Gaian’ over time.
+This mechanism was originally proposed in the framework of the
+Tangled Nature Model (TNM) Christensen et al. (2002). Originally
+designed to study co-evolution, we demonstrated in Arthur & Nichol-
+son (2017) that the TNM is closely related to the generalised Lotka-
+Volterra model. The TNM is based on the idea that the growth rate
+of a species is given by a fitness function that depends on the other
+species present. Any model making this assumption will look like the
+TNM close to equilibrium Arthur & Nicholson (2022). By studying
+the model with agent based dynamics we can incorporate mutation,
+giving us a very flexible, robust and general model of evolutionary
+ecology. Since the TNM is quite general, conclusions drawn in this
+framework are likely to have general applicability.
+Artificial life modelling has been used extensively to study Gaia.
+The original Daisy World Watson & Lovelock (1983) led to a large
+number of variants Wood et al. (2008) and there are a variety of
+other models such as the Guild Model Downing & Zvirinsky (1999),
+Greenhouse World Worden (2010), Flask Model Williams & Lenton
+(2007) and Exo-Gaia Nicholson et al. (2018) to name a few. We have
+previously discussed Gaian models based on the TNM in Arthur &
+Nicholson (2017, 2022). Here we propose a new variant on the TNM
+that is more similar to other Gaian models with a very simple abiotic
+(non-living) component.
+While previous Gaian models have included mutation (such as
+the Flask model and ExoGaia) the complexity of the biosphere in
+these models has been limited and different species within the mod-
+els only impact one another via the shared environment, e.g. via
+1 See Landi et al. (2018) for a thorough discussion of the relationship be-
+tween ecosystem complexity and stability, though most of these models don’t
+consider coupling to the external environment
+resource competition or via global parameters such as temperature.
+When we look at life on Earth it is clear that different species can
+have a large impact on each other beyond resource competition or
+changing global parameters like temperature. For example, there are
+complex interactions between worms, plants and soil that change the
+structure, chemistry, water retention and other properties of soil for
+the benefit of many species Le Bayon et al. (2021). These kinds of
+symbiotic (and also antagonistic) interactions are usually missing in
+Gaian models. We also observe that throughout Earth history there
+have been dramatic and spontaneous changes in the diversity and
+complexity of the biosphere, e.g. the Great Oxidation Event which
+allowed for aerobic respiration to become an important energy source
+for life Ligrone (2019). These types of events, crucial for the selec-
+tion mechanism discussed above, are absent in other Gaian models.
+In contrast, TNM species interact directly through antagonistic or
+symbiotic inter-species couplings, the population varies consider-
+ably due to spontaneously occurring ‘quakes’ and there is no rigid
+upper bound on the population. Thus by combining elements of the
+TNM with elements of earlier Gaian models we can explore how Ga-
+ian regulation emerges within a system that allows for more complex
+ecosystem dynamics.
+With this model we hope to show that the arguments for Entropic
+Gaia are robust by demonstrating how they work in a setting where
+life needs to interact with and regulate an external environment. At the
+same time we will explore how Gaia can inform the search for life in
+the universe, in particular how Gaia predicts a larger ‘habitable-zone’.
+In section 2 we describe the model and how we add temperature,
+which is a combination of abiotic and biotic components. In section 3
+we study the model at constant background temperature to understand
+how temperature is regulated and interacts with the spontaneous
+‘quakes’ that occur in the TNM. Section 4 discusses the changes to
+the habitable-zone in the presence of life and section 5 studies how
+life adapts to deteriorating abiotic conditions. Finally we conclude in
+section 6.
+2 MODEL DESCRIPTION
+2.1 The Tangled Nature Model
+We start, as in Arthur & Nicholson (2022), with the generalised
+Lotka-Volterra model
+𝑑𝑁𝑖
+𝑑𝑡 = 𝑁𝑖 𝑓𝑖(�𝑛, 𝑁)
+(1)
+𝑁𝑖 is the population of species 𝑖, 𝑁 is the total population and 𝑛𝑖 =
+𝑁𝑖
+𝑁 . 𝑓𝑖 is a fitness function that depends on the type and abundance
+of the other species present through �𝑛 = (𝑛1, 𝑛2, . . . , 𝑛𝐷) and 𝑁. We
+can expand 𝑓𝑖 to linear order around the equilibrium at 𝑁 = 0
+𝑑𝑁𝑖
+𝑑𝑡 = 𝑁𝑖
+�
+𝑓𝑖(�0, 0) +
+∑︁
+𝑗
+𝑑𝑓𝑖
+𝑑𝑛 𝑗
+(�0, 0)𝑛 𝑗 + 𝑑𝑓𝑖
+𝑑𝑁 (�0, 0)𝑁 . . .
+�
+(2)
+The summations here and for the rest of this paper are over all extant
+species. The three terms on the right hand side are the basic TNM
+variables.
+• 𝑟𝑖 ≡ 𝑓𝑖(�0, 0) is the growth rate of species 𝑖 in the absence of any
+other species. We set this to zero, meaning that one species’ growth
+depends entirely on the other species present. We could add some
+species with non-zero growth rates to represent primary producers
+but for simplicity and consistency with the rest of the TNM literature
+every species has 𝑟𝑖 = 0.
+MNRAS 000, 1–18 (222)
+
+A Gaian Habitable Zone
+3
+• 𝐽𝑖 𝑗 ≡ 𝑑 𝑓𝑖
+𝑑𝑛𝑗 (�0, 0) is the inter-species coupling matrix where 𝐽𝑖 𝑗
+is the effect of species 𝑗 on species 𝑖. As usual Christensen et al.
+(2002), we set the elements randomly from a symmetric distribution.
+Here each element 𝐽𝑖 𝑗 is randomly chosen from a standard normal
+product distribution times 𝑐 = 100. The exact functional form of the
+distribution is not important, only that it has infinite support Arthur
+et al. (2017).
+• −𝜇 ≡
+𝑑 𝑓𝑖
+𝑑𝑁 (�0, 0) is the inverse carrying capacity, controlling
+how much of the global ‘resource’ is consumed by each individual.
+The growth equation now looks like
+𝑑𝑁𝑖
+𝑑𝑡 = 𝑁𝑖 ��
+�
+∑︁
+𝑗
+𝐽𝑖 𝑗𝑛 𝑗 − 𝜇𝑁��
+�
+= 𝑁𝑖 𝑓 𝑇 𝑁 𝑀
+𝑖
+(3)
+In Arthur & Nicholson (2022) we added higher order terms to
+the fitness function and argued that these could be interpreted as
+species-environment interactions, since their net effect was to modify
+the 𝜇 term to create an “effective” carrying capacity. This kind of
+‘endogenous’ environment (e.g. roughly analogous to atmospheric
+composition or oceanic pH) is in contrast to most Gaian models
+which represent the environment through one or more ‘exogenous’
+parameters, which the model agents aim to regulate. Daisyworld is the
+paradigmatic example, where black and white daisies spontaneously
+regulate a rising global temperature. We want to study this type of
+regulation in the TNM framework and only deal with an abiotic
+environment so, for simplicity, we do not include the higher order
+terms.
+While this is a common approach in Gaian modelling it is worth
+some consideration. It was shown in Arthur & Nicholson (2022) and
+Arthur & Nicholson (2017) that selection in the TNM tends to pro-
+duce beneficial endogenous/biotic environments. If we included both
+an abiotic and a biotic environment, TNM agents would be subject to
+more selective pressure i.e. they would need to avoid degrading the
+external parameters (temperature) and internal parameters (∼ pH). In
+Arthur & Nicholson (2017) it was noted that environmental selection
+isrelatively weak, because whennewspecies occurtheystart with low
+populations and therefore minimal impact on the environment. This
+must also be the case for an abiotic environment. Ultimately the rela-
+tive weighting of each in the fitness function would determine which
+environmental parameters are most ‘optimised’. Studying these ef-
+fects is interesting but we leave it for future work, focusing here on
+understanding the model with a purely exogenous environment.
+2.2 Adding Temperature
+To add temperature to the TNM we let the global temperature 𝑇 be
+the sum of abiotic and biotic components:
+𝑇 = 𝑇0 + 𝑇𝑙𝑖 𝑓 𝑒
+(4)
+𝑇0 is the temperature in the absence of life and𝑇𝑙𝑖 𝑓 𝑒 is the effect of the
+extant species in the model on the temperature. Every individual of
+species𝑖 has an effect, 𝐻𝑖, on the global temperature. The values of 𝐻𝑖
+will be selected from a normal distribution with mean 0 and standard
+deviation 𝜎𝐻 , so species are equally likely to have a warming or
+cooling effect. The total effect of life on the temperature is
+∑︁
+𝑖
+𝐻𝑖𝑁𝑖
+(5)
+We describe how �
+𝑖 𝐻𝑖𝑁𝑖 is related to 𝑇𝑙𝑖 𝑓 𝑒 in the next section.
+We make the reproduction rate depend on the temperature by
+modifying the fitness function to
+𝑓 𝑇 𝑁 𝑀
+𝑖
+(𝑇) =
+∑︁
+𝑗
+𝐽𝑖 𝑗
+1 +
+�𝑇 −𝑇𝑃
+𝜏
+�2 𝑛 𝑗 − 𝜇𝑁
+(6)
+𝑇𝑃 is the preferred temperature and 𝜏 is a tolerance parameter. The
+functional form is chosen so that at temperatures, 𝑇, far from 𝑇𝑃
+the interaction strength is reduced, for example at 𝑇 = 𝑇𝑃 + 𝜏 the
+inter-species interaction strength is halved. The functional form
+1
+1+𝑥2
+is chosen for simplicity, any function that applies a smooth and
+symmetric temperature ‘window’ would work. We have chosen 𝑇𝑃
+and 𝜏 to be constant for all species and interactions. We could, for
+example, make the width different for every inter-species interaction:
+𝜏 → 𝜏𝑖 𝑗 and similarly for 𝑇𝑃. In the interest of keeping this work
+relatively brief and in line with other work such as the original
+Daisyworld model Watson & Lovelock (1983), Flask model Williams
+& Lenton (2007) and ExoGaia Nicholson et al. (2018), we use a
+constant 𝑇𝑃. By keeping 𝑇𝑃 constant for all species we can focus on
+and highlight life’s impact on its environment. If 𝑇𝑃 is kept constant,
+then any improvement to a “planet’s” survival rate when including
+life-environment interaction can only come from life improving its
+environment rather than life simply adapting to it. As this is the part
+of Gaia theory that is less well accepted Kirchner (2003) it makes
+sense to explore scenarios where this effect isn’t potentially obscured
+by species adaptation.
+2.3 Running the Model
+We solve the growth equation using agent based dynamics. This
+means that we generate individual agents whose reproduction rate
+is controlled by the fitness function 𝑓 𝑇 𝑁 𝑀
+𝑖
+(𝑇). Each agent is an
+individual of some species𝑖 and each agent’s reproduction probability
+is given by
+𝑝𝑜 𝑓 𝑓
+𝑖
+=
+1
+1 + 𝑒− 𝑓 𝑇 𝑁 𝑀
+𝑖
+(𝑇 )
+(7)
+The basic dynamics of the model are then (see also Arthur et al.
+(2017)):
+(i) Choose an individual and, with probability 𝑝𝑜 𝑓 𝑓
+𝑖
+, make a
+copy of that individual. The copying step is meant to mimic asexual
+reproduction. We take the 𝐿 = 20 bit binary representation of the
+species-index 𝑖 and copy one bit at a time, with a probability 𝑝𝑚𝑢𝑡 =
+0.01 to flip a bit during each copy operation.
+(ii) Chose a random individual and kill it with probability 𝑝𝑘𝑖𝑙𝑙 =
+0.1
+𝐿 is the genome length, where the value of 20 is standard Christensen
+et al. (2002), meaning that the model can generate 2𝐿 ∼ 106 unique
+species. A ‘generation’ is the time required to iterate over the basic
+reproduction/death loop above 𝑁/𝑝𝑘𝑖𝑙𝑙 times, where this number
+is recalculated at the end of each generation. This means in each
+generation every individual has had a chance to be selected once
+on average for a birth/death process. To update the temperature we
+perform the following steps after every generation
+• If required, update the abiotic temperature 𝑇0 (see Section 5).
+• Update 𝑇𝑙𝑖 𝑓 𝑒 using
+𝑇𝑙𝑖 𝑓 𝑒(𝑡) = 𝜆𝑇𝑙𝑖 𝑓 𝑒(𝑡 − 1) + (1 − 𝜆)
+∑︁
+𝑖
+𝐻𝑖𝑁𝑖
+(8)
+• Set 𝑇 = 𝑇0 + 𝑇𝑙𝑖 𝑓 𝑒
+MNRAS 000, 1–18 (222)
+
+4
+Arthur & Nicholson
+Variable
+Symbol
+Value
+Inverse carrying capacity
+𝜇
+0.1
+Mutation rate
+𝑝𝑚𝑢𝑡
+0.01
+Death rate
+𝑝𝑘𝑖𝑙𝑙
+0.1
+Lag parameter
+𝜆
+0.9
+Preferred temperature
+𝑇𝑃
+100
+Temperature tolerance
+𝜏
+2
+Temperature effect
+𝜎𝐻
+0.05
+Table 1. A list of all the key parameters in the model and the values we
+choose. The model has a large parameter space and the parameters are set
+to convenient values used in previous work on the TNM. The qualitative
+behaviour of the model is very robust to variations in these parameter values
+Christensen et al. (2002); Arthur et al. (2017).
+Here 𝑡 is the generation number, the timescale in this model and 𝜆
+is a lag-parameter that stops the temperature from changing instan-
+taneously. This mimics the real behaviour of the Earth-system, e.g.
+climate models have demonstrated a lag in the response of surface
+temperatures over the ocean due to changes in atmospheric 𝐶𝑂2
+Boucher et al. (2012). The model is initialised with 500 individuals
+of a randomly chosen species and all averages are taken over 1000
+model runs using different random seeds.
+3 CONSTANT TEMPERATURE EXPERIMENTS
+First we run the model with constant𝑇0. Figure 1 shows the behaviour
+of the population and temperature in one ‘run’ of the model for
+104 generations. The basic features of the standard TNM - quasi-
+stable states punctuated by sharp transitions - persist Christensen
+et al. (2002). The important features of ‘core’ and ‘cloud’ Becker
+& Sibani (2014) are are retained as can be seen in Figure 2. The
+core species are the only ones with significant population and these
+are the primary drivers of the temperature. The cloud species are
+mutants with small populations and random positive and negative
+effects on the temperature. These two runs show that life can move
+the temperature away from 𝑇𝑃 or towards it, the question is what
+happens on average, in the long run.
+Figures 3 (a) and (b) show the average population and average
+temperature for 𝑇0 = 100 = 𝑇𝑃 and 𝑇0 = 105 = 𝑇𝑃 + 2.5𝜏 respec-
+tively. For (a) 𝑇0 = 𝑇𝑃 and the temperature fluctuates close to the
+abiotic temperature while the population increases logarithmically.
+This behaviour, increasing population with constant temperature, in-
+dicates that the TNM agents are optimising their mutual interactions,
+�
+𝑗 𝐽𝑖 𝑗 𝑁 𝑗, as in the standard model, while keeping the temperature
+close to 𝑇𝑃. In (b) where 𝑇0 > 𝑇𝑃 we see that the population in-
+creases while the temperature decreases. Thus the TNM agents are,
+on average, simultaneously optimising their mutual interactions while
+improving the temperature.
+In Arthur & Nicholson (2022) we discussed Selection by Survival
+(SBS) and Sequential Selection with memory (SSM). SBS is just dif-
+ferential survival i.e. at late times we see systems with Gaian features
+because those are the only ones that could survive that long. SBS is a
+good null model, here it would predict that the average temperature
+tends towards 𝑇𝑃 because runs that don’t maintain 𝑇𝑃 go extinct,
+leaving a small number of surviving runs that happen to operate at
+𝑇𝑃. SSM would predict that the punctuations during individual runs
+drive the average temperature towards 𝑇𝑃. The numbers in the top
+row of Figure 3 (a) and (b) show the proportion of runs which survive
+up to that point in the experiment. In (b) for example, at 𝑇0 > 𝑇𝑃
+about 9% of the runs have gone completely extinct (𝑁 = 0) by 105
+generations compared to 3% when 𝑇0 = 𝑇𝑃. This is a relatively small
+increase in extinction rate compared to the relatively large decrease
+in the scaling factor 1/
+�
+1 +
+�𝑇0−𝑇𝑃
+𝜏
+�2�
+≃ 0.14.
+Figure 4 shows the model runs in more detail for 𝑇0 = 105 > 𝑇𝑃.
+(a), (b) and (c) demonstrate that the runs can be split into two types:
+low temperature, cooling core; and high temperature, heating core.
+We will loosely call these ‘Gaian’ and ‘non-Gaian’ respectively. (d)
+is the crucial plot. It shows the proportion of surviving runs over
+time (dashed line) and the proportion of the surviving runs that have
+𝑇 ≤ 𝑇𝑃. Here we see that while some runs do go extinct (SBS)
+in the surviving runs the proportion of Gaian states increases. This
+means that non-Gaian states transition to Gaian states, leading to
+more of them over time. This is exactly as sequential selection with
+memory predicts: (non-terminal) resets tend, on average, to improve
+conditions for life. We will discuss the exact mechanism in detail
+below.
+This mechanism, Sequential Selection with Memory (SSM) was
+discussed in Arthur & Nicholson (2022) and briefly in Secion 1.
+Each model run consists of multiple quasi-equilibria interrupted by
+quakes (Figure 1). These quakes completely reset the species which
+make up the core. These core species are (by definition) the ones
+with large populations which control the model dynamics, in this
+case the total population and temperature. As has been discussed
+in the TNM literature (especially Becker & Sibani (2014)), quakes
+occur spontaneously due to the evolution of a ‘parasite’ that disrupts
+the core. A parasite, 𝑎, is any species with significant reproduction
+probability that isn’t a member of the core. To have a large probability
+to reproduce, the sum of its interactions must be high enough that
+its reproduction rate is higher than its death rate. Solving for fitness
+gives:
+∑︁
+𝑗
+𝐽𝑎 𝑗𝑛 𝑗
+1 +
+�𝑇 −𝑇𝑃
+𝜏
+�2 ≥ 𝜇𝑁 +
+�
+1 − 1
+𝑝𝑘
+�
+(9)
+Lower total population makes it easier for a parasite to occur by
+decreasing the 𝜇𝑁 term. Low total population can occur either due
+to weak inter-species interactions in the core or unfavourable tem-
+peratures. However because of the smaller number of reproduction
+events at low 𝑁, fewer mutants are generated. On the other hand high
+populations raise the barrier and increase the number of mutation
+events.
+Crossing the barrier requires finding a mutant 𝑎 with sufficiently
+large, positive interactions with some or all species in the core.
+Large values of 𝐽𝑎 𝑗 are rare (for our choice of distribution, expo-
+nentially so) and the rate of generating new mutants is low. Con-
+sidering each reproduction event as 𝐿 = 20 Bernoulli trials, the
+expected number of mutations in a reproduction is given by a Bino-
+mial distribution 𝐵(𝐿, 𝑝𝑚𝑢𝑡) with mean 𝐿𝑝𝑚𝑢𝑡 = 0.2 and variance
+𝐿𝑝𝑚𝑢𝑡 (1 − 𝑝𝑚𝑢𝑡) ≃ 0.2. Thus the rate of exploration of the genetic
+space is quite slow. Ultimately the barrier height is more important
+than the increased rate of reproduction and is what explains the trend
+of (slowly) increasing population and stability in the TNM. For much
+more on this see Becker & Sibani (2014).
+Here we have to analyse how the temperature interacts with this
+mechanism. Assume we have a case where 𝑇0 > 𝑇𝑃 as in Figure
+4. Temperatures far from 𝑇𝑃 make a quake more likely by reducing
+the total population and hence the barrier height. When a quake
+occurs a new core is selected on the basis of strong inter-species
+interactions that allow it to quickly ‘use up’ the carrying capacity.
+This new core has an equal chance to be warming or cooling, because
+of the symmetry of 𝐻𝑖. If it is warming we stay in a non-Gaian state,
+if not we move to a Gaian state. In a Gaian state the barrier can
+MNRAS 000, 1–18 (222)
+
+A Gaian Habitable Zone
+5
+Figure 1. The column (a) shows the population (top row) and temperature (bottom row) where the background temperature is 𝑇0 = 𝑇𝑃 = 100. Column (b) shows
+the population and temperature where 𝑇0 = 105. The temperature in (a) is above 𝑇0 and 𝑇𝑃 while the temperature in (b) is below both 𝑇0 and 𝑇𝑃.
+be significantly higher, leading to a much more stable, long lived
+core. In a non-Gaian state the barrier is low, meaning the state will
+be relatively short lived, being vulnerable to parasites and to large
+population fluctuations which may result in total extinction. As shown
+in Figure 4 (d) over time this leads to more and more model runs in
+a Gaian state.
+To summarise: both mechanisms, SBS and SSM operate. Ga-
+ian states have temperatures close to 𝑇𝑃, and thus high populations
+which, in this model, makes them more stable. Non-Gaian states are
+far from 𝑇𝑃 and have low populations. This makes them vulnerable
+to total extinction (SBS) and punctuation which can take a non-Gaian
+to a Gaian state (SSM). In this model, for this particular temperature,
+SSM is a more important mechanism than SBS, though the ratio can
+vary with 𝑇0, as we will explore in the next section.
+These ideas can help explain why the Earth today is in a habit-
+able state. Since its conception the Gaia hypothesis has been defined
+in numerous ways and ranging from a strong hypothesis that self-
+regulating feedback loops are an expected property of a life-planet
+coupled system, known as ‘probable Gaia’ Lenton & Wilkinson
+(2003), to a weaker hypotheses that suggests that while the Earth
+MNRAS 000, 1–18 (222)
+
+(a)
+(b)
+1400
+1000
+1200
+800-
+1000
+800
+600
+N
+N
+600-
+400
+400-
+200
+200
+0
+0
+2000
+4000
+6000
+8000
+10000
+0
+2000
+4000
+6000
+8000
+10000
+t (generations)
+t (generations)
+108
+To
+108
+Tp
+Tp
+106
+106
+104
+104
+Temperature
+102
+102
+100
+100
+98
+98 -
+96 
+96
+94 
+94 
+0
+2000
+4000
+6000
+8000
+10000
+0
+2000
+4000
+6000
+8000
+10000
+t (generations)
+t (generations)6
+Arthur & Nicholson
+Figure 2. Model snapshot at 𝑡 = 9000 generations for the runs (a) and (b) from Figure 1. Each node represents a different species, with the size of the node an
+indication of species’ population (upper and lower limits are applied to the point sizes for clarity). The colour of the nodes indicates the heating or cooling effect,
+𝐻𝑖. The width of the arrows indicates the interaction strength 𝐽𝑖 𝑗𝑛𝑗. Only interactions with core species are shown. In (a) the red (bottom-right) core species
+has a strong enough heating effect to overwhelm the cooling effect of the other core species, so this configuration has a net heating effect, as seen in Figure 1(a).
+In (b) both core species have a (weak) cooling effect, reducing the temperature, as seen in Figure 1(b).
+does have self-regulating feedback loops, these emerged merely by
+chance and that Gaia is not an expected feature of a planet hosting
+life, known as ‘lucky Gaia’ Watson (2004). As Figure 5 shows, in our
+model the fraction of Gaian states is increasing over time. This sug-
+gests that for early life starting out on a planet, a large amount of luck
+might be needed to initially start off in a Gaian state, but for surviv-
+ing runs over time the probability of being in a Gaian state increases.
+This would suggest that when observing a biosphere ‘lucky Gaia’
+may be the case for young planets but ‘probable Gaia’ is operating
+for older ones.
+The experiments in Figure 5 have considered systems with only
+internal perturbations, that is, those generated by the biosphere. How-
+ever, real planets experience many abiotic perturbations, both rapid
+and slower, such as changes in volcanic activity, changes in solar
+luminosity or impacts by large objects Covey et al. (1994); Overpeck
+& Cole (2006); Goldblatt & Zahnle (2011). Life is thought to have
+emerged early on Earth during a time when debris left over from
+the formation of the solar system was frequently colliding with the
+Earth. Biospheres in a non-Gaian state will be more susceptible than
+Gaian biospheres to perturbations and will have a higher risk of going
+extinct. This is closely related to the ‘Gaian bottleneck’ hypothesis
+Chopra & Lineweaver (2016) that proposes that early on in a planet’s
+history, if life emerges it must quickly establish self-regulatory feed-
+back loops to stabilise the climate of its planet in order to persist.
+If the biosphere fails then life goes extinct, the planet’s abiotic pro-
+cesses take over and the planet reverts to an inhospitable state. What
+is novel here is the idea that apart from total extinction, a planet can
+have a ‘near death experience’ where a mass extinction clears out a
+large fraction of the extant species. These mass extinctions are cru-
+cial for the exploration of the space of possible ecosystems Arthur
+& Sibani (2017) and ultimately lead to the emergence of long-lived
+stable states. Population diversity is known to significantly increase
+the resilience of ecosystems to perturbations Peterson et al. (1998);
+Luck et al. (2003), and additionally yeast Guan et al. (2012) and
+bacteria Lambert & Kussell (2014) have been shown to develop in-
+creased resilience to environmental stressors if exposed to them in
+the past. It is possible that large perturbations that do not eliminate
+all life are actually beneficial for evolving Gaia. Indeed, there may
+be evidence of this in Earth history, as it is thought that a period of
+global glaciation may have triggered the evolution of multi-cellular
+life Hoffman et al. (1998); Hedges (2004); Vincent et al. (2004);
+Boyle et al. (2007).
+4 HABITABLE ZONE EXPERIMENTS
+The habitable zone around a star is defined as the distance from a star
+where liquid water could exist on the surface of a planet Kasting et al.
+(1993). Models demonstrate that the habitable zone is impacted by
+several factors, including the age and class of the host star Ramirez &
+Kaltenegger (2016), planetary mass Kopparapu et al. (2014), plane-
+tary atmospheric composition Pierrehumbert & Gaidos (2011), and
+the surface water content of the planet Abe et al. (2011). Additionally
+a planet being within the habitable zone doesn’t guarantee habitabil-
+ity, as a planet may have more than one possible climate state for
+the same stellar and orbital parameters, e.g. a temperate Earth versus
+a frozen Earth Goldblatt & Zahnle (2011). For a more extreme ex-
+ample, it is thought that Venus and Earth might represent alternate
+end states for the same planetary system, with small perturbations
+occurring early on in their history influencing their modern day states
+Lenardic et al. (2016).
+Existing exoplanet surveys and models have identified that rocky
+MNRAS 000, 1–18 (222)
+
+(a)
+(b)
+0.04
+0.02
+0.00
+-0.02
+-0.04A Gaian Habitable Zone
+7
+Figure 3. (a) shows the average (over all surviving model runs) of the population (top row) and temperature (bottom row) where the background temperature
+𝑇0 = 100 = 𝑇𝑃. Column (b) shows the population and temperature where 𝑇0 = 105. The numbers next to the vertical dashed lines in the top row are the
+proportion of runs which have survived for that number of generations.
+planets can exist at a range of distances from their host star Domagal-
+Goldman et al. (2016). Thus, it is a natural question to ask about the
+stability and persistence of Gaia across a range of background tem-
+peratures, some more conducive to life, some less. In this section we
+run many experiments where we vary the background temperature
+𝑇0 and look at averages over 1000 model histories. To mimic the
+idea of a habitable zone with and without biotic influence we com-
+pare two versions of the model: one where life cannot influence the
+temperature, 𝜎𝐻 = 0, and one where life can influence it 𝜎𝐻 = 0.05.
+In Figure 5 we show the fraction of runs which survive for 105
+generations in both scenarios. Perhaps surprisingly, the distributions
+are roughly similar. As the background temperature changes, a similar
+number of model runs survive for 105 generations whether life can
+effect the environment or not. This shows, at least, that species-
+environment interactions have little effect on the probability of total
+extinction and therefore on the presence or absence of life. However,
+as we saw in the previous section, the model runs can be split into
+Gaian and non-Gaian states. Figure 5 also shows the proportion of
+MNRAS 000, 1–18 (222)
+
+(a)
+(b)
+400
+400
+350-
+350
+10.972
+i0.909
+300 -
+300-
+10.98
+10.989
+≥ 250
+≥ 250
+0.9
+i0.992
+10.981
+200 -
+10.99
+150 -
+150-
+100
+100
+102
+103
+104
+105
+102
+103
+104
+105
+t (generations)
+t (generations)
+108
+108
+Tp
+Tp
+T
+106 
+106
+Temperature
+104 -
+Temperature
+104
+102
+102
+100
+100
+98
+98
+102
+103
+104
+105
+102
+103
+104
+105
+t (generations)
+t (generations)8
+Arthur & Nicholson
+Figure 4. 𝑇0 = 105. (a) shows the temperature at 𝑡 = 105 generations versus population. Colour corresponds to the heating (red) or cooling (blue) effect of the
+core. There are clearly two distinct clusters: one with (potentially) high population and low temperature and one with low population and high temperature. (b)
+and (c) show histograms of the temperature and population respectively. (d) shows the proportion of surviving runs at each generation as well as the proportion
+that have 𝑇 ≤ 𝑇𝑃.
+runs that have 𝑁 > 200. The value of 200 is not itself significant,
+what is important is the comparison between 𝜎𝐻 = 0 and 𝜎𝐻 = 0.05.
+Far from 𝑇𝑃, only the Gaian states can have large populations, in the
+other cases the total population is low and life is simply ‘clinging
+on’. Importantly for exoplanet astronomy, a small pocket of life that
+is clinging on to existence is unlikely to produce a detectable bio-
+signature.
+Figure 6 shows the population of the model runs as a function of
+𝑇0. We see that when 𝜎𝐻 = 0 the total population at 𝑇𝑃 is larger.
+At 𝜎𝐻 = 0 the TNM agents are only attempting to optimise inter-
+species interactions, not interactions and temperature and thus can
+find a better maxima. For example, strongly symbiotic cores may
+have a detrimental effect on the temperature which is only relevant
+in the 𝜎𝐻 = 0.05 case. However, the population falls rather rapidly
+with 𝑇0 at 𝜎𝐻 = 0 compared to the 𝜎𝐻 = 0.05 case. We also see
+(from the colour gradient) that at 𝜎𝐻 = 0.05, for 𝑇0 far from 𝑇𝑃 only
+MNRAS 000, 1–18 (222)
+
+(a)
+(b)
+110.0
+110.0
+107.5
+107.5
+105.0
+105.0
+102.5
+102.5
+100.0
+100.0
+97.5
+97.5
+95.0
+95.0
+92.5
+92.5
+0
+200
+400
+600
+800
+1000
+1200
+1400
+0.000
+0.025
+0.050
+0.075
+0.100
+0.125
+0.150
+0.175
+N
+P(T)
+(c)
+(d)
+0.0000
+1.0
+0.0005
+0.0010
+0.8 -
+0.0015
+0.6
+0.0020
+0.0025
+0.4 -
+0.0030
+0.2
+0.0035
+Fraction Surviving
+0.0040
+0.0
+Fractionofsurvivors<Tp
+0
+200
+400
+600
+800
+1000
+1200
+1400
+102
+103
+104
+105
+N
+t (generations)A Gaian Habitable Zone
+9
+Figure 5. We run 1000 experiments at a variety of different 𝑇0 from 80 to 120 in steps of 2. As usual 𝑇𝑃 = 100 and 𝜏 = 2 for 105 generations. The figure
+shows proportion of surviving runs and the proportion of large population runs for a model with no temperature feedback 𝜎𝐻 = 0 and a model with feedback
+𝜎𝐻 = 0.05. (Note the anomaly at 𝑇0 = 118 for 𝜎𝐻 = 0.05 is a consequence of very low statistics, of the 7/1000 surviving runs, 2 happen to have 𝑁 > 200).
+those runs which heat or cool as appropriate are capable of having
+large populations. Figure 7 demonstrates that the runs split into two
+clusters, as also shown in Figure 4, which can be labelled by their
+temperature, in combination with Figure 6 this demonstrates that
+large population Gaian states may be observed when 𝑇0 is far from
+𝑇𝑃.
+This simple Gaian model would therefore predicts that if life plays
+only a minimal role in shaping its planet and we were looking at an
+abiotic habitable zone, that there would be a narrow range of radii
+around the host star where we might expect a detectable biospheres.
+Outside this narrow range the chance of finding an inhabited planet
+drops dramatically. If however life does play a strong role in shaping
+and regulating its host planet then we would expect to observe a much
+larger habitable zone. In the centre of this zone where conditions are
+‘ideal’ large population states, and so therefore potentially detectable
+biospheres, will be most probable but as we move towards the edges
+of the habitable zone the probability of detectable biospheres will
+be much higher than an abiotic habitable zone would predict. Our
+MNRAS 000, 1–18 (222)
+
+OH= O
+OH= 0.05
+TemperatureWindow
+TemperatureWindow
+Proportionnotextinct
+Proportionnotextinct
+1.2
+FractionofsurvivingrunswithN>2o0
+1.2 -
+FractionofsurvivingrunswithN>2o0
+1.0 -
+1.0-
+0.8 
+0.8 -
+0.6
+0.6-
+0.4 -
+0.4 
+0.2
+0.2 
+0.0 -
+0.0 -
+80
+90
+100
+110
+120
+80
+90
+100
+110
+120
+To
+To10
+Arthur & Nicholson
+Figure 6. As Figure 5, 1000 experiments at different 𝑇0 values. The figure shows the population of each of the model runs. Some jitter in the x-direction is
+applied to the points for clarity. The left hand shows the case where there is no species environment interaction 𝜎𝐻 = 0 and the right shows 𝜎𝐻 = 0.05 where
+the colour of the points reflects a heating (red) or cooling (blue) core.
+model suggests that there is a chance to detect biosignatures quite far
+from the abiotic habitable zone, provided life can affect the global
+temperature. Our model also predicts that looking at planets outside
+the abiotic habitable zone will be more informative for testing ideas
+of Gaia Theory, since within it we expect to see habitable planets
+whether Gaia is operating or not. Our model also demonstrates that
+finding a non-Gaian state within the biotic habitable zone is not
+incompatible with Gaia theory. Where life can shape its planet there
+remains the possibility for it to push its planet towards inhospitable
+conditions.
+5 INCREASING TEMPERATURE
+Geological evidence on Earth suggests that life emerged on our planet
+very soon after surface conditions allowed Nisbet & Sleep (2001)
+implying that the probability for the emergence of life might be high
+for planets with the correct prerequisites, however no alien life has yet
+MNRAS 000, 1–18 (222)
+
+1400
+.
+1400
+0
+.
+.
+..
+·
+.
+.
+1200
+:
+1200
+！
+:
+:
+:
+:
+1000
+1000
+0
+800
+800
+N
+N
+600
+.
+0
+600
+400
+400
+200
+200
+9
+.
+.
+0
+0
+80
+90
+100
+110
+120
+80
+90
+100
+110
+120
+To
+ToA Gaian Habitable Zone
+11
+Figure 7. As Figure 5, showing the temperature in each of the model runs. The colour of the points reflects a heating (red) or cooling (blue) core. Only
+𝜎𝐻 = 0.05 is shown, when 𝜎𝐻 = 0, 𝑇 = 𝑇0.
+been detected. The Gaian bottleneck hypothesis suggests an answer to
+this apparent contradiction and proposes that for newly emerged life
+on a young planet, there is a small window of opportunity whereby
+life can establish self-regulatory feedback loops to maintain habitable
+conditions. If the biosphere succeeds, then planetary habitability can
+be maintained for long time spans, however if the biosphere fails,
+surface conditions on the planet will rapidly become inhospitable,
+causing life to go extinct. This hypothesis is closely tied to ideas of an
+inhabitance paradox Goldblatt (2016) - that the long term habitability
+of a planet depends directly on whether or not it is inhabited. In this
+section we investigate aspects of the inhabitance paradox in the TNM
+setting.
+The classic Daisyworld experiment studies temperature regulation
+by life in the face of increasing solar luminosity. We can perform
+a similar experiment by increasing 𝑇0 over the course of the model
+runs. Figure 8 shows population and temperature for individual model
+runs where the background temperature, 𝑇0, increases linearly from
+𝑇𝑖𝑛𝑖𝑡 = 𝑇𝑃 = 100 up to 𝑇0 = 105 over the course of 104 genera-
+tions. The key observation is that the actual temperature 𝑇 (bottom
+row of Figure 8) increases more slowly than 𝑇0 - meaning that life
+is regulating the temperature. The only way the TNM can regulate
+without changing the composition of the core is by altering the pop-
+ulations of the core species. In Figure 8 we can see the temperature
+increase during an equilibrium is slowed by increasing or decreasing
+the population, and thus life’s contribution to the total temperature.
+Figure 9 shows the configuration of the model agents at a partic-
+ular time in the history of the simulation where the core - the group
+of species with significant reproduction probability - is stable and
+life is adapting to the temperature change. There are two different
+cases shown in (a) and (b). In case (a), between roughly 𝑡 = 4000
+and 𝑡 = 10000, the total population is increasing, which has the
+effect of slowing the temperature increase. Figure 9 (a) shows that
+the cloud (by definition species not in the core) has a roughly equal
+number of heating and cooling species, and each of these species has
+a small population, thus the cloud (i.e. the majority of species) does
+not participate in temperature regulation. Of the 4 species making up
+the core, 2 have a cooling effect, one is heating and one is approxi-
+mately neutral. The upper left and lower right species happen to have
+𝐻𝑖 = −0.047 and 𝐻𝑖 = 0.046 respectively, as well as roughly equal
+populations, so their effects cancel out, resulting in a net cooling by
+increasing the core population.
+Note that in Figure 8 (a) during this period the temperature is
+below 𝑇𝑃 = 100. As 𝑇0 increases it will push 𝑇 towards 𝑇𝑃, the
+fitness of all species
+𝑓𝑖 =
+∑︁
+𝑗
+𝐽𝑖 𝑗𝑛 𝑗
+1 +
+�𝑇 −𝑇𝑃
+𝜏
+�2 − 𝜇𝑁
+increases and therefore the population increases, which increases the
+cooling effect to (partially) offset the abiotic temperature increase.
+Figures 8 (b) and 9 (b) shows the opposite case. The core has a net
+heating effect and the temperature is above 𝑇𝑃. Increasing 𝑇0 moves
+the temperature further from 𝑇𝑃, reducing the fitness and also the
+population, therefore reducing the heating effect of life.
+This is a regulation mechanism known as ‘rein-control’ where the
+temperature of the system can be thought of as being ‘pulled’ in
+two different directions by different reins, in this case 𝑇0 and the
+heating or cooling effect of life. As all species share the same 𝑇𝑃
+it is the overall heating or cooling impact of the TNM community
+that is important for temperature regulation. Looking at the case of
+a cooling community first, Figure 8 (a), after 𝑡 ≈ 4000 generations
+has 𝑇 < 𝑇𝑃 < 𝑇0. In this case when 𝑇 < 𝑇𝑃, as 𝑇0 increases, this
+moves 𝑇 closer to 𝑇𝑃 and boosts the growth rate and hence the size
+of the cooling core, slowing the rate of heating. Once 𝑇 ≃ 𝑇𝑃 the
+𝑇0 rein is pulling away from 𝑇𝑃, limiting further growth and so the
+system stabilises. These feedback loops for an overall cooling TNM
+community are shown in Figure 10.
+MNRAS 000, 1–18 (222)
+
+120
+115
+110 -
+105 -
+← 100 -
+ S6
+90 -
+85 -
+80 -
+80
+85
+90
+95
+100
+105
+110
+115
+120
+To12
+Arthur & Nicholson
+Figure 8. Showing a single model run where background temperature 𝑇0 is increasing over time. The population is shown in the top row and temperature in the
+bottom row. The two columns show the two different types of temperature regulation by the core. On the left, after 4000 generations the temperature is regulated
+by increasing the population. On the right, between 4000 and 8000 generations, temperature is regulated by decreasing the population.
+In Figure 8 (b) by 𝑡 ≈ 2000 the TNM community is overall heating
+and 𝑇 > 𝑇0 > 𝑇𝑃. In this scenario any further growth of the com-
+munity would increase 𝑇 which would decrease the growth rate. On
+the other hand a reduction in population reduces its heating effect,
+which partially offsets the increase in 𝑇0 and so the real temperature
+𝑇 increases more slowly. Even though 𝑇0 > 𝑇𝑃 the heating TNM
+community and 𝑇0 are still ‘pulling’ the temperature in opposite di-
+rections as a reduction in the population will cool the environment
+which will move 𝑇 closer to 𝑇𝑃. When 𝑇0 > 𝑇𝑃 a heating TNM
+community can never achieve a 𝑇 close to 𝑇𝑃.
+At 𝑡 ≈ 9000 we see that there is a quake and 𝑇 rapidly drops
+below 𝑇𝑃 as the TNM community switches from an overall heating
+one to overall cooling. This example demonstrates that a biosphere
+in a non-Gaian state can become ‘unstuck’ and transition to a Ga-
+ian state if life can cling on for long enough. Twice during Earth’s
+history it is thought that the planet was covered in ice from poles to
+MNRAS 000, 1–18 (222)
+
+(a)
+(b)
+1750
+1400
+1500
+1200
+1250-
+1000
+1000
+800
+N
+750
+600
+500
+400
+250
+200
+0-
+.
+2000
+4000
+6000
+8000
+10000
+12000
+2000
+4000
+6000
+8000
+10000
+12000
+t (generations)
+t (generations)
+108
+108
+IP
+106
+106
+104 
+104
+emperature
+Temperature
+102
+102
+100
+100
+98 -
+98 -
+96
+96
+94 -
+94 -
+0
+2000
+4000
+6000
+8000
+10000
+12000
+0
+2000
+4000
+6000
+8000
+10000
+12000
+t (generations)
+t (generations)A Gaian Habitable Zone
+13
+Figure 9. Model snapshot at 𝑡 = 7000 generations for the runs (a) and (b) from Figure 8. (a) The core has an overall cooling effect (b) the core has a heating
+effect.
+Figure 10. Feedback loops between community population, N, and environment temperature, T, for an overall cooling TNM community. A + symbol (also
+indicated with a solid arrow) indicates an increase in the source leads to an increase in the sink, e.g. an increase in population leads to an increase in temperature.
+A - symbol (also indicated with a dashed arrow) indicates an increase in the source leads to a decrease in the sink, e.g. an increase in the temperature leads to a
+decrease in the total population. A feedback loop with an overall positive sign (determined by multiplying each sign in the loop) indicates a runaway feedback
+loop, whereas a feedback look with an overall negative sign indicates a stable feedback loop. Therefore for a cooling TNM community, temperature regulation
+occurs below 𝑇𝑃.
+equator - known as a Snowball Earth Hoffman et al. (1998). These
+Snowball Earth states persisted for millions of years and although
+there is evidence that a diversity of life survived these states, a frozen
+planet would present fewer niches for life than a thawed planet would
+(indeed, on Earth the biodiversity is lowest at the poles Rutherford
+et al. (1999)). Such a state could represent a non-Gaian biosphere
+clinging on, and both Earth history and our experiments demonstrate
+that observing a planet in a non-Gaian state doesn’t mean that it will
+always remain so.
+5.1 Averages
+Again, we are interested in what happens in the long run on average.
+Figure 11 shows that for our setup where 𝑇0 > 𝑇𝑃, on average the
+temperature is regulated below 𝑇0. Only those communities that have
+strong mutually symbiotic interactions and a cooling effect are likely
+to survive. If the rate of heating is not too strong (top row) most of
+the runs survive and the population grows logarithmically over time
+while the proportion of runs at or below𝑇𝑃 falls at a much slower rate
+than the increasing background temperature. Since most of the runs
+survive we can’t have Selection by Survival, so Sequential Selection
+with Memory must be responsible for this behaviour. The 𝑁 versus
+MNRAS 000, 1–18 (222)
+
+N
+N
++
+for T < Tp
+for T > T.(a)
+(b)
+0.04
+0.02
+0.00
+-0.02
+-0.0414
+Arthur & Nicholson
+Figure 11. Top row is the scenario where we heat from 𝑇0 = 100 = 𝑇𝑝 to 𝑇𝑓 𝑖𝑛 = 105 over the course of 104 generations. The first column is the average
+population, the numbers in black are the proportion of runs which have survived, cyan italic shows the proportion of survivors which have 𝑇 ≤ 𝑇𝑃. The second
+column shows the average temperature and 𝑇0. The final column shows 𝑁 versus 𝑇 for all the model runs after 104 generations when 𝑇0 = 𝑇𝑓 𝑖𝑛. The second
+row is the same as the first but with 𝑇𝑓 𝑖𝑛 = 120.
+𝑇 plot in the top row of Figure 11 shows that we still have a split
+between model runs with a heating core and a cooling core, where
+only those with a cooling core can have large 𝑁.
+The bottom row of Figure 11 shows the case where the heating is
+much more aggressive with 𝑇𝑓 𝑖𝑛 = 120. With a constant background
+𝑇0 = 120 around 27% of model runs survive for 104 generations.
+Figure 11 shows that once the temperature goes above ∼ 110 the
+runs start to go extinct though a larger proportion, 68%, survive until
+𝑡 = 104. Surviving runs are divided into two groups: runs where a
+small population is ‘clinging on’ at high 𝑇 and runs where a large,
+cooling population can be maintained.
+We investigate this further in Figure 12, where we directly compare
+runs with an constant background temperature 𝑇0 = 110 = 𝑇𝑃 + 5𝜏
+to runs where the temperature gradually increases to 𝑇0 = 110 over
+104 generations. At 104 generations, when both systems experience
+the same 𝑇0, the runs which have been heated gradually are doing
+better i.e. more of them survive, they have higher populations and
+lower temperatures. This simple observation has a few implications.
+First it suggests that if life occurs earlier, as soon as conditions are
+optimal for it, then it can survive longer and it can have a greater
+influence on the long term habitability of its planet. Second it sug-
+gests that more realistic models aiming to map out the habitable zone
+around a star should consider if the planet has ever been hospitable
+for life. In that case planets which would have inhospitable abiotic
+parameters, like 𝑇0, at the time of observation may have been able to
+maintain habitable temperatures. This phenomena - where life is key
+in preserving habitability is known as the inhabitance paradox - that
+long term habitability of a planet isn’t possible without life main-
+taining habitability Goldblatt (2016). It also ties closely to the Gaian
+Bottleneck hypothesis Chopra & Lineweaver (2016) - life emerging
+during a window of opportunity can prevent the environment from
+degrading, even as 𝑇0 changes.
+Finally, Figure 13 studies the effect of the rate of heating by
+comparing two scenarios where 𝑇0 is increased from 𝑇𝑃 = 100
+to 𝑇𝑓 𝑖𝑛 = 110 over 104 versus 105 generations. The ‘slow’ heating
+scenario could be thought to mimic something like the gradually
+increasing solar luminosity while the fast heating scenario is akin
+to something like the rapid onset of global glaciation Overpeck &
+Cole (2006). Figure 13 shows that, in general, slower heating leads
+to more Gaian states. The population is higher and the final temper-
+ature is lower. The average population in fact stops increasing in the
+fast heating case, as abiotic conditions degrade faster than SSM can
+operate, while the slow heating case shows a continuously increas-
+ing population up to 105 generations. Thus, if SSM is to operate
+the larger the separation between abiotic and biotic timescales (e.g.
+geologic versus evolutionary) then the more likely we are to observe
+a Gaia.
+MNRAS 000, 1–18 (222)
+
+t= 104
+300
+107
+To
+110.0
+106
+280 
+Tp
+(T)
+107.5
+:0.47
+10.43
+105
+260 
+105.0
+104 -
+10.99
+10.98
+240 
+10.50
+102.5
+M
+220-
+10.99
+102
+100.0
+101
+97.5
+180 
+100
+95.0
+66
+92.5
+160
+98
+102
+103
+104
+2000
+4000
+6000
+8000
+10000
+0
+250
+500
+750
+1000
+1250
+1500
+1750
+t (generations)
+N
+t (generations)
+t= 104
+120
+280 -
+120
+(T)
+10.45
+260
+115 
+115
+10.99
+10.48
+240 -
+ature
+110
+10.99
+220 -
+105
+200
+:0.24
+105 -
+10.68
+100
+180 -
+160 -
+100
+95
+102
+103
+104
+2000
+4000
+6000
+8000
+10000
+0
+200
+400
+600
+800
+1000
+1200
+1400
+t (generations)
+t (generations)
+NA Gaian Habitable Zone
+15
+Figure 12. Comparing increasing 𝑇0 = 100 → 110 to constant 𝑇0 = 110. Top left is average population, the numbers show the number of surviving runs at each
+time-step. Top right shows average temperature and 𝑇0. Bottom left shows all of the runs (see Figure 4) for the increasing temperature case, and bottom right
+shows the runs for the constant temperature case.
+6 CONCLUSIONS
+Models such as the one described here help us to understand how
+planetary regulation arises from ‘selfish’ individuals. Gaia is a prime
+example of an emergent system - one where the whole has properties
+its parts do not. However Gaia was first discussed some years before
+emergence and complexity thinking were common. Lovelock and
+others discussing Gaia at the macro level, for example talking about
+her health with the notion of Geophysiology Lovelock (1989), have
+been harshly criticised. There have been two primary criticisms: the
+first argues that Gaia is simply a metaphor and not a scientific theory
+Kirchner (1989) and the second argues that episodes from earth
+history where life generates hostile conditions is strong evidence
+against Gaia Ward (2009). We believe the notion of Entropic Gaia
+Arthur & Nicholson (2022) and our discussion of selection principles
+answers both of these criticisms.
+First to address the charge that Gaia is ‘just a metaphor’ it is in-
+structive to discuss some other emergent systems. A gas is ‘just’ a
+collection of individual atoms. However emergent properties, like
+MNRAS 000, 1–18 (222)
+
+300
+112
+Tinit = 100
+Tinit = 100
+Tinit = 110
+Tinit = 110
+:0.99
+110
+250 -
+10.95
+10.99
+108-
+200
+10.81
+106
+≤ 150
+10.94
+104
+100
+10.99
+102 
+50
+100 
+98
+102
+103
+104
+2000
+4000
+6000
+8000
+10000
+t (generations)
+t (generations)
+Tinit = 100, t = 104
+Tinit = 110, t = 104
+115.0
+112.5
+112.5
+110.0
+110.0
+107.5
+107.5
+105.0-
+105.0
+102.5
+102.5
+100.0-
+100.0
+97.5
+97.5
+95.0 -
+95.0 
+92.5
+0
+250
+500
+750
+1000
+1250
+1500
+1750
+2000
+o
+200
+400
+600
+800
+1000
+1200
+N
+N16
+Arthur & Nicholson
+Figure 13. Increasing 𝑇0 = 100 → 110 over 104 generations and 105 generations. Top left is average population, the numbers show the fraction of surviving
+runs at each time-step. Top right shows average temperature and 𝑇0 (note the log scale makes the linearly increasing 𝑇0 look exponential). The numbers give the
+average temperature at the end of the experiment. Bottom left shows all of the runs at the end of the fast heating experiment, and bottom right shows the runs in
+the slow heating scenario.
+pressure and temperature, are not features of individual atoms but
+are still very much real. An organism is a ‘just’ a system of chemical
+reactions. However biology is not just applied chemistry, it is legiti-
+mate and useful to reason about cells. The economy is a phenomenon
+that emerges out of the production and consumption patterns of mil-
+lions of individuals. Depression, recession, asset bubbles and so on
+are properties of the whole system that have real explanatory power.
+Of course we can have incorrect theories about gases, cells or eco-
+nomics, but these do not make it illegitimate to reason about whole
+systems. When talking about life at planetary scale, we talk about
+something called ‘Gaia’. This is a metaphor in the same sense as
+an organism or an economy, a metaphor that admits rigorous micro
+foundations and which can be very productive for understanding a
+system or a collection of systems. In the context of bio-signature
+detection, where we may have potentially very many ‘Gaias’ and
+MNRAS 000, 1–18 (222)
+
+400
+112
+tmax =10000
+tmax = 10000
+tmax=100000
+tmax = 100000
+110
+350
+10.88
+108 
+300
+10.98
+105.55
+M
+10.98
+10.99
+104-
+103/79
+250 -
+10.99
+i0.95
+102
+200寸
+100
+150
+98
+102
+103
+104
+105
+102
+103
+104
+105
+t (generations)
+t (generations)
+To=110,t=104
+To=110, t= 105
+115.0
+115
+112.5
+110.0-
+110
+107.5
+105.0
+105
+102.5
+100 
+100.0
+97.5
+95 
+95.0 
+0
+250
+500
+750
+1000
+1250
+1500
+1750
+2000
+o
+500
+1000
+1500
+2000
+N
+NA Gaian Habitable Zone
+17
+very limited information about the processes going on inside them,
+a holistic theory is crucial.
+The second class of criticisms is directly addressed by our idea that
+Gaia arises due to a selection principle operating on species networks.
+To briefly re-iterate - sequential selection posits a type of punctuated
+equilibrium Eldridge & Gould (1972), characterised by stable periods
+interrupted by catastrophes. Models of co-evolution such as the TNM
+and others (e.g. Kauffman & Weinberger (1989)) show exactly this
+kind of behaviour. Entropic Gaia is the argument that these stable
+periods get longer over time. In the TNM this is for the simple reason
+that each punctuation is not a complete reset, the next stable period
+emerges from the debris of the previous equilibrium. The species
+networks that can establish themselves must have high population
+growth rates so they saturate the carrying capacity, while also not
+self limiting. High populations mean more diversity, which means
+even more ‘debris’ during the next reset. In this view, periods of
+disregulation are not evidence against Gaia, they are an integral part
+of how she arises.
+To show the use of such a theory in this work we have, within a
+concrete and fairly general modelling framework, investigated some
+pressing questions of astro-biology through Gaia theory. In section 3
+we studied the effect of life on the ability of a planet to sustain life in
+suboptimal abiotic conditions. This leads us to propose the idea of the
+Gaian habitable zone versus the standard abiotic habitable zone.
+Our results predict that Gaia extends the habitable zone around a star
+while making the abiotic habitable zone slightly less hospitable. This
+has a straightforward and testable implication - search for life outside
+the abiotic habitable zone as a signature of Gaia.
+In section 5 we study the effect of a deteriorating abiotic envi-
+ronment to address the idea of the Gaian bottleneck. Life’s chances
+of long term survival, and the emergence of Gaia, are both more
+likely if life can ‘catch’ a window of high habitability (in this model
+where 𝑇0 = 𝑇𝑃). Life can then, on average, maintain better condi-
+tions. Again this has implications in the search for life - planets which
+were once inside but are currently outside the abiotic habitable zone
+may host life. Again Gaia expands the boundary of habitability and
+inhabitance.
+Both Selection by Survival (SBS) and Sequential Selection with
+Memory (SSM) play a role in determining the likelihood of a Gaian
+planet. Nearer the centre of the abiotic habitable zone, SSM is the
+main mechanism for generating Gaias and towards the edges SBS
+becomes more important. Finding a non-Gaian planet at the center
+of the abiotic habitable zone is not incompatible with Gaia theory.
+If life can strongly influence its environment it can degrade it. The
+results of this model suggest that if life can cling on, and abiotic
+conditions do not degrade too much, then the planet can become
+‘unstuck’ through the evolution of species which regulate the tem-
+perature. To map out the Gaian habitable zone around a particular
+star, or class of star, will require fusing detailed abiotic models with
+models of biogeochemistry. Some steps in this direction were taken
+in Nicholson et al. (2022), where the fine-details, such as lifespan
+or maintenance energy requirements of the biosphere were shown
+not to affect the general conclusion about life’s effect on potential
+bio-signatures. If this is the case generally, and this framework can
+be expanded to cover a range of biotic scenarios, then we may be able
+to produce detailed predictions of the Gaian habitable zone without
+needing to know the population-level details of any alien life. Iden-
+tifying potential metabolic pathways and limiting abiotic factors on
+microbial growth (e.g. resource limitation) would be sufficent for
+robust biosignauture predictions.
+In summary, we propose a statistical theory of planetary habitabil-
+ity, with strong and testable implications on the search for alien life.
+Our model, as well as Earth history, teaches us that a Gaian planet
+can emerge from periods of disregulation and low habitability. Ulti-
+mately, this suggests a wider range of habitable and inhabited planets
+than abiotic models would predict.
+ACKNOWLEDGEMENTS
+This work was supported by a Leverhulme Trust research project
+grant [RPG-2020-82].
+DATA AVAILABILITY
+The code is available on request from the authors.
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+

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+An Analysis of the Automatic Bug Fixing
+Performance of ChatGPT
+Dominik Sobania
+Johannes Gutenberg University Mainz
+Email: [email protected]
+Carol Hanna
+University College London
+Email: [email protected]
+Martin Briesch
+Johannes Gutenberg University Mainz
+Email: [email protected]
+Justyna Petke
+University College London
+Email: [email protected]
+Abstract—To support software developers in finding and fixing
+software bugs, several automated program repair techniques have
+been introduced. Given a test suite, standard methods usually
+either synthesize a repair, or navigate a search space of software
+edits to find test-suite passing variants. Recent program repair
+methods are based on deep learning approaches. One of these
+novel methods, which is not primarily intended for automated
+program repair, but is still suitable for it, is ChatGPT. The
+bug fixing performance of ChatGPT, however, is so far unclear.
+Therefore, in this paper we evaluate ChatGPT on the standard
+bug fixing benchmark set, QuixBugs, and compare the perfor-
+mance with the results of several other approaches reported in
+the literature. We find that ChatGPT’s bug fixing performance is
+competitive to the common deep learning approaches CoCoNut
+and Codex and notably better than the results reported for the
+standard program repair approaches. In contrast to previous
+approaches, ChatGPT offers a dialogue system through which
+further information, e.g., the expected output for a certain input
+or an observed error message, can be entered. By providing such
+hints to ChatGPT, its success rate can be further increased, fixing
+31 out of 40 bugs, outperforming state-of-the-art.
+Index Terms—Automated program repair, automatic bug fix-
+ing, ChatGPT, Codex, language models.
+I. INTRODUCTION
+Complex software usually contains undiscovered bugs in its
+source code. The later these are found, the more far-reaching
+consequences these can have. Uncorrected bugs in software
+can lead to failures of essential systems, which can result in
+high economic costs [1].
+In order to support programmers in finding and fixing
+software errors, automated program repair (APR) systems have
+been introduced that automatically suggest software patches
+to correct the detected errors [2], [3]. For instance, Haralds-
+son et al. [4] suggest an approach based on genetic improve-
+ment (GI) [5] that tracks emerging bugs during a workday and
+searches for potential fixes for them overnight. The following
+morning the programmers get a list of suggestions which
+should help fix the detected bugs.
+Standard methods for automated program repair can be
+classified into two categories: the generate-and-validate ap-
+proaches mutate software guided by a search strategy, while
+semantics-driven (or synthesis-based) approaches use a con-
+straint solver to synthesize repairs [3]. The generate-and-
+validate ones have first seen industrial uptake [4]. One of
+the key disadvantage of standard approaches to APR is their
+running cost. The generate-and-validate ones usually rely on
+test suites to verify program correctness, while synthesis-based
+ones on calls to a constraint solver. Both validation strategies
+are costly, making typical APR tools hours to run before a
+viable patch is presented to the developer.
+Most recently, program repair tools based on deep learn-
+ing (DL) approaches have been introduced [6]. These learn bug
+fixing patterns from existing databases and treat the automated
+program repair problem as a neural machine translation task,
+producing a ranking of, sometimes hundreds of, patches.
+Unlike standard approaches, such generated patches are not
+usually evaluated against a test suite, or other automated
+verification strategy, so may not even compile. Nevertheless,
+DL-based program repair has shown competitive results to
+standard approaches [6].
+In recent years, several large-scale language models based
+on the Transformer architecture [7] have been introduced, such
+as CodeBERT [8], PyMT5 [9], and Codex [10], which can also
+process and extend source code and achieve comparable results
+to standard approaches on various coding tasks [11]. A large-
+scale language model based on the Transformer architecture
+that has recently received great attention is ChatGPT.1 With
+ChatGPT not only text input can be extended, but it is even
+possible to have a conversation with the language model and
+the previous chat history is taken into account for answer
+generation. In addition to very general or subject-specific
+topics, ChatGPT can also be used to discuss source code, e.g.,
+to ask for a suggestion for a fix of incorrect code. However,
+the quality of these suggestions is still unclear.
+Therefore, in this work we evaluate and analyse the au-
+tomatic bug fixing performance of ChatGPT. Moreover, we
+provide a comparison with results reported in the literature
+obtained using state-of-the-art APR approaches and Codex.
+We chose the QuixBugs [12] benchmark set for our study, as
+it contains small, yet challenging programs for current APR
+1https://openai.com/blog/chatgpt/ (accessed January 18, 2023).
+arXiv:2301.08653v1  [cs.SE]  20 Jan 2023
+
+approaches. We consider all Python problems from QuixBugs,
+i.e., 40 overall.
+We first ask ChatGPT for bug fixes for the selected bench-
+marks and manually check whether the suggested solution is
+correct or not. We repeat the query four times, to account for
+the heuristic nature of ChatGPT. Next, we compare its perfor-
+mance with that of Codex and dedicated APR approaches. For
+the standard APR approaches, we take the results from a recent
+paper [13] that examines the performance of several methods
+on the QuixBugs benchmark set. For dedicated APR methods
+based on deep learning, we take results from CoCoNut [14].2
+For the large-scale language model Codex, we take the results
+from [15]. Furthermore, we study and categorize ChatGPT’s
+answers to gain a deeper understanding of its behavior. Given
+that ChatGPT provides a unique opportunity for a conversation
+with the model, we provide a small hint to the model (e.g., a
+failing test input with an error it produces) to see if it improves
+ChatGPT’s fix rate.
+We find that ChatGPT’s program repair performance is
+competitive to the results achieved with CoCoNut and Codex
+(19 vs. 19 vs. 21 instances solved, respectively). Compared
+to the standard program repair approaches, ChatGPT achieves
+notably better results. With ChatGPT, we could fix bugs in
+19 out of 40 problems while with the standard approaches
+only 7 can be fixed, even though we give ChatGPT only the
+incorrect code snippet without any additional information and
+without using the chat option in a conversational way. If the
+chat function is actively used, we can fix even more instances.
+This shows the power of providing manual hints to a program
+repair system. All our experimental data is available online.3
+II. CHATGPT FOR AUTOMATED PROGRAM REPAIR
+In this section we present our methodology for assessing
+ChatGPT’s program repair performance.
+A. Benchamrk
+To evaluate the automatic bug fixing performance of Chat-
+GPT, we use the QuixBugs [12] benchmark set. Unlike
+many other benchmark suites for automated program repair,
+QuixBugs contains relatively small problems (small number of
+code lines). These are thus suitable for use in a dialogue sys-
+tem. For each of the 40 benchmark problems from QuixBugs,
+we take the erroneous Python code, remove all contained
+comments4, and ask ChatGPT if the code contains a bug and
+how it can be fixed. For each benchmark problem, we make
+several independent requests to ChatGPT and manually check
+whether the given answer is correct or not. We standardize our
+procedure by using the same format for each query. We ask:
+“Does this program have a bug? How to fix it?” followed by
+an empty line and the buggy code without comments. Figure 1
+shows an example request to ChatGPT for the BITCOUNT
+problem. Lines 1-2 contain the question to ChatGPT where
+2Although more recent approaches exist, we found this work is the most
+recent providing sufficient patch ranking detail.
+3https://gitlab.rlp.net/dsobania/chatgpt-apr.
+4This was necessary, as sometimes the comments contain the solution.
+1
+Does
+t h i s
+program
+have a bug ? How to
+2
+f i x
+i t ?
+3
+4
+def
+b i t c o u n t ( n ) :
+5
+count = 0
+6
+while n :
+7
+n ˆ= n − 1
+8
+count += 1
+9
+r e t u r n
+count
+Fig. 1: Request to ChatGPT for the BITCOUNT problem.
+we ask how the bug can be fixed and starting from line 4
+we present the erroneous code snippet. For this example, we
+would expect from ChatGPT an answer that addresses the
+bug in line 7, where n ˆ= n - 1 should be replaced with
+n &= n - 1, either with a response containing the complete
+code snippet with the fixed bug (correctly addressed) or by
+giving an exact and correct description how to change the
+affected code lines.
+B. Comparison Study
+We ran four independent requests to ChatGPT for each
+problem from the QuixBugs dataset. In order to compare the
+results of ChatGPT with the standard APR methods, we take
+the results from a comprehensive study from the literature
+[13] that reports the performance of ten different methods
+(Arja [16], Cardumen [17], Dynamoth [18], JGenProg [19],
+JKali [19], JMutRepair [19], Nopol [20], NPEfix [21], RSRe-
+pair [16], and Tibra [19]) on the problems from QuixBugs. For
+dedicated APR approaches based on deep learning we chose
+recent results reported by Lutellier et al. [14].5 In Table I
+we report a fix only if the correct patch was ranked first by
+Lutellier et al.’s proposed approach, CoCoNut. For the large-
+scale language model Codex, we take the results from a recent
+paper [15]. We ran this experiment on ChatGPT versions from
+December 15, 2022 and January 9, 2023.
+C. Dialogue Study
+Given that ChatGPT provides a unique opportunity of a
+dialogue with the model, we also conduct a study where we
+provide ChatGPT with a hint, based on ChatGPT’s response. If
+ChatGPT does not provide a correct answer to the first request
+(described in the previous paragraph), we tell ChatGPT in a
+standardized way that the function is not working correctly and
+additionally provide an input example that shows that the func-
+tion is not working properly. If ChatGPT incorrectly claimed
+the program was correct, we replied: “The function does not
+work. E.g., for the input <input> it should return <output>.”
+or “The function does not work. E.g. for the input <input>
+I get the following error message: <output>”, depending
+on whether the failing test case from the QuixBugs dataset
+returned an incorrect answer or threw an error. In the case of
+5CoCoNut, solves overall only 2 instances less than best reported thus far
+on the QuixBugs Python dataset [15], though details on patch ranking for
+each program were missing from the later work.
+
+more complex inputs we made the following response: “The
+function does not work. E.g., given the following call: <code
+snippet> The following should be the output: <output>.”6 We
+only provide one such hint and report results. This experiment
+was run on the ChatGPT version from January 9, 2023.
+III. RESULTS AND DISCUSSION
+In this section, we present the results of the comparison
+of ChatGPT, Codex, CoCoNut, and the standard APR ap-
+proaches. We classify ChatGPT’s answers and report on short
+discussions with the model. Furthermore, we describe what
+we noticed while working with ChatGPT.
+A. Automatic Bug Fixing Performance
+Table I shows the achieved results of ChatGPT, Codex,
+CoCoNut, and the dedicated APR approaches on the bench-
+mark problems from QuixBugs. For the ChatGPT results, a
+checkmark () indicates that a correct answer was given in
+at least one of the four runs for a benchmark problem. A
+cross () indicates that no correct answer was given in any
+of the runs. In parentheses we additionally report the number
+of runs that led to a successful solution. For the results from
+the literature, a checkmark indicates that a correct bug fix is
+reported. A cross means that no successful bug fix is reported.
+We see that the results achieved by ChatGPT are similar
+to Codex in performance and outperform the standard APR
+approaches. Overall, we find bug fixes for 19 benchmark
+problems with ChatGPT, 21 are reported for Codex, 19 for
+CoCoNut, and only 7 for the standard approaches.
+The large gap in performance between the language model
+based approaches and the standard APR approaches can be
+explained by the fact that the latter usually just use a small test
+suite to define the problem, which can be easily overfitted. The
+authors of [13] also report this problem. If only the test suite
+is considered for evaluation, the standard approaches would
+solve a total of 16 benchmark problems. However, as in real-
+world applications only programs that work also on unseen
+inputs are usable, we have only adopted the 7 generalizing
+problems from [13] as correct.
+If we take a closer look at the results for ChatGPT, we
+see that benchmark problems are often only solved in one or
+two runs. Only for the problems BUCKETSORT and FLATTEN
+ChatGPT finds a bug fix in all four runs. So ChatGPT seems
+to have a relatively high variance when fixing bugs. For an
+end-user, however, this means that it can be helpful to execute
+requests multiple times.
+Furthermore, it is not surprising that ChatGPT solves about
+the same number of problems as Codex, as ChatGPT and
+Codex are from the same family of language models.7 How-
+ever, we still see potential for improvement for ChatGPT, as
+the given responses are often close to the correct solution
+(for a detailed classification of ChatGPT’s responses see
+Section III-B).
+6The third case only appeared once. All queries are available online.
+7https://beta.openai.com/docs/model-index-for-researchers (accessed Jan-
+uary 18, 2023).
+Nevertheless, we are very strict in our evaluation and
+consider only patches as correct if the bug introduced by
+QuixBugs is actually identified and corrected. E.g., for some
+problems, ChatGPT suggests a complete re-implementation
+which is then bug-free. However, these are probably no real
+bug fixes, since the introduced bug is not localized. We assume
+that ChatGPT simply reproduced what it has learned here.
+Furthermore, we do not count a bug as fixed if additional
+changes suggested by ChatGPT introduce new errors that
+prevent the program from running properly. Moreover, by
+sending just a single request in this evaluation, we are not
+using the full potential of the dialogue system. Consequently,
+we take a closer look at how ChatGPT behaves when we
+interact more with the system and give it more information
+about the bug in Section III-C.
+B. A Classification of ChatGPT’s Answers
+While working with ChatGPT, we noticed different types
+of responses that ChatGPT gave to our requests, especially
+when a bug could not be found. Therefore, we identified the
+different types of answers from ChatGPT for the benchmark
+problems from QuixBugs and analyzed their frequency. We
+identified the following classes of ChatGPT answers:
+• More information required: Asks for more information
+on the program behavior to identify the bug.
+• No bug found: Does not find a bug and states the program
+is working correctly.
+• Correct fix provided: Provides the correct fix for the
+correct bug.
+• Tries to fix something else: Does not find the intended
+bug and tries to fix or advise on something else that is
+not really a bug or adjusts for edge cases.
+• Provides fix but introduces new bug: Provides the
+correct fix for the target bug but introduces a new bug
+somewhere else.
+• Alternative implementation: Does not fix the bug but
+gives a working alternative implementation.
+Figure 2 shows the number of occurrences of identified
+classes of ChatGPT answers given for the problems from
+QuixBugs.
+We see that for most of our requests, ChatGPT asks for more
+information about the problem and the bug. With the second
+most number of answers given, we observe ChatGPT claiming
+that the given code snippet does not seem to have a bug. In
+both cases it might be useful to fully utilize the possibilities
+of the dialogue system ChatGPT offers, as further information
+might lead to a correct bug fix.
+Less often than the request for more information, we
+observe that ChatGPT fixes the bug but at the same time
+introduces new errors, or we see that ChatGPT not really
+addresses the bug correctly but suggests a completely new
+working re-implementation for the problem.
+C. A Discussion with ChatGPT
+In order to be able to compare ChatGPT with other systems
+in a standardized form, we have so far studied how ChatGPT
+
+TABLE I: Results achieved by ChatGPT, Codex, CoCoNut, and the standard APR approaches on the problems from the
+QuixBugs benchmark set. For ChatGPT, we also report the number of successful runs in brackets.
+Benchmark problem
+ChatGPT
+Codex [15]
+CoCoNut [14]
+Standard APR [13]
+bitcount
+ (0 / 4)
+
+
+
+breadth-first-search
+ (2 / 4)
+
+
+
+bucketsort
+ (4 / 4)
+
+
+
+depth-first-search
+ (0 / 4)
+
+
+
+detect-cycle
+ (0 / 4)
+
+
+
+find-first-in-sorted
+ (2 / 4)
+
+
+
+find-in-sorted
+ (3 / 4)
+
+
+
+flatten
+ (4 / 4)
+
+
+
+gcd
+ (0 / 4)
+
+
+
+get-factors
+ (1 / 4)
+
+
+
+hanoi
+ (0 / 4)
+
+
+
+is-valid-parenthesization
+ (2 / 4)
+
+
+
+kheapsort
+ (0 / 4)
+
+
+
+knapsack
+ (1 / 4)
+
+
+
+kth
+ (0 / 4)
+
+
+
+lcs-length
+ (0 / 4)
+
+
+
+levenshtein
+ (0 / 4)
+
+
+
+lis
+ (0 / 4)
+
+
+
+longest-common-subsequence
+ (0 / 4)
+
+
+
+max-sublist-sum
+ (0 / 4)
+
+
+
+mergesort
+ (1 / 4)
+
+
+
+minimum-spanning-tree
+ (0 / 4)
+
+
+
+next-palindrome
+ (1 / 4)
+
+
+
+next-permutation
+ (0 / 4)
+
+
+
+pascal
+ (1 / 4)
+
+
+
+possible-change
+ (1 / 4)
+
+
+
+powerset
+ (0 / 4)
+
+
+
+quicksort
+ (1 / 4)
+
+
+
+reverse-linked-list
+ (2 / 4)
+
+
+
+rpn-eval
+ (0 / 4)
+
+
+
+shortest-path-length
+ (1 / 4)
+
+
+
+shortest-path-lengths
+ (0 / 4)
+
+
+
+shortest-paths
+ (1 / 4)
+
+
+
+shunting-yard
+ (2 / 4)
+
+
+
+sieve
+ (0 / 4)
+
+
+
+sqrt
+ (1 / 4)
+
+
+
+subsequences
+ (1 / 4)
+
+
+
+to-base
+ (0 / 4)
+
+
+
+topological-ordering
+ (0 / 4)
+
+
+
+wrap
+ (0 / 4)
+
+
+
+ΣΣΣ (Solved)
+19
+21
+19
+7
+
+0
+10
+20
+30
+40
+50
+# of occurrences
+other
+alternative implementation
+provides fix but introduces new bug
+tries to fix something else
+correct fix provided
+no bug found
+more information required
+Fig. 2: Number of occurrences of identified classes of ChatGPT answers given for the problems from QuixBugs.
+1
+Request
+( 1 ) :
+2
+Does
+t h i s
+program
+have a bug ? How to
+f i x
+i t ?
+3
+4
+def
+gcd ( a ,
+b ) :
+5
+i f
+b == 0:
+6
+r e t u r n
+a
+7
+e l s e :
+8
+r e t u r n
+gcd ( a % b ,
+b )
+9
+10
+Response
+( 1 ) :
+11
+This
+i s
+the
+Euclidean
+a l g o r i t h m
+f o r
+f i n d i n g
+the
+g r e a t e s t
+common d i v i s o r
+(GCD)
+of two
+12
+i n t e g e r s ,
+a and b .
+I t
+appears
+to
+be
+implemented
+c o r r e c t l y .
+I f
+you
+are
+e x p e r i e n c i n g
+13
+i s s u e s
+with
+t h i s
+program ,
+i t
+i s
+l i k e l y
+t h a t
+the
+problem
+l i e s
+elsewhere
+in
+your
+code .
+14
+Without more
+i n f o r m a t i o n
+about
+the
+bug you
+are
+experiencing ,
+i t
+i s
+d i f f i c u l t
+to
+say
+15
+how to
+f i x
+i t .
+16
+17
+Request
+( 2 ) :
+18
+No ,
+the
+f u n c t i o n
+i s
+not
+c o r r e c t . E . g . ,
+f o r
+the
+i n p u t
+a = 4 and b = 5 I
+get
+the
+f o l l o w i n g
+19
+e r r o r
+message :
+” RecursionError : maximum r e c u r s i o n
+depth
+exceeded
+in
+comparison ” .
+20
+21
+Response
+( 2 ) :
+22
+The
+i s s u e
+i s
+t h a t
+the
+f u n c t i o n
+c a l l s
+i t s e l f
+r e c u r s i v e l y ,
+and
+in some cases ,
+i t
+never
+23
+r e a c h e s
+the
+base
+case
+( b == 0) ,
+causing
+i t
+to
+r e c u r s e
+i n d e f i n i t e l y .
+The base
+case
+of
+24
+the
+gcd
+a l g o r i t h m
+i s
+when the
+remainder
+i s
+zero ,
+but
+in
+t h i s
+implementation ,
+i t ’ s when
+25
+b
+i s
+zero .
+For example ,
+when a=4 and b=5 , gcd (4 ,
+5)
+i s
+c a l l e d ,
+which
+then
+c a l l s
+26
+gcd (4 % 5 ,
+5)
+which
+i s
+gcd (4 ,
+5 ) .
+This
+causes
+an
+i n f i n i t e
+loop .
+27
+28
+You can
+f i x
+t h i s
+by swapping
+the
+parameters
+in
+the
+r e c u r s i v e
+c a l l .
+29
+30
+def
+gcd ( a ,
+b ) :
+31
+i f
+a == 0:
+32
+r e t u r n
+b
+33
+e l s e :
+34
+r e t u r n
+gcd ( b % a ,
+a )
+35
+36
+This way ,
+i f
+a
+i s
+zero ,
+the
+f u n c t i o n
+r e t u r n s
+b ( which
+i s
+the
+gcd
+of
+a and b ) ,
+o t h e r w i s e
+37
+i t
+c a l l s
+i t s e l f
+with
+the new values
+of b % a and a .
+Fig. 3: A discussion with ChatGPT about the GCD problem from QuixBugs. We inserted request and response markers in bold
+font to make the chat history more readable.
+behaves when making exactly one request (see Section III-A).
+However, a potentially powerful advantage of ChatGPT is
+that we can interact with the system in a dialogue to specify
+a request in more detail. This is useful, as both, a textual
+description as well as a definition given by a test suite, are
+possibly ambiguous and could be misleading.
+
+1
+Request
+( 1 ) :
+2
+Does
+t h i s
+program
+have a bug ? How to
+f i x
+i t ?
+3
+4
+def
+b i t c o u n t ( n ) :
+5
+count = 0
+6
+while n :
+7
+n ˆ= n − 1
+8
+count += 1
+9
+r e t u r n
+count
+10
+11
+Response
+( 1 ) :
+12
+I ’m unable
+to
+t e l l
+i f
+the
+program
+has a bug
+without
+more
+i n f o r m a t i o n
+on
+the
+expected
+13
+behavior
+and
+the
+i n p u t
+t h a t
+i s
+causing
+the
+i s s u e .
+[ . . . ]
+14
+15
+Request
+( 2 ) :
+16
+The program
+should
+count
+the
+number
+of 1− b i t s
+in
+the
+bin ary
+encoding
+of n .
+17
+18
+Response
+( 2 ) :
+19
+The
+f u n c t i o n
+b i t c o u n t ( n )
+appears
+to
+be
+counting
+the
+number
+of 1− b i t s
+[ . . . ]
+20
+21
+To confirm
+t h i s ,
+you can
+t e s t
+the
+f u n c t i o n
+with some
+t e s t
+cases
+to
+check
+i f
+22
+i t
+produces
+the
+expected
+output :
+23
+24
+t e s t
+c a s e s
+= [0 ,
+1 ,
+2 ,
+3 ,
+4 ,
+5 ,
+6 ,
+7 ,
+8 ,
+9 ,
+10 ,
+11 ,
+12 ,
+13 ,
+14 ,
+15]
+25
+exp ect ed output = [0 ,
+1 ,
+1 ,
+2 ,
+1 ,
+2 ,
+2 ,
+3 ,
+1 ,
+2 ,
+2 ,
+3 ,
+2 ,
+3 ,
+3 ,
+4]
+26
+27
+Request
+( 3 ) :
+28
+The
+f u n c t i o n
+does
+not work .
+For
+i n p u t
+1
+i t
+seems
+to
+run
+i n t o
+an
+i n f i n i t e
+loop .
+29
+30
+Response
+( 3 ) :
+31
+You are
+c o r r e c t
+t h a t
+the
+f u n c t i o n
+w i l l
+run
+i n t o
+an
+i n f i n i t e
+loop when the
+i n p u t
+i s
+1 .
+32
+[ . . . ]
+33
+34
+def
+b i t c o u n t ( n ) :
+35
+i f
+n == 0:
+36
+r e t u r n
+0
+37
+count = 0
+38
+while n :
+39
+n ˆ= n − 1
+40
+count += 1
+41
+r e t u r n
+count
+42
+43
+def
+b i t c o u n t ( n ) :
+44
+count = 0
+45
+while n > 0:
+46
+n ˆ= n − 1
+47
+count += 1
+48
+r e t u r n
+count
+49
+50
+def
+b i t c o u n t ( n ) :
+51
+r e t u r n
+bin ( n ) . count ( ’1 ’)
+52
+53
+def
+b i t c o u n t ( n ) :
+54
+count = 0
+55
+while n :
+56
+i f
+n & 1:
+57
+count += 1
+58
+n >>= 1
+59
+r e t u r n
+count
+Fig. 4: A discussion with ChatGPT about the BITCOUNT problem from QuixBugs. We inserted request and response markers
+in bold font to make the chat history more readable. In addition, we shortened ChatGPT’s responses in some places to keep
+the example as small as possible. When we left out large text passages, we marked this in the text.
+
+Consequently, we investigate for two benchmark problems
+how ChatGPT behaves in a conversation and if it is possible
+to find a working solution by discussing the problem with the
+system. We choose GCD and BITCOUNT as benchmark prob-
+lems because in our previous experiments the contained bug
+could not be fixed correctly for both problems. Furthermore,
+the problems consist of a relatively small number of code lines
+which allows us to discuss these problems in detail.
+Figure 3 shows an example discussion with ChatGPT about
+the GCD problem (lines 1–8). In the first response (lines
+10–15), ChatGPT does not present any solution. It asks for
+more information about the bug (we observed this behavior
+for many other problems, see Section III-B). Since the given
+function causes recursion issues for many possible inputs,
+we give ChatGPT an exact input example and the resulting
+error message from Python (lines 17–19). By mentioning the
+recursion issue, the final response goes in the right direction
+and we get a correctly working patched version (lines 30–34).
+In Figure 4 we see an example discussion with ChatGPT
+about the BITCOUNT problem (lines 1–9). Again, ChatGPT
+asks for more information about the problem and for an input
+that causes an error (lines 11–13). As follow-up request, we
+give ChatGPT a description of what the function should do
+(based on a code comment from QuixBugs) and ignore the
+request for an example input to see how ChatGPT reacts (lines
+15 and 16). We can see in the following answer (lines 18–25)
+that there is clearly a relation between ChatGPT’s first and
+second answer because now we get an explanation of how
+we can test the function with some test inputs. We respond
+with a problem description for a test input and describe that
+there is probably an issue with an infinite loop (lines 27 and
+28). ChatGPT responds with four code snippets where the first
+two (lines 34–48) do not solve the problem with the infinite
+loop and the last two (lines 50–59) are complete but working
+re-implementations which, however, not directly address the
+contained bug. It seems that ChatGPT simply returns functions
+here that somehow fit the content of the problem discussion,
+even though the test cases mentioned by ChatGPT show
+that the first two functions cannot work correctly. Also the
+bug is not simply fixed by replacing n ˆ= n - 1 with
+n &= n - 1 in the given function, but ChatGPT, as al-
+ready mentioned, returns two complete re-implementations.
+However, both observations are not particularly surprising for
+a language model based approach. Nevertheless, the given
+answers would be useful for a programmer as they help to
+solve the problem.
+D. Systematic Follow-up Requests for ChatGPT
+Next, we conducted a study where we systematically discuss
+with ChatGPT. For those programs for which the contained
+bug was not correctly addressed by ChatGPT (see Table I),
+we provide ChatGPT with a follow-up request giving a hint,
+as specified in Section II-C. We report our results in Table II.
+We use the same notation as before with the addition that a
+checkmark with an asterisk (*) defines that a solution was
+found without a follow-up request being necessary in this run.
+TABLE II: Results achieved by ChatGPT with additional
+information given in a follow-up request for the unsolved
+benchmark problems (see Table I).
+Benchmark problem
+ChatGPT
+bitcount
+
+depth-first-search
+*
+detect-cycle
+*
+gcd
+
+hanoi
+
+kheapsort
+
+kth
+
+lcs-length
+
+levenshtein
+
+lis
+
+longest-common-subsequence
+
+max-sublist-sum
+
+minimum-spanning-tree
+
+next-permutation
+
+powerset
+
+rpn-eval
+
+shortest-path-lengths
+
+sieve
+*
+to-base
+
+topological-ordering
+
+wrap
+
+ΣΣΣ (Solved)
+9 (12)
+For 9 benchmark problems we see that a more detailed
+description of the bug is helpful for ChatGPT. For 3 bench-
+mark problems no follow-up request was necessary in this run,
+since the bug was correctly addressed in the response given
+on our first request. Overall, adding a hint to ChatGPT vastly
+improves its performance, with 31 out of 40 problems solved.
+ChatGPT thus offers an exciting new way of approaching
+automated program repair.
+IV. THREATS TO VALIDITY
+It is worth noting that ChatGPT is currently under active
+development. During our study there was a major update to
+it, which might have influenced our results. Although we
+observed repairability rates before and after the update to be
+similar. However, future releases might yield different results.
+Furthermore, ChatGPT allows for conversation with its users.
+Asking a different question than the ones presented in this
+study could potentially have a different impact on results.
+To mitigate this threat to validity, we conducted a pre-study,
+varying the questions asked. We noted no significant influence
+on the results. Moreover, the results might vary depending
+on the programming language, size of the benchmarks, and
+
+the number of queries issued. To mitigate these threats, we
+chose a standard benchmark set and targeted Python – the
+most popular programming language.8 The classification of
+the results was done manually and therefore represents the
+subjective assessment of the authors. To enable a verification
+of our results, we made our conversations with ChatGPT
+available online.
+V. CONCLUSIONS AND FUTURE WORK
+To support programmers in finding and fixing software
+bugs, several automated program repair (APR) methods have
+been proposed. ChatGPT, a recently presented deep learning
+(DL) based dialogue system, can also make suggestions for
+improving erroneous source code. However, so far the quality
+of these suggestions has been unclear. Therefore, we compared
+in this work the automatic bug fixing performance of ChatGPT
+with that of Codex and several dedicated APR approaches.
+We find that ChatGPT has similar performance to Codex
+and dedicated DL-based APR on a standard benchmark set. It
+vastly outperforms standard APR methods (19 vs. 7 out of 40
+bugs fixed). Using ChatGPT’s dialogue option and giving the
+system more information about the bug in a follow-up request
+boosts the performance even further, giving an overall success
+rate of 77.5%. This shows that human input can be of much
+help to an automated APR system, with ChatGPT providing
+means to do so.
+Despite its great performance, the question arises whether
+the mental cost required to verify ChatGPT answers outweighs
+the advantages that ChatGPT brings. Perhaps incorporation
+of automated approaches to provide ChatGPT with hints as
+well as automated verification of its responses, e.g., through
+automated testing, would yield ChatGPT to be a viable tool
+that would help software developers in their daily tasks.
+We hope our results and observations will be helpful for
+future work with ChatGPT.
+ACKNOWLEDGMENTS
+This work was partially supported by UKRI EPSRC grant
+no. EP/P023991/1.
+REFERENCES
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+

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@@ -0,0 +1,1535 @@
+Salient Object Detection for Images Taken by People With Vision Impairments
+Jarek Reynolds*, Chandra Kanth Nagesh*, and Danna Gurari
+* denotes equal contribution
+University of Colorado Boulder
+Abstract
+Salient object detection is the task of producing a bi-
+nary mask for an image that deciphers which pixels be-
+long to the foreground object versus background. We in-
+troduce a new salient object detection dataset using images
+taken by people who are visually impaired who were seek-
+ing to better understand their surroundings, which we call
+VizWiz-SalientObject. Compared to seven existing datasets,
+VizWiz-SalientObject is the largest (i.e., 32,000 human-
+annotated images) and contains unique characteristics in-
+cluding a higher prevalence of text in the salient objects
+(i.e., in 68% of images) and salient objects that occupy a
+larger ratio of the images (i.e., on average, ∼50% cover-
+age). We benchmarked seven modern salient object detec-
+tion methods on our dataset and found they struggle most
+with images featuring salient objects that are large, have
+less complex boundaries, and lack text as well as for lower
+quality images. We invite the broader community to work on
+our new dataset challenge by publicly sharing the dataset at
+https://vizwiz.org/tasks-and-datasets/salient-object.
+1. Introduction
+Locating the most prominent foreground object in an im-
+age is a core computer vision problem, often referred to
+as salient object detection (as well as salient object seg-
+mentation and foreground object detection/segmentation)
+[8,12,32,40]. This work is motivated by the desire to have
+salient object detection models work well for images taken
+by people who are blind or with low vision1 (i.e., people
+with vision impairments). Such a feature could offer sev-
+eral benefits to this community. For example, it could con-
+tribute to privacy-preservation for photographers who rely
+on visual assistance technologies to learn about objects in
+their daily lives, using mobile phone applications such as
+Microsoft’s Seeing AI, Google Lookout, and TapTapSee.2
+1For people with low vision, solutions do not exist to correct their vi-
+sion (e.g., by wearing glasses, surgery).
+2Many companies record submitted data as evidence that potentially
+could be needed for legal reasons.
+Figure 1.
+Example images demonstrating unique features of
+our new VizWiz-SalientObject dataset when compared to other
+datasets. The salient objects commonly contain text and occupy
+a larger portion of the image (i.e., high coverage).
+All content except the foreground content of interest could
+be obfuscated, which is important since private information
+is often inadvertently captured in the background of images
+taken by these photographers [24]. Additionally, localiza-
+tion of the foreground object would empower low vision
+users to rapidly magnify content of interest and also enable
+quick inspection of smaller details [21,39].
+Many salient object detection datasets have been created
+to enable progress in algorithm development [7,8,22,42]. A
+limitation of existing datasets is they are typically built us-
+ing high-quality images collected from photo-sharing web-
+sites on the Internet. As we will show in Section 3.2, such
+images commonly lack many characteristics that can be ob-
+served in real-world settings, especially for visual media
+taken by visually impaired photographers who are trying
+to learn about the content they photograph [24], often pho-
+tographing distinct types of content such as objects showing
+text [25], and cannot verify visual quality [13].
+To fill this gap, we introduce a new salient object de-
+tection dataset based on images captured in an authentic
+use case where visually impaired photographers shared their
+images to solicit assistance in learning about the visual con-
+tent. We created this dataset by crowdsourcing the collec-
+1
+arXiv:2301.05323v1  [cs.CV]  12 Jan 2023
+
+rCableSales
+SUPPLY
+ext Present
+口
+WASYOURTRIP?
+ANTTOHEARFR
+OUR ON-LINB SUR
+pse hcwyou want to take the
+XILLL
+uter
+gli.ols.sgizmo.com/s3l
+pne:
+KUCLO LCINNO CICMO G
+gli.olsiphone.sgizmo.co
+Pad
+LOLIUILKIUtion of salient object annotations for nearly 40,000 images
+taken from the VizWiz-Captions dataset [25]. Examples
+of resulting annotated images are shown in Figure 1. Af-
+ter applying quality control filtration steps, our final dataset
+consists of 32,000 annotated images. We call our dataset
+VizWiz-SalientObject (or VizWiz-SO). We conduct a de-
+tailed analysis revealing how this new dataset relates to ex-
+isting datasets. When comparing our salient objects to the
+visual evidence needed to answer questions the photogra-
+phers asked about their images (i.e., taken from the VizWiz-
+VQA-Grounding dataset [11]), we observe that over half
+the time the necessary visual evidence is the salient ob-
+ject. When comparing our dataset to seven existing datasets,
+we observe VizWiz-SalientObject is the largest (i.e., 32,000
+human-annotated images) and is unique in its higher preva-
+lence of text in the salient objects (i.e., in 68% of images) as
+well as salient objects occupying a larger ratio of the images
+(i.e., on average, ∼50%).
+We also benchmark modern salient object detection al-
+gorithms on our new dataset to uncover open challenges
+for the research community. Experiments with seven al-
+gorithms reveal that they struggle most for images with
+salient objects that are large, have less complex bound-
+aries, and lack text as well as for lower quality images.
+To facilitate progress on these challenging problems, upon
+publication, we will publicly-share the dataset and an
+evaluation server with leaderboard at the following link:
+https://vizwiz.org/tasks-and-datasets/salient-object.
+In summary, our new dataset supports the development
+of more generalized algorithms that not only address the in-
+terests of people with vision impairments but also can ben-
+efit related applications that encounter similar real world
+challenges observed in our dataset. Relevant applications
+include robotics, lifelogging, and privacy protection.
+2. Related Work
+Salient Object Detection Datasets.
+Over the past cou-
+ple of decades, many datasets were introduced to facili-
+tate improving the design of algorithms that address salient
+object detection problems. Several survey papers provide
+comprehensive characterizations of the tens of datasets de-
+signed for this task [7, 8, 22, 42]. A common observation
+is that datasets were artificially constructed around high-
+quality images which often feature salient objects in the
+center of the images with a high contrast against the back-
+ground. This is a mismatch from many real-world settings,
+especially for visual media taken by visually impaired pho-
+tographers who often photograph distinct types of content,
+such as objects showing text [25], with the aim to learn
+about that content. We introduce the first salient object de-
+tection dataset based on images taken by visually impaired
+people in an authentic use case where they were trying to
+learn about their visual surroundings. Compared to seven
+modern datasets, our dataset is larger, has a high prevalence
+of salient objects containing textual information, and shows
+objects that occupy larger portions of the images.
+Salient Object Detection Algorithms.
+Researchers have
+designed novel algorithms to automatically perform salient
+object detection for over 20 years, with the status quo since
+2015 being that state-of-the-art methods employ neural net-
+works trained on large-scale annotated datasets.
+Several
+survey papers provide comprehensive characterizations of
+the hundreds of algorithms for this task [7,8,22,42]. While
+convolutional neural network (CNN) based models became
+the mainstream method [1, 33, 43] in 2015, transformer
+based models [30, 44] have become the mainstream ap-
+proach over the past few years. To assess how well mod-
+ern methods perform on our new dataset, we benchmark
+seven modern methods. We observe that existing methods
+fall below human performance and struggle most for salient
+objects that lack text and occupy a larger ratio of the image.
+Visual Assistance Technologies.
+Visually impaired peo-
+ple can share their visual media (images and videos) with
+various technologies [3, 4, 6, 14, 18, 27, 32, 40] in order to
+receive assistance for daily tasks such as deciding what to
+eat, wear, and buy [10,24]. The widespread impact of such
+technologies for real users is exemplified by reports from
+some of these companies that the technologies have 10s to
+100s of thousands of users who have submitted millions of
+assistance requests [5,9,14,17]. The most common reported
+goal for using such technologies is to learn about a (salient)
+object [9,10,23,28,47]. Given this common use case, salient
+object detection models could help for privacy preservation.
+Specifically, images (or video frames) could be edited be-
+fore being shared with companies, by obfuscating the back-
+ground, in order to reduce inadvertent disclosures of pri-
+vate content that often appears in the background of images
+taken by visually impaired photographers [24].
+3. VizWiz-SalientObject Dataset
+We now introduce our new salient object detection
+dataset, we call VizWiz-SalientObject (VizWiz-SO).
+3.1. Dataset Creation
+Image Source.
+We focus on images taken by visually im-
+paired people who shared them in an authentic use case
+where they were soliciting visual assistance. Specifically,
+we leverage the 39,181 labeled images from the VizWiz-
+Captions dataset, each of which is paired with five crowd-
+sourced captions [25]. Observing that images from these
+photographers can have severe quality issues resulting in no
+detectable salient object (e.g., extreme blur or inadequate
+illumination), we did not use the images which were cap-
+tioned as follows by at least four of the five crowdworkers:
+2
+
+“Quality issues are too severe to recognize visual content.”
+We also did not use the small images (i.e., both the height
+and width were less than 300 pixels) because of the chal-
+lenges of collecting precise annotations for such images.
+This left us with 37,120 images for our annotation task.
+Task Design.
+Our task interface for segmenting salient
+objects begins with a comprehensive instruction set at the
+top detailing both how to navigate the interface and how to
+complete challenging annotation scenarios. Next, it shows
+an image alongside two preliminary questions for verifying
+there is a single, unambiguous foreground object. The first
+question asks “Is the image showing a screenshot?” If the
+answer is “yes”, we conclude the image lacks a salient ob-
+ject. Next, we ask the more general, direct question of “Is
+there a single unambiguous foreground object?” An anno-
+tator is only prompted to segment the foreground object for
+images deemed by these preliminary questions to show a
+single, unambiguous foreground object.
+To demarcate the boundary of the salient object, the in-
+terface collects a series of points that are connected into
+polygon(s). When segmenting the salient object, the an-
+notator is required to remove any holes (e.g., donut) as well
+as capture all object parts when occlusions break a salient
+object into more than one polygon (e.g., hand obfuscates a
+pencil into two parts). The annotator also has an option to
+select a button indicating that the salient object occupies the
+full image. We provide more details about the task interface
+as well as a screenshot of it in the Supplementary Materials.
+Annotation Collection.
+We leveraged the benefits of
+an around-the-clock distributed workforce by crowdsourc-
+ing annotations via Amazon’s crowdsourcing marketplace,
+Amazon Mechanical Turk (AMT).
+Although AMT can support our large-scale annotation
+needs, it brings concerns about annotation quality due to the
+anonymous nature of the crowdsourced workforce. Con-
+sequently, we implemented several measures to ensure the
+collection of high-quality annotations, as summarized be-
+low. First, we restricted who were potential candidates for
+our task.
+We only accepted workers who had at least a
+98% acceptance rate while having completed at least 500
+Human Intelligence Tasks (HITs) on AMT. Moreover, to
+encourage understanding of our initial and ongoing task in-
+structions, we opted for crowdworkers only from the United
+States since that provided us confidence that they have
+English-language proficiency. In addition, we also required
+crowdworkers to pass a qualification assessment covering
+five challenging annotation scenarios documented in our in-
+structions. The qualification images feature foreground ob-
+jects consisting of complex boundaries, holes within the ob-
+ject, and occlusions obfuscating portions of the foreground
+object. Consequently, the task required crowdworkers to
+demonstrate an understanding for how to generate multi-
+ple polygons, annotate holes, handle occlusions, and draw
+complex boundaries.
+We employed 40 AMT crowdworkers who completed
+our qualification task to complete annotations of all images.
+For each of the 37,120 images, we collected two annotations
+from the crowdworkers.3 During annotation collection, we
+monitored ongoing quality by tracking each worker’s per-
+formance with respect to their frequency of indicating the
+presence of full-screen annotations or no prominent fore-
+ground object as well as the level of detail they provided in
+their segmentations (e.g., high prevalence of triangles). Cu-
+mulatively, the crowdworkers took 1,290 annotation hours
+over 11 days to complete annotating the 37,120 images.
+Annotation Post-Processing.
+We next analyzed the re-
+dundant annotations per image to determine how to use each
+annotated image in the final dataset.
+First, we removed
+3,662 images for which workers agreed there was no sin-
+gle, unambiguous salient object, which occurred when both
+annotators either answered “Yes” to “Is the image a screen-
+shot?” or “No” to “Is there a single most prominent fore-
+ground object?” Next, we manually inspected 7,443 images
+for which workers disagreed on the answers to either of the
+two preliminary questions and determined whether there is
+indeed a single, unambiguous object. Finally, with all im-
+ages deemed to have a single, unambiguous salient object,
+we determined which annotation to assign as ground truth.
+To assist in this process, we computed the intersection over
+union (IoU) score between the two segmentations for all
+images with two or more segmentations. With IoUs ≥ 0.90,
+we deemed both annotations high quality and randomly se-
+lected one as ground truth. For the remaining 2,951 images
+with IoUs< 0.90, we manually reviewed the annotations to
+decide whether one was correct or whether the image should
+be discarded due to foreground object ambiguity.
+3.2. Dataset Analysis
+We now characterize the VizWiz-SalientObject (VizWiz-
+SO) dataset and how it relates to existing datasets.
+3.2.1
+Salient Objects vs Answer Groundings for VQA
+We first explore how the target content the photographers
+were asking about relates to an image’s salient object. To
+do so, we compare the annotations of the visual evidence
+needed to answer questions about the images, i.e., an-
+swer groundings provided in the VizWiz-VQA-Grounding
+dataset [11], to the annotations of the salient objects in our
+dataset. We first identified all annotated images that were in
+3For a subset of images, we collected four annotations to support fur-
+ther analysis of human annotation performance, which we describe in the
+Supplementary Materials.
+3
+
+Figure 2. The histogram summarizes for 6,540 images the fre-
+quency of observing different levels of similarity between two
+segmentations per image, which show the salient object and the
+visual evidence needed to answer the photographer’s question re-
+spectively. These findings reveal that visually impaired photogra-
+phers often want to learn about the salient objects in their images.
+common across the two datasets, yielding a total of 6,540
+images. For each image, we then measured the similarity
+between the answer grounding and salient object segmenta-
+tions using the IoU metric. We visualize our results using a
+histogram where we categorize each image into one of ten
+interval bins starting with IoU=[0.0, 0.1), incrementing in
+intervals of 0.1, and ending with IoU=[0.9, 1.0). Results
+are shown in Figure 2.
+We observe that about half of the images have a high sim-
+ilarity between the salient object and VQA answer ground-
+ing; e.g., 46% had an IoU ≥ 0.9. This reveals that visually
+impaired photographers often are trying to learn about the
+salient object in their images when trying to get answers to
+their visual questions.
+We also observe that roughly one quarter of the images
+have a very low similarity between the salient object and
+VQA answer grounding; i.e., 25.7% of images had an IoU
+< 0.1. We manually reviewed these 1,680 images with IoUs
+less than 0.1 to understand the reasons for this finding. We
+discovered that 95% (i.e., 1,599) of these images have a
+salient object featuring a full-screen or large region while
+the VQA answer grounding captures a small aspect of the
+salient object. Examples include expiration dates on food
+packages or the current page number of an open book. The
+remaining 5% (i.e., 81) of these images featured a VQA an-
+swer grounding unrelated to the salient object.
+More generally, we observe that the IoU scores follow a
+U-shaped distribution with only a small portion of images
+having middling scores; e.g., 7.9% (i.e., 511) of images had
+an IoU ≥ 0.3 and < 0.7. Among these images, we found the
+salient object contained the VQA answer grounding region
+100% of the time. There are two primary trends that led to
+these less common IoU scores. The first trend is that larger
+VQA answer grounding regions occur with smaller salient
+objects. Examples include brands of cereal, types of soda,
+and denominations of currency. The second trend was for
+salient objects featuring holes. That is because the VizWiz-
+VQA-Grounding dataset did not account for holes in their
+annotation task. The absence of annotated holes in only one
+of the two segmentations led to lower IoU scores.
+Altogether, these findings highlight that a valuable step
+for tackling many of this population’s VQA goals is to ini-
+tially locate the salient object. That is because the answer
+will likely only be grounded in the salient object or the
+background rather than their intersection.
+3.2.2
+VizWiz-SO vs Existing Datasets
+We next compare our dataset to seven datasets:
+• DUTS [41]: the most commonly used dataset to train
+state-of-the-art algorithms (e.g., [1,30,33,38,43,44]) due
+to its large size paired with diverse saliency challenges.
+• DUT-OMRON [46]: consist of images showing multiple
+salient objects, often with complex backgrounds. This is
+a useful reference when considering extending our dataset
+to when photographs taken by visually impaired photog-
+raphers showing multiple salient objects. We share our
+collected metadata indicating when this occurs to facili-
+tate this line of future research.
+• ECSSD [45]: consists of images featuring complex scenes
+that present textures and structures expected to be com-
+mon in real-world salient object detection scenarios.
+• PASCAL-S [29]: derived from PASCAL VOC’s [16] val-
+idation set, it is designed to facilitate salient object seg-
+mentation generalization on realistic images.
+• HRSOD [48]: explicitly designed for salient object de-
+tection on high-resolution images; this is relevant for our
+real-world application since images taken by people with
+vision impairments often are relatively high resolution.
+• UHRSD [44]: currently the largest ultra-high resolution
+salient object detection dataset, which is relevant to our
+work since images taken by people with vision impair-
+ments can be ultra high resolution.
+• DAVIS-S [48]: derived from DAVIS [36], a densely an-
+notated video segmentation dataset. This is relevant for
+our real-world application to analyze implications for
+video frames since visually impaired photographers often
+stream live video with their cameras when using visual
+assistance technologies [4,18].
+Of note, images in six of these datasets originate from the
+Internet on photo-sharing websites such as Flickr [29, 41,
+44–46, 48], and so likely are high quality since they were
+deemed of sufficient quality to upload to the Internet.4
+4The origins of the images for the final dataset is not reported [48].
+4
+
+50%
+46.0%
+3.40%
+Image
+30%
+25.7%
+20%
+9.0%
+10%
+4.4%
+4.8%
+2.4%
+2.1%
+1.7%
+1.7%
+2.2%
+[0.8, 0.9]
+[0.9.
+[O.
+[o.
+0
+4.
+0.6)
+,1.0)
+0.3)
+IoU SimilarityDAVIS-S [48]
+PASCAL-S [29]
+HR [48]
+ECSSD [45]
+DUT-O [46]
+UH [44]
+DUTS [41]
+Ours
+Images
+92
+850
+2,010
+1,000
+5,168
+5,920
+15,572
+32,000
+Text
+13%
+24%
+15%
+15%
+11%
+19%
+13%
+68%
+MR
+22%
+31%
+25%
+9%
+17%
+35%
+19%
+1%
+Holes
+82%
+50%
+62%
+29%
+28%
+75%
+41%
+4%
+Table 1. Characterization of our VizWiz-SO dataset and seven existing salient object detection datasets with respect to how many images
+are included (“Images”), the percentage of images that have text present in the salient objects (“Text”), the percentage of images that have
+salient objects consisting of more than one region (“MR”), and the percentage of images that have salient objects containing any holes
+(“Holes”). As shown, our dataset is distinct in that it contains more images, more salient objects with text present, more salient objects
+consisting of one region, and fewer salient objects containing holes. (HR=HRSOD; UH=UHRSD)
+Figure 3. Summary statistics for ours and seven other datasets with respect to four measures. Each box reveals statistics about all salient
+objects in a particular dataset with the central mark capturing the median value, box edges the 25th and 75th percentiles values, whiskers
+the most extreme data points not considered outliers, and individually plotted points the outliers. Our dataset is unique in that salient objects
+tend to have less complex boundaries, occupy larger portions of an image, and exhibit a greater diversity of sizes relative to the image.
+For each salient object in every dataset, we characterize
+it in six ways. Three measures focus on detecting the pres-
+ence versus absence of particular properties for the salient
+object. These are whether the salient object contains text 5,
+consists of multiple regions 6, or contains any hole(s). The
+remaining three measures characterize the salient region it-
+self. First, we identify the position of an object within an
+image by measuring its center of mass relative to the im-
+age coordinates, resulting in x and y coordinate values in
+the range between 0 to 1. Next, we characterize the ob-
+ject’s boundary complexity by computing its isoperimetric
+inequality, which is the ratio of the object’s area to the
+length of its perimeter.
+Values range from 0 to 1, with
+larger values indicating simpler boundaries that are less
+jagged/dented (e.g., a circle). Finally, to gauge the relative
+size of a salient object in the image, we compute its cover-
+age ratio, meaning the fraction of all image pixels that are
+occupied by the salient object’s pixels.
+We show summative statistics of our findings per dataset
+in Table 1 and Figure 3. In particular, in Table 1, we re-
+5We obfuscate all image content but the salient object and then check
+whether Microsoft Azure’s OCR API returns text.
+6Multiple regions means there are multiple separate polygons. This can
+occur either because multiple salient objects were annotated or because of
+occlusions that lead to more than one region for a single salient object.
+port how many images are in each dataset paired with what
+percentage of those images have salient objects with text,
+multiple regions, and holes. In Figure 3, we visualize statis-
+tics summarizing the values for each dataset’s salient ob-
+jects with respect to center of mass, boundary complexity,
+and coverage ratio using boxplots.
+While our findings highlight that our VizWiz-SO dataset
+has many distinct characteristics, one commonality it has
+with most existing salient object detection datasets is that
+the salient objects typically occupy centered positions
+within an image. Specifically, in Figure 3, we observe this
+trend for all datasets except HRSOD. We found this some-
+what surprising since visually impaired photographers can-
+not visually inspect their images to verify they are conform-
+ing to the common photographer’s bias of centering con-
+tents of interest they are trying to photograph. Yet, given
+our findings from Section 3.2.1 that photographers often are
+interested in learning about an image’s salient object, our
+findings suggest these photographers have skills in center-
+ing contents of interest in pictures they take.
+A unique aspect of our VizWiz-SO dataset is that it fea-
+tures more salient objects with textual data. Specifically,
+68% of salient objects in VizWiz-SO contain text while the
+dataset with the next highest prevalence of text, PASCAL-
+S [29], only has it for 24% of the images (Table 1). A gap of
+5
+
+Center of mass Y-axis
+Center of mass X-axis
+ Boundary Complexity
+Coverage Ratio
+1.0
+0.8
+0.6
+0.4
+0.2
+0.0
+DAVIS-S
+PASCAL-S
+HRSOD
+ECSSD
+DUT-OMRON
+UHRSD
+DUTS
+Ours: VizWiz-SOthis magnitude (i.e., 44 percentage points) suggests that our
+new dataset offers a considerable domain shift in the salient
+object detection problem space. We suspect part of this shift
+stems from the types of salient objects included, with many
+more daily objects such as products (e.g., food packages)
+included in our VizWiz-SO dataset.
+Another unique aspect of VizWiz-SO is that far fewer
+images feature salient objects that consist of multiple re-
+gions; i.e., only 1% of images (Table 1).
+We suspect
+this distinction stems from our unique approach of adopt-
+ing a rigorous annotation preprocessing step, where we re-
+quire crowdworkers to verify images have one unambigu-
+ous salient object before allowing them to annotate images
+for use in our final dataset. Any remaining objects in our
+dataset with multiple regions are therefore highly likely a
+result of occlusions breaking a single salient object into
+multiple polygons, which evidently is incredibly rare.
+VizWiz-SO is also unique due to the rarity in which
+salient objects contain holes; i.e., only observed for 4%
+of images (Table 1). From visual inspection, we suspect
+this finding reflects a domain shift in the types of content
+found in the datasets. For example, examples from other
+datasets of objects with holes include people riding bikes,
+people dancing, and animals in intricate poses.
+In con-
+trast, in VizWiz-SO, objects with holes include retail pack-
+aging made to hang from hooks, pairs of scissors, and coffee
+mugs. We posit the lower prevalence of holes in VizWiz-SO
+stems from the fact that images originate from an authentic
+use case where photographers primarily photograph house-
+hold and retail items, which naturally feature fewer holes.
+A further distinction of our VizWiz-SO dataset is that the
+salient objects tend to have less complex boundaries (Fig-
+ure 3). We suspect this is again because of a domain shift in
+the types of objects in our dataset, with many more human-
+made items, such as food packaging boxes and cans, that by
+design are typically more structured in shape.
+A final distinction of salient objects in our VizWiz-SO
+is how much of the image they occupy (Figure 3). First,
+they tend to occupy a much larger amount of the image than
+observed in other datasets. Specifically, they on average oc-
+cupy roughly half of all image pixels, with a mean coverage
+ratio of 0.5 and a median of 0.46. In contrast, the dataset
+with the next highest coverage ratio statistics is PASCAL-
+S [29], and over 75% of its images contain salient objects
+that occupy less than half of the image pixels. We attribute
+this distinction to the authentic use case of our dataset,
+where visually impaired photographers attempting to learn
+about the salient objects they are photographing seem to be
+taking zoomed-in or close-to-camera images of the content
+of interest. Another unique aspect of our salient objects, is
+that they exhibit a larger range of sizes, as shown by the
+gaps between the 25 and 75 percentile values of each box.
+For example, PASCAL-S features the next largest interquar-
+tile range with a 23% gap(i.e., 19% to 42%). In contrast,
+the gap for VizWiz-SO is more than twice as large at 56%
+(i.e., 22% to 78%). Consequently, a unique challenge of
+our dataset for algorithms is that they no longer can assume
+a strong bias regarding a salient object’s relative size.
+4. Algorithm Benchmarking
+We benchmark modern salient object detection algo-
+rithms to show how they perform on our new dataset. We
+conducted all experiments on a Nvidia A100 GPU.
+4.1. Experimental Design
+Dataset Splits.
+We use the existing splits available for the
+VizWiz-Captions dataset [25], which translates to approxi-
+mately a 60/20/20 training, validation and test split for our
+VizWiz-SO dataset. In particular, from the 32,000 anno-
+tated images, the number of images in each split respec-
+tively is 19,116, 6,105, and 6,779.
+Evaluation Metrics.
+We evaluate each model with re-
+spect to five popular metrics for salient object detection
+models: Mean Absolute Error (MAE), Structure Measure
+(Sm), Mean F-Measure (Fm), Enhanced Alignment Mea-
+sure (Em), and Intersection over Union (IoU).
+Algorithms.
+We benchmark the following seven methods
+from the past three years to assess the difficulty of our new
+dataset for modern salient object detection models:
+• Boundary Aware Segmentation Network (BASNet) [38]:
+an appealing model for real-time applications like our tar-
+get use case because it can achieve 70fps during inference
+time while achieving competitive performance (i.e., was
+a top-performer in 2019).
+• Fusion, Feedback and Focus Network (F3Net) [43]: state-
+of-the-art performing model on five datasets in 2020.
+• U2 Network (U2Net) [1]: an appealing model for real-
+world applications like our target use case because it has
+a very light footprint (4.7MB), and so is more suitable
+for resource-constrained devices such as smartphones. It
+achieved competitive performance in 2020.
+• Visual Saliency Transformer (VST) [30]: achieved state-
+of-the-art performance in 2021, and is based purely on a
+transformer architecture.
+• Pyramidal Feature Shrinking Network (PFSNet) [33]:
+achieved state-of-the-art performance on five datasets in
+2021; it consists of a decoder that aims at using aggre-
+gated adjacent feature nodes hierarchically to avoid the
+problem of leaping feature fusion.
+• Pyramid Grafting Network (PGNet) [44]: introduced in
+2022, it is a one-stage framework based on a transformer
+6
+
+HP
+BASNet
+F3Net
+U2Net
+VST
+PFSNet
+PGNet
+DIS
+VST-FT
+VST-S
+[38]
+[43]
+[1]
+[30]
+[33]
+[44]
+[37]
+Attr.
+Backbone
+-
+R-34
+R-50
+-
+T2T-ViT
+R-50
+R-18+SWIN
+U2Net
+VST
+ViT
+Training set
+-
+D
+D
+D
+D
+D
+D+HR
+DIS5K
+D+VW
+VW
+Input size
+-
+2562
+3522
+3202
+2242
+3522
+2242, 10242
+10242
+2242
+2242
+Size (MB)
+-
+333
+98
+4.7
+171
+120
+280
+169
+171
+171
+VizWiz-SO
+MAE ↓
+0.02
+0.28
+0.28
+0.26
+0.17
+0.32
+0.21
+0.36
+0.19
+0.21
+Sm ↑
+0.92
+0.59
+0.55
+0.61
+0.65
+0.48
+0.62
+0.46
+0.64
+0.63
+Fm ↑
+0.96
+0.77
+0.74
+0.80
+0.83
+0.70
+0.79
+0.61
+0.74
+0.72
+Em ↑
+0.97
+0.64
+0.65
+0.65
+0.76
+0.60
+0.74
+0.55
+0.77
+0.70
+IoU ↑
+0.94
+0.62
+0.53
+0.63
+0.73
+0.48
+0.67
+0.49
+0.70
+0.69
+Table 2. Analysis of existing algorithms that we benchmark on our VizWiz-SO dataset, including both off-the-shelf models (which are
+cited) as well as those fine-tuned (-FT) and trained from scratch (-S). We first report differentiating attributes between the algorithm
+architectures and then present the model performance with respect to five widely-used metrics. (HP=Human Performance; R=ResNet [26];
+ViT=Vision Transformer [15]; Swin=Shifted window transformer [31]; D=DUTS-TR [41]; VW=VizWiz-SO; HR=HRSOD [48])
+and CNN backbone that achieves state-of-the-art perfor-
+mance on five benchmark datasets. [41,44,46,48].
+• Dichotomous Image Segmentation (DIS) [37]: also in-
+troduced in 2022 as the state-of-the-art method for the
+DIS5K [37] dataset; it is designed for detecting salient
+object in high resolution images, which makes it relevant
+for our use case where many images coming from people
+with vision impairments are relatively high resolution.
+We further characterize each model by identifying the
+backbone architecture used in the architecture, datasets used
+for training, image size used for training, and model foot-
+print. These characteristics are reported in Table 2.
+All models predict saliency maps that represent the
+brightness of certain pixels within the same spatial reso-
+lution as the input image; e.g., ∈ [0, 1] or alternatively
+∈ [0, 255]. The predictions generated by salient object de-
+tection models are converted into binary masks.
+Humans.
+We also evaluate human performance to estab-
+lish an upper bound for what we should strive for from au-
+tomated methods. Since, we get two human annotations per
+image in our dataset, we calculate human performance by
+comparing the two annotations in cases where the IoU is
+greater than 0.90.
+4.2. Performance for Off-The-Shelf Models
+We first evaluate each of the algorithms as is in their orig-
+inal design. Results are shown in Table 2.
+We observe that VST [30] is the top-performing model.
+Yet, it still falls short of human performance. For exam-
+ple, the gap in performance is 0.15 in terms of MAE, 0.211
+in terms of IoU, 0.26 for Sm, and 0.2 for Em.
+Conse-
+quently, this dataset offers a new challenging benchmark
+for the community.
+A further observation is that the models perform poorly
+on the VizWiz-SO dataset in comparison to their perfor-
+mance on the original datasets for which they were bench-
+marked. For example the MAE and Sm performance ob-
+served by PGNet [44] on DUTS-TE is 0.028 and 0.912 re-
+spectively versus 0.2123 and 0.6233 respectively for our
+dataset. We hypothesize that part of the reason for this poor
+performance is that models trained and evaluated on other
+datasets are not able to learn how to generalize to the real-
+world challenges that arise for images taken by visually im-
+paired photographers.
+4.3. Performance When Training on VizWiz-SO
+We next explore whether training the top-performing al-
+gorithm, VST [30], on our new dataset will lead to improved
+performance. To do so we analyze two additional models:
+(1) the pretrained VST [30] model fine-tuned on VizWiz-
+SO (VST-FT) and (2) the pretrained VST [30] algorithm
+trained from scratch on VizWiz-SO (VST-S). We use the
+default hyperparameters reported in the VST [30] paper for
+model training. Results are shown in Table 2.
+We observe that both models, i.e., created by training
+from scratch and fine-tuning on our VizWiz-SO dataset,
+achieve worse results than the baseline of not training the al-
+gorithm on our dataset. This suggests that the training data
+used by algorithms is not the only culprit for what makes
+our new dataset challenging. Rather, our findings suggest
+that new algorithmic frameworks are also needed to achieve
+strong generalization performance on our new dataset.
+4.4. Fine-grained Analysis
+We next conduct fine-grained analysis to better isolate
+what makes our dataset challenging for modern algorithms.
+To do so, we divide our VizWiz-SO test set according to the
+following four factors, with the first three based on metadata
+collected in Section 3.2 to characterize our dataset:
+7
+
+BASNet
+F3Net
+U2Net
+VST
+PFSNet
+PGNet
+DIS
+VST-FT
+VST-S
+[38]
+[43]
+[1]
+[30]
+[33]
+[44]
+[37]
+Text Presence
+True
+0.23
+0.22
+0.22
+0.13
+0.25
+0.16
+0.32
+0.16
+0.17
+False
+0.35
+0.38
+0.32
+0.24
+0.42
+0.29
+0.40
+0.24
+0.26
+Coverage
+Small
+0.06
+0.16
+0.07
+0.11
+0.16
+0.12
+0.10
+0.09
+0.11
+Medium
+0.15
+0.20
+0.15
+0.09
+0.24
+0.15
+0.25
+0.09
+0.10
+Large
+0.60
+0.47
+0.54
+0.30
+0.54
+0.35
+0.70
+0.38
+0.39
+Complexity
+High
+0.15
+0.21
+0.15
+0.12
+0.24
+0.16
+0.21
+0.11
+0.12
+Low
+0.38
+0.34
+0.35
+0.21
+0.38
+0.25
+0.48
+0.26
+0.27
+Image Quality
+Good
+0.22
+0.23
+0.21
+0.14
+0.26
+0.17
+0.30
+0.16
+0.17
+Poor
+0.44
+0.43
+0.41
+0.27
+0.47
+0.34
+0.50
+0.30
+0.31
+Table 3. Fine-grained analysis of existing algorithms with respect to presence of text on the salient object (“Text Presence”), relative size
+of the salient object in the image (“Coverage”), relative complexity of the salient object’s boundary (“Complexity”), and image quality
+(“Image quality”). As shown the algorithms perform worse when there is salient objects lack text, occupy a large portion of the image,
+have less complex boundarys as well as when the image quality is poor.
+• Text Presence:
+two groups based on whether text is
+present in the salient object.
+• Coverage Ratio (Coverage): three groups based on the
+33rd and 66th quartile values in our dataset. All images
+with coverage ratio less than 0.32 has small coverage, be-
+tween 0.32 and 0.62 has medium coverage, and greater
+than 0.62 has large coverage.
+• Boundary Complexity (Complexity): two groups by split-
+ting them around the mean score for boundary complex-
+ity (i.e., 0.66) with high boundary complexity when the
+score is less than the mean and low boundary complexity
+otherwise.
+• Image Quality: leveraging metadata from prior work [25],
+which indicates how many of the five crowdworkers in-
+dicated an image as insufficient quality to recognize the
+content, we split the images into groups with good qual-
+ity being when none of the crowdworkers indicate insuf-
+ficient quality and poor otherwise.
+Due to space constraints, we only report results in the main
+paper with respect to the Mean Absolute Error [35]. Results
+for all benchmarked models are shown in Table 3.
+In terms of text presence, we see that the models perform
+better when there is text present as opposed to when there
+is none. For example, the performance drops by 0.11 for
+the best performing model, VST. We suspect visual patterns
+that arise with text may serve as a valuable cue to models in
+locating salient objects.
+Next, we see that as the coverage ratio of the salient ob-
+jects increase, the models tend to perform worse. For in-
+stance, the best performing model, VST, has a performance
+dropoff of 0.19 when predicting images with small cover-
+age ratios as opposed to large coverage ratios. We see an
+even greater performance dropoff from other models such
+as 0.60 for DIS. We suspect this performance gap arises in
+part from the fact that existing datasets largely lack such
+large salient objects, which both could have affected what
+algorithms were designed to handle as well what they could
+learn from the data they observed.
+Further observed trends are that performance drops for
+salient objects with lower boundary complexity and for
+poorer quality images. These are two additional factors that
+reflect domain shifts between our dataset and prior datasets
+that could have affected the design of algorithms as well
+what they could learn from the data training data.
+5. Conclusions
+We introduce the VizWiz-SalientObject dataset to en-
+courage the community to design more generalized salient
+object detection models that can handle a larger range of
+challenges motivated by our authentic use case that also can
+occur in many real-world applications. We offer our exper-
+imental findings from benchmarking modern salient object
+detection algorithms as a valuable starting point for iden-
+tifying valuable future research directions. To summarize,
+new models are needed to better handle salient objects that
+are large, have less complex boundaries, and lack text as
+well as work well in the presence of lower quality images.
+We now close with a discussion of some ethical impli-
+cations of our work. While we are motivated to better as-
+sist a population that is traditionally marginalized in society,
+we acknowledge our work can lead to potentially adverse
+social effects. Our concern is primarily centered on bad-
+actor behaviors intended to exploit the privacy, autonomy,
+and livelihoods of a population demographic inherently sus-
+ceptible to such behavior. Bad actors could use our work to
+deceive visually impaired individuals in harmful ways, such
+as through fraud, scams, and other deceptive practices, by
+for example intercepting their visual media and replacing
+automatically detected salient objects with misinformation.
+8
+
+Acknowledgments. This project was supported in part by
+a National Science Foundation SaTC award (#2148080)
+and Amazon Mechanical Turk. We thank Leah Findlater
+and Yang Wang for contributing to this research idea.
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+Appendix
+This document supplements the main paper with additional
+information concerning:
+1. Dataset Creation (supplements Section 3.1)
+• Annotation Task Interface
+• Worker Qualification Task
+• Analysis of Workers’ Annotation Differences
+2. Dataset Analysis: VizWiz-SO vs Existing Datasets
+(supplements Section 3.2.2)
+3. Experimental Design (supplements Section 4.1)
+A. Dataset Creation
+A.1. Annotation Task Interface
+The task interface displays five images within a tabbed
+container on the left and preliminary questions with task
+instructions on the right. A screenshot of the task interface
+(without instructions) is shown in Figure 4.
+To account for occlusions and holes while keeping the
+task simple for annotators, we permitted annotators to gen-
+erate multiple polygons. For occlusions, annotators could
+use as many polygons as necessary for demarcating fore-
+ground objects partitioned into multiple polygons.
+For
+holes, we apply an even-odd fill rule to images featuring
+foreground objects with holes. With an even-odd fill rule,
+every area inside an even number of enclosed areas be-
+comes hollow, and every region inside an odd number of
+enclosed areas becomes filled [34]. By treating the image’s
+four corners as the first enclosed area, the outermost bound-
+ary of the foreground object becomes the second enclosed
+area. Moreover, holes within foreground objects represent
+the third layer of enclosed areas and become filled, allowing
+annotators to demarcate foreground objects featuring holes.
+In practice, annotators first trace the outermost boundary of
+the foreground object and close the path by clicking the first
+point a second time. We then instructed annotators to trace
+any holes within the foreground object, and so those holes
+end up in odd-numbered layers.
+A.2. Worker Qualification Task
+We administered a qualification task for workers to sup-
+port our collection of high-quality ground truth annotations.
+The qualification task required annotating five images, each
+of which features a distinct challenging annotation scenario.
+All five images are shown in Figure 5. The first two images
+show a table and a bench, offering examples with complex
+boundaries and holes. The next two images feature a per-
+son holding a coffee mug, to support educating a crowd-
+worker about our expectations for annotating objects with
+10
+
+Figure 4. A screenshot of our annotation task interface.
+Figure 5. The five images used for the worker qualification task.
+Each was selected to demonstrate a challenging annotation sce-
+nario such as complex boundaries, holes, and occlusions.
+complex geometries that have many curves and occlusions
+that require annotating multiple polygons. The final image
+is a spatula. This task verified a crowdworker’s ability to
+correctly identify and annotate multiple holes that can arise
+within the salient object.
+After crowdworkers annotated each qualification image,
+the backend code of our website checked if their annotation
+was sufficiently similar to the GT annotation (i.e., IoU sim-
+ilarity of at least 0.90). Crowdworkers could only proceed
+to the following image after they obtained an IoU ≥ 0.90
+on the current image. Crowdworkers obtaining an IoU ≥
+0.90 on all five qualification assessment images on a per-
+image basis gave us substantial confidence that they would
+be able to successfully handle complex and challenging out-
+lier cases within the original VizWiz Dataset.7
+A.3. Analysis of Workers’ Annotation Differences
+We collected a larger number of redundant annotations
+per image for a random subset of images to better explore
+when and why annotation differences are observed from dif-
+ferent workers. Specifically, for this analysis, we collected
+four annotations as opposed to two for a subset of 1,237 im-
+ages. Examples of the redundant annotations collected per
+image are shown in Figure 6.
+The first example (i.e., row 1 of Figure 6) highlights that
+annotation differences can stem from challenging annota-
+tion scenarios where objects contain holes (e.g., in mug han-
+dle) or are occluded (e.g., by the straw). For instance, the
+hole was not annotated in the third annotation. Addition-
+ally, only the fourth annotation captured the occlusion that
+arises from the straw.
+The second example (i.e., row 2 of Figure 6) highlights
+that annotation differences can stem from ambiguity regard-
+7Some crowdworkers did not pass the qualification assessment due to
+time constraints. In these cases, crowdworkers would contact us with the
+images they annotated. If we were confident in their annotation abilities,
+we manually added these crowdworkers to the qualified worker pool.
+11
+
+Image 1
+Image 2
+Image 3
+Image 4
+Image 5
+Work may be rejected for not following instructions
+Step 1: Is the image showing a screenshot?
+O Yes
+O No
+WASYOURTRIP?
+Step 2: Is there a single prominent foreground object?
+OYes
+ONO
+ANTTOHEARFRO
+OUR ON-LINE SURVEYTONACHANCETO WIN SU
+Step 3: Demarcate the prominent foreground object
+osehcwyou wantto takethesunvey
+typethatweb address inyourbrowse
+puter
+one:
+gli.ols.sgizmo.com/s3/
+Prev Image
+Next Image
+Pad
+gli.olsiphone.sgizmo.com/s3
+gli.olsipad.sgizmo.com/s3/
+Select Full
+Undo Last
+Clear All
+Image
+Point
+PolygonsFigure 6. Example annotations from our random subset where we
+collected four annotations as opposed to two. We find worker dif-
+ferences primarily occur in challenging annotation scenarios such
+as holes, occlusions, complex boundaries, and object saliency.
+ing what is the salient object. As shown, the first two an-
+notations flag the image as lacking a foreground object,
+the third annotation identifies the child holding the cup as
+the salient object, and the fourth annotation identified the
+child’s cup as the salient object.
+The third example (i.e., in row 3 of Figure 6) highlights
+that annotation differences also can arise for objects that
+simultaneously have complex boundaries and holes. In an-
+notation one, the worker did not fully annotate the salient
+object, cutting out part of the object from the annotation.
+Only the third and fourth annotations accurately annotate
+all holes that are present in the salient object’s boundary
+while also having tight boundaries in the annotation.
+In summary, we found occlusions, holes, and saliency
+ambiguity to be the primary factors contributing to annota-
+tion differences. In the case of occlusions, worker differ-
+ences can arise when deciding whether to include objects
+that are a composite part of the salient object. In the case
+of holes, annotation differences can arise regarding which
+holes to annotate. Last, we found that it can be ambiguous
+as to which object is the most salient.
+Figure 7. Example ground truth annotations from the HRSOD
+dataset which exemplify that salient objects are not usually not
+centered in the image. This is a common trend in the dataset.
+B. Dataset Analysis
+B.1. VizWiz-SO vs Existing Datasets
+We present finer-grained details about typical image res-
+olutions for the different salient object detection datasets
+to expand upon discussions in the main paper about how
+VizWiz-SO relates to other datasets. Specifically, we report
+the median image width (Med. W), median image height
+(Med. H), and whether the dataset supports high resolu-
+tion images (High Res.) as defined by whether the median
+image height and width are greater than 1080 and 1920 re-
+spectively. Results are reported in Table 4. We observe that
+our new dataset, overall, provides higher resolution images
+than most datasets.
+We also expand on a surprising finding reported in our
+main paper that the HRSOD dataset is the only one for
+which salient objects do not occupy the typical center po-
+sitions. To do so, we visualize the ground truth masks of
+some non-centered objects in Figure 7. In row one, we see
+that objects are horizontally distributed to left and right po-
+sitions of the images. Similarly, we observe in row two that
+the salient objects are vertically distributed to the top and
+bottom positions of the images.
+C. Algorithmic Benchmarking
+C.1. Experimental Design
+We compute the five metrics used in the benchmarking
+section using the following definitions:
+Mean Absolute Error [35] represents the average abso-
+lute difference between the predicted saliency map and its
+12
+
+Annotation 1
+Annotation 2
+Annotation 3
+Annotation 4
+WRKDAVIS-S [48]
+PASCAL-S [29]
+HR [48]
+ECSSD [45]
+DUT-O [46]
+UH [44]
+DUTS [41]
+Ours
+Med. W
+1080
+375
+2704
+300
+300
+3612
+300
+1296
+Med. H
+1920
+500
+3264
+400
+400
+5000
+400
+968
+High Res.
+
+
+
+
+
+
+
+
+Table 4. Characterization of our VizWiz-SO dataset and seven existing salient object detection datasets with respect to metrics showcasing
+the image resolution. This includes median image width (“Med. W”), median image height (“Med. H”), and flag indicating if high
+resolution (“High Res.”). (HR=HRSOD; UH=UHRSD)
+ground truth per pixel. It can be given as:
+MAE =
+1
+H ∗ W
+H
+�
+r=1
+W
+�
+c=1
+|pred(r, c) − gt(r, c)|
+(1)
+where pred represents the predicted saliency map, gt repre-
+sents the ground truth, (H, W) represents the height and
+width of the image, and (r, c) represents the pixel co-
+ordinates for the given image.
+Structure Measure [19] is used to measure the similarity
+between the predicted saliency map and the ground truth.
+Since, we convert both the predictions and ground truths
+into the [0, 1] range, we apply the formula directly to the
+predictions and maps. It can defined as follows:
+Sm = (1 − α)Sr + αSo
+(2)
+where, Sr is defined as the region aware similarity score, So
+is defined as the object aware similarity score, and α repre-
+sents the weight that is used to sum up the values. We set
+α = 0.5, therefore making sure that we see equal contribu-
+tion from both region and object aware scores.
+F-Measure [2] represents the precision and recall ratio
+for the given prediction. It can be represented as:
+Fm = (1 + β2) ∗ Precision ∗ Recall
+β2 ∗ Precision + Recall
+(3)
+Here precision =
+T P
+T P +F P and recall =
+T P
+T P +F N on the
+entire prediction image by pixels. We set β2 = 0.3 and re-
+port the average of all F-measures as Fm similar to previous
+works.
+Enhanced Alignment Measure [20] is used as the met-
+ric to measure the effectiveness of the saliency prediction
+against the ground truth. It captures the pixel-level match-
+ing information and image-level statistics into one single
+metric by the means of an enhanced alignment matrix φ. It
+is defined as follows:
+Em =
+1
+H ∗ W
+H
+�
+r=1
+W
+�
+c=1
+φF M(r, c)
+(4)
+where, φF M represents the enhanced alignment matrix for
+the foreground map, (H, W) represents the height and
+width of the image, and (r, c) represents the pixel co-
+ordinates for the given image.
+Intersection over Union also known as Jaccard Index is
+used to determine the similarity between sample sets. In
+this case it captures the overlap between the ground truth
+and prediction map of the salient object. We convert the
+predictions in binary map and compute the Jaccard Index
+over two classes. It can be defined as follows:
+IoU = J(A, B) = |A ∩ B|
+|A ∪ B|
+(5)
+where, A and B are images of same size, consisting of inte-
+ger class values {0, 1}.
+We further show how the models performed on VizWiz-
+SO with qualitative examples shown in Figure 8. These ex-
+amples feature a variety of challenges we observed for the
+models, such as a large salient object, less complex bound-
+aries, lack of text on the salient object, and lower qual-
+ity images. For example, we observe how the models fail
+to perform adequately in identifying larger salient objects
+(rows 4 and 5). We also observe the models perform bet-
+ter when salient objects contain text (rows 1 and 2) versus
+lack text (rows 5 and 6). Further, we see models perform
+worse for images that are lower quality (rows 3, 4, and 5).
+Our fine-grained analysis in the main paper suggests each
+of these factors offer unique challenges for modern salient
+object detection models.
+13
+
+Figure 8. Examples of difficult images present in VizWiz-SO, with characteristics such as high coverage ratio, presence of text, less
+complex boundaries, and lower image quality. We show how the seven models perform on these cases as compared to the human annotation
+(GT=Ground Truth). We see that models such as PFSNet [33], DIS [37], and F3Net [43] do not always give us the correct salient objects
+or sometime no predictions at all. We also notice that VST [30] usually predicts salient objects with better accuracy compared to other
+models, but also suffer from not detecting the correct salient object.
+14
+
+Image
+GT
+BASNet
+F3Net
+U2Net
+VST
+PFSNet
+PGNet
+DIS

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+TRAINING WITH MIXED-PRECISION FLOATING-POINT ASSIGNMENTS
+Wonyeol Lee 1 Rahul Sharma 2 Alex Aiken 1
+ABSTRACT
+When training deep neural networks, keeping all tensors in high precision (e.g., 32-bit or even 16-bit floats)
+is often wasteful. However, keeping all tensors in low precision (e.g., 8-bit floats) can lead to unacceptable
+accuracy loss. Hence, it is important to use a precision assignment—a mapping from all tensors (arising in
+training) to precision levels (high or low)—that keeps most of the tensors in low precision and leads to sufficiently
+accurate models. We provide a technique that explores this memory-accuracy tradeoff by generating precision
+assignments that (i) use less memory and (ii) lead to more accurate models at the same time, compared to the
+precision assignments considered by prior work in low-precision floating-point training. Our method typically
+provides > 2× memory reduction over a baseline precision assignment while preserving training accuracy, and
+gives further reductions by trading off accuracy. Compared to other baselines which sometimes cause training
+to diverge, our method provides similar or better memory reduction while avoiding divergence.
+1
+INTRODUCTION
+In deep neural network training, floating-point formats are
+usually used to represent tensors and it is worthwhile to use
+the smallest bitwidth format that gives acceptable results.
+For example, it is common to replace tensors using 32-bit
+floats with tensors that use 16-bit floats (Micikevicius et al.,
+2018; Kalamkar et al., 2019). The benefits are easy to under-
+stand: computations using lower-precision floats not only
+use less memory but are also faster (due to improved vec-
+tor parallelism, locality, and reduced data movement). The
+downside is that there is generally some loss of training accu-
+racy, and in the worst case training may not even converge.
+For such low-precision floating-point training, the most
+common approaches use two floating-point formats—one
+for lower-precision floats (e.g., 8-bit floats) and the other
+for higher-precision floats (e.g., 16-bit floats)—and assign
+one of the two formats to each tensor (including weights,
+activations, and their gradients). The precision assignments
+studied in previous work fall into one of two assignment
+schemes (which both have several variants): the uniform
+assignment uses low precision for almost all tensors
+(often excepting those in the first and/or last few layers)
+(Micikevicius et al., 2018), while the operator-based
+assignment limits low precision to the input tensors of
+certain operators (e.g., convolutions) (Sun et al., 2019).
+Prior work has shown that both precision assignment
+schemes (with well-chosen low-bitwidth floating-point
+formats) can match the accuracy of 32-bit-float training
+1Stanford University, USA
+2Microsoft Research, India.
+Correspondence to: Wonyeol Lee <[email protected]>.
+Preprint. Under review.
+(Micikevicius et al., 2018; Kalamkar et al., 2019; Wang
+et al., 2018; Sun et al., 2019; Chmiel et al., 2021; Drumond
+et al., 2018; Cambier et al., 2020; Fox et al., 2021).
+There is an important limitation in all prior approaches
+to low-precision floating-point training: they use very
+few precision assignments (most often just one) for a
+given set of models, but there are some other models and
+inputs where the chosen precision assignment (i) results
+in noticeably worse accuracy than 32-bit-float training,
+(ii) causes training to even diverge, or (iii) admits a more
+efficient assignment that achieves similar training accuracy
+(see Figures 1, 3, and 4).
+In this paper, we present a new, automated method for choos-
+ing precision assignments that removes the limitations de-
+scribed above. To do so, we formally introduce the memory-
+accuracy tradeoff problem: given a dataset, a model, and
+two floating-point precision levels (i.e., bitwidths; high and
+low), find a mixed precision assignment (a mapping from
+all tensors arising in training to high/low precision) for the
+model that maximizes training accuracy subject to a given
+upper bound on the model aggregate (i.e., the total number
+of bits of all tensors appearing in training). The model aggre-
+gate is a proxy for the memory and time required for training,
+as it is roughly proportional to memory footprint and also
+well-correlated with training time (since training is often
+dominated by data movement) (Micikevicius et al., 2018).
+We prove that the memory-accuracy tradeoff problem
+is theoretically difficult (namely NP-hard) partly due
+to the exponential number of possible mixed precision
+assignments (which we often refer to simply as precision
+assignments for brevity). The large number of possible
+assignments makes the problem difficult in practice as
+arXiv:2301.13464v1  [cs.LG]  31 Jan 2023
+
+Training with Mixed-Precision Floating-Point Assignments
+(a) SqueezeNet
+(b) ShuffleNet-v2
+(c) MobileNet-v2
+Figure 1: Training trajectory of various models on CIFAR-100. Colors denote precision assignments: all-32-bit πfp32 (red),
+uniform πunif (yellow), and operator-based πop (blue) (see §3.1); the latter two use the 8-bit (and 16-bit) floats in (Sun et al.,
+2019) as low (and high) precision numbers. Markers denote the “width multiplier” of a model, which controls the capacity
+of the model (see §5.3): 1.0 (•), 0.5 (■), 0.25 (▲), and 0.1 (
+). Some lines of πunif are missing as they converge to small
+values or diverge. Observe that neither πunif nor πop works best for all models: in some models, πop has a similar accuracy
+to πfp32; but in other (and all) models, the accuracy drop of πop (and πunif) from πfp32 are noticeably large (i.e., >1%).
+well: there is no known analytical method for predicting
+the training accuracy of a given precision assignment, and
+for any practical model there are far too many precision
+assignments to simply test them all.
+We propose a simple (heuristic) approach to the tradeoff
+problem that prioritizes tensors for low-precision formats
+based on the tensor’s size (with an additional step described
+below). More specifically, our algorithm takes as input
+a single parameter giving a desired upper bound on the
+model aggregate. Starting with the largest tensor in the
+model, tensors are assigned low precision in size order
+(from largest to smallest) until the model aggregate falls
+below the given upper bound; all remaining tensors are
+assigned high precision. Our main result is that this method
+discovers mixed precision assignments that use less memory
+while achieving higher training accuracy than previous
+approaches. While we cannot show that our method finds
+Pareto-optimal memory-accuracy tradeoffs, we do show that
+our results are closer to Pareto-optimal than prior methods.
+Some precision assignments initially generated by our
+algorithm cause training to diverge due to an excessive
+number of overflows. To address this issue, we propose an
+overflow handling technique that promotes tensors causing
+too many overflows from low precision to high precision
+during training.
+In our experiments, these promotions
+consume only a small amount of additional memory (< 3%
+of the maximum model aggregate) and prevent training
+from diverging. The overflow handling technique is not
+specific to our algorithm and can be applied to other
+precision assignment methods as well.
+We evaluate a PyTorch implementation of our method using
+experiments on standard image classification tasks. We first
+demonstrate that the precision assignments computed by our
+method alleviate the limitations of existing methods: they
+indeed explore the tradeoff between memory and accuracy
+and exhibit a better tradeoff than the uniform and operator-
+based assignments. We then show the two main components
+of our method (i.e., precision demotion of larger tensors
+and precision promotion of overflowing tensors) are both
+important to produce competitive precision assignments.
+We also provide some guidance on how users may apply
+our method to navigate the memory-accuracy tradeoff.
+To summarize, this work makes four main contributions:
+• We formally introduce the memory-accuracy tradeoff
+problem to explore better mixed precision assignments
+for low-precision floating-point training and prove the
+NP-hardness of the problem.
+• We present a novel precision assignment technique, as a
+heuristic solution to the tradeoff problem, that proposes
+assignments based on a single parameter denoting a
+desired upper bound on the model aggregate.
+• We present a novel technique that can handle an exces-
+sive number of overflows arising in training while using
+a small amount of additional memory. The technique can
+be applied to any (not just our) precision assignments.
+• We demonstrate that the mixed precision assignments
+found by our method do explore the tradeoff between
+memory and training accuracy, and outperform existing
+precision assignment methods.
+We remark that this work focuses on low-precision
+floating-point training, not fixed-point training (which uses
+fixed-point formats), since we want to target upcoming
+hardware (e.g., (Andersch et al., 2022)) with native support
+for low-precision floats (e.g., 8-bit floats) and their oper-
+ations. Also, this work focuses on low-precision training
+(which trains a model from scratch), not inference (which
+assumes a pre-trained model). More discussion is in §2.
+Our precision assignment method typically provides > 2×
+memory reduction over the operator-based assignment
+while maintaining similar training accuracy and gives
+further reductions by trading off accuracy. Our method
+also provides similar memory reduction to the uniform
+assignment, while avoiding the divergence of training often
+caused by a uniform assignment.
+
+70
+accuracy (%)
+60
+50
+test 
+40
+0
+50
+100
+150
+200
+epoch75
+70
+test accuracy (%)
+65
+60
+55
+50
+45
+0
+50
+100
+150
+200
+epoch75
+test accuracy (%)
+70
+65
+60
+55
+50
+45
+0
+50
+100
+150
+200
+epochTraining with Mixed-Precision Floating-Point Assignments
+The paper is organized as follows.
+After discussing
+related work (§2), we define the memory-accuracy tradeoff
+problem and study its hardness (§3). We then describe our
+algorithm for the problem (§4) and our evaluation (§5). We
+conclude with limitations and future work (§6).
+2
+RELATED WORK
+Low-precision floating-point training has been exten-
+sively studied since the work of (Micikevicius et al., 2018).
+One active research direction is to select appropriate floating-
+point formats (or their variants) for low- and high-precision
+numbers in training. Various floating-point formats have
+been proposed, including FP16 (Micikevicius et al., 2018),
+BF16 (Kalamkar et al., 2019), FP8 (Wang et al., 2018),
+HFP8 (Sun et al., 2019), and FP6 (Chmiel et al., 2021),
+along with some variants such as HBFP (Drumond et al.,
+2018), S2FP8 (Cambier et al., 2020), and BM (Fox et al.,
+2021). Recently, the problem of automatically selecting
+such floating-point formats has been considered: e.g., (Yang
+et al., 2022). Another research direction is to develop algo-
+rithmic techniques that improve training accuracy under low
+precision: e.g., (Sa et al., 2018; Yang et al., 2019a; Zamirai
+et al., 2020; Björck et al., 2021). Our work is orthogonal
+and complementary to all these prior works: they consider
+various floating-point formats or training algorithms but use
+a fixed precision assignment, which is either the uniform
+or operator-based assignment (or their variants); our work
+explores various precision assignments once floating-point
+formats and training algorithms are fixed (e.g., based on
+the prior works). The tradeoff between memory and accu-
+racy in training is also considered in (Yang et al., 2022),
+but the work differs from ours: they vary floating-point for-
+mats when a precision assignment is fixed, while we vary
+precision assignments when floating-point formats are fixed.
+Low-precision fixed-point training uses fixed-point for-
+mats as a low-precision representation instead of a floating-
+point format. Some works use fixed-point formats for for-
+ward tensors and floating-point formats for backward ten-
+sors: e.g., (Courbariaux et al., 2015; Jacob et al., 2018; Choi
+et al., 2018; Yang et al., 2019b; Sun et al., 2020). Other
+works use only fixed-point formats for all tensors: e.g.,
+(Gupta et al., 2015; Zhou et al., 2016; Wu et al., 2018; Das
+et al., 2018; Banner et al., 2018; Sakr & Shanbhag, 2019;
+Zhang et al., 2020; Rajagopal et al., 2020). Among all these
+works, some consider various mixed precision assignments
+with different bitwidth (fixed-point) formats (e.g., (Sakr &
+Shanbhag, 2019; Zhang et al., 2020)); but they are not ap-
+plicable to our context (i.e., floating-point training) since
+they rely on some properties of fixed-point formats that do
+not hold for floating-point formats (e.g., all numbers in a
+given format are equally distributed). The approach taken in
+(Rajagopal et al., 2020) is orthogonal and complementary
+to ours: they use only the uniform precision assignment, but
+change the underlying low-precision formats during train-
+ing; we consider various mixed precision assignments, but
+fix the underlying low-precision formats during training.
+Low-precision inference, often called neural network quan-
+tization (in a narrow sense), aims at reducing the latency
+or memory of neural network inference (instead of train-
+ing) by using low-precision numbers (Nagel et al., 2021).
+Existing approaches typically assume a pre-trained model
+and try to find low-precision formats for each part of
+the inference computation, either by retraining the model
+(called quantization-aware training) or without any retrain-
+ing (called post-training quantization); see, e.g., (Gholami
+et al., 2022; Qin et al., 2022) for surveys. Some works on
+inference consider various mixed precision assignments, but
+they are not applicable to our context: they focus on making
+inference more efficient and usually assume a pre-trained
+model; we focus on making training more efficient and aim
+at learning a model from scratch.
+Floating-point tuning is another related topic, which con-
+siders the following problem: given a program, assign ap-
+propriate formats (among given candidates) to the program’s
+floating-point variables such that the program’s output has
+an error smaller than a given threshold for all given inputs,
+while also maximizing performance (Rubio-González et al.,
+2013; 2016; Chiang et al., 2017; Guo & Rubio-González,
+2018; Menon et al., 2018). This problem is different from
+the problem we consider: the former is concerned with the
+floating-point error after a single run of a program, while
+we are concerned with the training accuracy after a large
+number of runs of a program (i.e., a gradient computation)
+where each run affects the next run; further, the former con-
+siders general-purpose programs, while we consider deep
+learning programs and exploit their unique features.
+3
+PROBLEM
+In this section, we first provide background on low-
+precision floating-point training (§3.1), based on which
+the memory-accuracy tradeoff problem is introduced (§3.2).
+We then prove the NP-hardness of the problem (§3.3). Our
+approach in §3–4 is more formal than most related works
+for two reasons: (i) we show the problem is NP-hard, which
+has not been considered in prior work; and (ii) to clearly
+describe the precision assignments to be considered.
+3.1 Background: Low-Precision Floating-Point Training
+Let T be the set of real-valued tensors and let [n] denote the
+set {1, · · · , n}. For a supervised learning task, we usually
+consider a model network M = (f1, · · · , fn) parameter-
+ized by θ = (θ1, · · · , θn) ∈ Tn, and a loss network L =
+(fn+1, · · · , fm), where fi : T2 → T is a primitive operator
+on tensors (e.g., convolution, batch normalization, maxpool,
+and softmax). Given an input-output pair (x, y) ∈ T2, the
+
+Training with Mixed-Precision Floating-Point Assignments
+𝑓!
+𝑑𝑓!,!
+𝑑𝑓!,#
+𝑓$%!
+𝑑𝑓$%!,!
+⋯
+𝑓$
+𝑑𝑓$,#
+𝑑𝑓$,!
+𝑓&
+𝑑𝑓&,!
+⋯
+⋯
+⋯
+#
+𝑑𝑣&
+%𝑣#
+&𝜃!
+#
+𝑑𝑣#
+#
+𝑑𝜃!
+%𝑣$%!
+%𝑣$%#
+#
+𝑑𝑣$%!
+#
+𝑑𝑣$%#
+&𝜃$
+#
+𝑑𝜃$
+%𝑣&%!
+#
+𝑑𝑣&%!
+𝑦
+: forward computation
+: backward computation
+%𝑣!
+%𝑣$
+%𝑣&
+#
+𝑑𝑣!
+#
+𝑑𝑣$
+Figure 2: A diagram showing the tensors and operators
+used in a gradient computation; see Eq. (1) for details. For
+brevity, rounding functions rndπ(·) are omitted.
+model M computes a predicted output y′ of x by iteratively
+applying fi(·, θi) to x (i ∈ [n]), and L computes a loss
+from y′ by iteratively applying fi′(·, y) to y′ (i′ ∈ [m]\[n]).
+A standard way to train M is to minimize the loss value
+using the gradient descent algorithm: iteratively update θ
+by following the gradient of the loss with respect to θ.
+Floating-point training. In practice, we perform a gradient
+computation usually with tensors represented in floating-
+point formats. Let π : TS → FP be a precision assignment
+giving the floating-point format of each tensor, where TS =
+∆
+{vi, dv i, θj, dθj | i ∈ [m + 1], j ∈ [n]} is the set of tensors
+arising in a gradient computation (explained below), and
+FP =
+∆ {fp(e, m, b) | e, m ∈ N, b ∈ Z} is the set of floating-
+point formats. Here fp(e, m, b) denotes a floating-point
+format that consists of a 1-bit sign, an e-bit exponent, and
+an m-bit mantissa, and has an (additional) exponent bias
+of b ∈ Z. A common choice of π is πfp32(t) =
+∆ fp32 for
+all t ∈ TS, where fp32 =
+∆ fp(8, 23, 0) is the standard 32-bit
+floating-point format.
+Given a precision assignment π, a gradient computation is
+typically performed by the backpropagation algorithm: with
+ˆv1 = rndπ(v1)(x) and ˆdv m+1 = rndπ(dv m+1)(1), compute
+ˆvi+1 = rndπ(vi+1)(fi(ˆvi, ˆui)),
+ˆθj = rndπ(θj)(θj),
+ˆdv i = rndπ(dv i)(df i,1( ˆdv i+1, ˆvi, ˆui)),
+ˆ
+dθj = rndπ(dθj)(df j,2( ˆdv j+1, ˆvj, ˆθj)),
+(1)
+for i ∈ [m] and j ∈ [n]; see Figure 2 for a diagram. Here
+rnd : FP × T → T is a function rounding a given input
+to a given floating-point format, df i,1, df i,2 : T3 → T are
+the backward operators of fi with respect to its first and
+second arguments, respectively, and ˆui = ˆθi if i ∈ [n] and
+y otherwise. We call vi and θj the forward tensors, and
+dv i and dθj the backward tensors. We put a hat over each
+tensor to emphasize that its value is the output of a rounding
+function to a possibly low-precision format; remark that
+such a rounding function is not used within fi, df i,1, and
+df i,2, since they typically use large bitwidth floats (e.g.,
+fp32) and no low-precision floats internally (Kalamkar
+et al., 2019; Cambier et al., 2020). After the computation,
+ˆ
+dθj stores the gradient of the loss value with respect to θj.
+The overall picture of floating-point training is now de-
+scribed as follows. In each iteration of the gradient descent
+algorithm, we compute ˆ
+dθj via Eq. (1) using a given preci-
+sion assignment π, training data (x, y), and current weights
+θ. We then update each θj by θj ← rndfp32(θj − η · ˆ
+dθj)
+given a learning rate η > 0, and proceed to the next iteration
+until the training ends. Here we use fp32 to represent θj
+by following the convention in low-precision floating-point
+training: a “master copy” of weights (i.e., θj) is stored sep-
+arately from the weight values (i.e., ˆθj) used in a gradient
+computation, and is usually represented by fp32 (Micike-
+vicius et al., 2018; Kalamkar et al., 2019; Cambier et al.,
+2020). The memory overhead of this master copy is very
+small compared to the memory required to store other ten-
+sors (e.g., activation tensors vi) (Micikevicius et al., 2018).
+Low-precision floating-point training. In low-precision
+training, we use a precision assignment π where some
+tensors have a smaller bitwidth than fp32.
+Particularly
+well-studied are π that use two predetermined floating-
+point bitwidths (which are different) and optionally vary
+the rest of the format from tensor to tensor.
+We call
+C : TS × {lo, hi} → FP a precision-candidate assignment
+if C(t, lo) has the same bitwidth for all t ∈ TS, the same
+holds for hi, and the bitwidth for lo is smaller than that for hi.
+We also define Π(C) =
+∆ {π : TS → FP | ∀t ∈ TS. π(t) ∈
+{C(t, lo), C(t, hi)}} to be the set of precision assignments
+that conform to C.
+Among various precision assignments in Π(C), two have
+received the most attention:
+the uniform assignment
+πunif,C (Micikevicius et al., 2018) and the operator-based
+assignment πop,C (Sun et al., 2019). The former assigns
+low-precision formats to all tensors uniformly1, and the
+latter to (most of) the input tensors of GEMM operators (in
+both forward and backward passes):
+πunif,C(t) =
+∆ C(t, lo) for all t ∈ TS,
+πop,C(t) =
+∆
+�
+�
+�
+�
+�
+�
+�
+C(t, lo) if t ∈ {vi, θi, dv i+1} for some i
+and fi is a GEMM operator
+(but not the first/last one)
+C(t, hi) otherwise,
+(2)
+where a GEMM operator refers to a general matrix multi-
+plication operator which arises in, e.g., fully-connected or
+convolutional layers. A particular variant πop′,C of πop,C
+has received much attention as well (Kalamkar et al., 2019;
+PyTorch, 2022), which assigns low-precision formats to
+(most of) the input and output tensors of GEMM operators:
+it is defined as πop,C except that {vi, θi, dv i+1} in Eq. (2)
+is replaced by {vi, θi, vi+1, dv i, dθi, dv i+1}. For several
+choices of C, these assignments have been shown to produce
+training accuracy similar to that by πfp32 on many datasets
+and models (see §1–2).
+1For simplicity we define πunif,C without the common
+exceptions for tensors near v1 and/or vm+1.
+
+Training with Mixed-Precision Floating-Point Assignments
+3.2
+Memory-Accuracy Tradeoff Problem
+We now introduce the following problem based on §3.1,
+to address the limitation of existing approaches for
+low-precision floating-point training discussed in §1:
+Problem 3.1 (Memory-accuracy tradeoff). Given training
+data {(xi, yi)}, a model and loss network M and L, a
+precision-candidate assignment C, and a lower bound
+r ∈ [0, 1] on the low-precision ratio, find π ∈ Π(C) that
+maximizes acc(π) subject to ratiolo(π) ≥ r.
+Here acc(π) denotes the accuracy of the model M when
+trained with π on {(xi, yi)}, and ratiolo(π) denotes the
+low-precision ratio of π, i.e., the portion of the tensors
+represented in low-precision under π, among all tensors
+appearing in a gradient computation:2
+ratiolo(π) =
+∆ size({t ∈ TS | π(t) = C(t, lo)})
+size(TS)
+∈ [0, 1]
+where size(T) =
+∆
+�
+t∈T size(t) denotes the total size
+(i.e., number of elements) of all tensors in T ⊆ TS. For
+instance, ratiolo(πhi) = 0 and ratiolo(πlo) = 1 for the
+all-high-precision assignment πhi and the all-low-precision
+assignment πlo. The problem asks for a precision assign-
+ment that maximizes training accuracy under a memory
+constraint, which is expressed as a fraction of the memory
+required to train the model using πhi.
+3.3
+NP-Hardness of the Problem
+We prove that the memory-accuracy tradeoff problem from
+§3.2 is NP-hard by showing that there is a polynomial-time
+reduction from the knapsack problem to this problem:
+Theorem 3.2. Problem 3.1 is NP-hard.
+Proof sketch. Recall the knapsack problem: given n items
+with weights wi ∈ N and profits pi ∈ N (i ∈ [n]), find a
+subset of the items that maximizes the total profit while its
+total weight does not exceed a given threshold W ∈ N.
+Given an instance (w, p, W) of the knapsack problem, we
+construct an instance of Problem 3.1 such that we get the
+following (informal) correspondence between the two: wi
+corresponds to the size of the parameter tensor θi; pi to the
+i-th component of the input data; W to the lower bound r
+on the low-precision ratio (in an inverse way); and selecting
+the i-th item corresponds to assigning a high-precision
+format to the tensor θi (and related tensors), which roughly
+decreases the low-precision ratio by wi while increasing
+the accuracy of the model (after training) by pi. Based
+2As explained in §1, the low-precision ratio is a proxy for
+the reduction in memory as well as training time (because the
+low-precision ratio increases as the model aggregate decreases).
+Note that it is not always possible to simply measure training time,
+as some floating-point bitwidths of interest (e.g., 8-bit) are not
+supported natively by current hardware.
+on this informal correspondence, we formally prove that
+an optimal solution to the above instance of Problem 3.1
+can be converted in linear time to an optimal solution to
+the given knapsack problem (w, p, W). That is, we have
+a linear-time reduction from the knapsack problem (which
+is NP-hard (Karp, 1972)) to Problem 3.1 which is therefore
+NP-hard. For a detailed proof, see Appendix A.
+Intuitively, the proof relies on two aspects of Problem 3.1:
+the size of the search space (i.e., |Π(C)|) is exponential in
+the size of the problem (especially |TS|), and some values
+representable in a high-precision format underflow to 0 in
+a lower-precision format. Note that underflows are relevant
+in low-precision training: they frequently arise in practice,
+degrading the results of training (Micikevicius et al., 2018).
+The NP-hardness result indicates that it is unlikely any
+polynomial-time algorithm solves the problem exactly.
+4
+ALGORITHM
+In this section, we propose a novel (heuristic) algorithm for
+the memory-accuracy tradeoff problem (§4.1), and a new
+technique to handle overflows arising in training (§4.2).
+4.1
+Precision Demotion for Saving Memory
+Consider an input to the memory-accuracy trade-off prob-
+lem (Problem 3.1): a model and loss network M = (f1,
+· · · , fn) and L = (fn+1, · · · , fm), a precision-candidate
+assignment C, and a lower bound r on the low-precision
+ratio. Given the input, our algorithm works in two steps.
+Tensor grouping. We first group tensors in TS such that
+each group consists of all the tensors between two “adjacent”
+GEMM operators (see below for details). This grouping re-
+duces the search space over precision assignments, from all
+of Π(C) to a subset in which the same precision is assigned
+to the tensors in the same group. This specific grouping
+strategy is based on two observations: a majority of floating-
+point operations are carried out in GEMM operators, and it
+is standard (e.g., in PyTorch) to use the same precision for a
+forward tensor and its corresponding backward tensor.
+Formally, we group tensors as follows. Let fk and fk′
+(k < k′) be GEMM operators that are “adjacent”, i.e., there
+is no GEMM operator in {fk+1, · · · , fk′−1}. For each such
+(fk, fk′), we create a group {vi, dv i, θj, dθj | i ∈ (k, k′] ∩
+[m + 1], j ∈ (k, k′] ∩ [n]}. After that, we create two more
+groups for the remaining tensors: one for the tensors near v1
+and the other for tensors near vm+1. As a result, we obtain
+a set of disjoint groups of tensors {T1, T2, · · · } ⊆ 2TS.
+Precision demotion. Given the groups of tensors, T1, T2,
+· · · , we construct a precision assignment π as follows: ini-
+tialize π to the all-high-precision assignment and update
+π by demoting the precision of all tensors in a group to
+low precision, one group at a time, until the low-precision
+
+Training with Mixed-Precision Floating-Point Assignments
+ratio of π becomes greater than r. We demote the pre-
+cision of groups in decreasing order of their sizes (i.e.,
+the total number of elements in tensors); that is, the pre-
+cision of a larger size group is demoted earlier. Formally,
+let {T ′
+1, T ′
+2, · · · } be the reordering of {T1, T2, · · · } such
+that size(T ′
+1) ≥ size(T ′
+2) ≥ · · · . After initializing π by
+π(t) = C(t, hi) for all t, we iterate over i ∈ N and update π
+to π(t) = C(t, lo) for all t ∈ T ′
+i, until ratiolo(π) ≥ r is first
+satisfied. The resulting π is the output of our algorithm.
+The intuition behind using group size as the priority order
+for precision demotion is based on the fact that it is actually
+optimal in a very simplified setting. Suppose that an input
+x to the model M stores a quantity of information I and
+the forward computation of M is nothing but a process of
+extracting the information in the input into a small number
+of values, i.e., the tensor vn+1. Assume that passing through
+each group Oi = {fk+1, · · · , fk′} of operators (corre-
+sponding to the group Ti of tensors) reduces the amount of
+information by a factor αi ∈ (0, 1), and using low precision
+on the group Ti further reduces the amount of information
+by a constant factor β ∈ (0, 1) for all i. Then, the amount
+of information left in vn+1 becomes I × (α1α2 · · · ) × βl,
+where l is the number of groups in low precision. In this
+simplified setting, maximizing the amount of information in
+vn+1 is equivalent to minimizing the number of groups in
+low precision, which is achieved precisely by demoting the
+largest groups first (when there is a constraint on the low-
+precision ratio). We show empirically (§5.4) that using the
+decreasing size order in precision demotion indeed produces
+better precision assignments than using other orders.
+4.2
+Precision Promotion for Handling Overflows
+While our algorithm in §4.1 exerts a constraint on memory
+usage, it places no explicit constraint on training accuracy,
+and so not surprisingly for some models and datasets
+the resulting precision assignment causes training to
+diverge—accuracy decreases significantly and remains low
+after some point. We observe that when training begins to
+diverge (and a bit before that), many overflows occur in the
+rounding function of some tensors ˆvi, i.e., an input tensor to
+the function rndπ(vi)(·) in Eq. (1) contains many elements
+whose magnitude is larger than the maximum representable
+number of the format π(vi) (Figure 6(a-b); §5.4). This
+rapid increase in overflows in individual tensors is a signal
+that training may diverge.
+Precision promotion. Based on this observation, after each
+gradient computation we update the current precision as-
+signment π by promoting to high precision (i.e., C(t, hi))
+any forward tensor t whose overflow ratio is greater than a
+given threshold Θ ∈ (0, 1); this updated precision assign-
+ment is used in the next gradient computation. Here the
+overflow ratio of t ∈ TS denotes the number of overflows
+arising in the rounding function of ˆt in Eq. (1), divided by
+the number of elements in ˆt. We show empirically (§5.4)
+that training always converges using this technique and the
+additional memory cost of promotion is small (in our exper-
+iments, < 3% of the maximum model aggregate). For the
+experiments, we use Θ = 0.01; in fact we found that a wide
+range of values for Θ (0.1, 0.01, and 0.001) all work well.
+Note that this technique is not specific to our algorithm and
+can also be applied to other precision assignment methods.
+We apply precision promotion only to forward tensors for
+two reasons. First, dynamic loss scaling (Micikevicius
+et al., 2018; Sun et al., 2019; Nvidia, 2019; PyTorch, 2022)
+already handles overflows in backward tensors, but not in
+forward tensors: loss scaling multiplies the backward loss
+tensor dv m+1 by a constant before performing backward
+computation, to scale up all backward tensors; the dynamic
+version adjusts the constant during training in a way that
+avoids overflows in backward tensors. Note that dynamic
+loss scaling does not affect forward tensors at all. Second,
+we cannot use a similar idea to handle overflows in forward
+tensors, because forward tensors are not linear in the input
+tensor v1 whereas backward tensors are linear in the back-
+ward loss tensor dv m+1 (by the linearity of differentiation).
+Precision promotion incurs little if any computational over-
+head: checking whether a single rounding operation over-
+flows is cheap, and we only apply rounding functions to the
+output tensor of an arithmetic-intensive operator (e.g., con-
+volution and batch normalization), amortizing the cost of the
+overflow checks over a large number of other operations.
+5
+EXPERIMENTS
+In this section, we evaluate our precision assignment
+technique (developed in §4) on standard training tasks to
+answer three research questions:
+• Does our technique explore the tradeoff between
+memory and accuracy and achieve a better tradeoff than
+existing (fixed) precision assignments (§5.3)?
+• Are the two main components of our technique,
+precision demotion/promotion of larger/overflowing
+tensors, important for good performance (§5.4)?
+• How can we choose the parameter r in our technique
+(i.e., a lower bound on the low-precision ratio) (§5.5)?
+5.1
+Implementation
+We have implemented our precision assignment technique
+using PyTorch (Paszke et al., 2019). Given a model and loss
+network, and a dataset, our implementation takes as param-
+eters a precision-candidate assignment C and a lower bound
+r on the low-precision ratio; it then automatically assigns
+precisions to tensors (appearing in training) according to
+our technique and uses those assigned precisions in gradient
+computations. To make these procedures automatic, our
+implementation works as follows:
+
+Training with Mixed-Precision Floating-Point Assignments
+• For each PyTorch class for a primitive operator (e.g.,
+torch.nn.Conv2d), our implementation provides its
+wrapped version (e.g., mpa.nn.Conv2d) which records
+auxiliary information for our technique (e.g., floating-
+point format of input/output tensors) and applies proper
+rounding functions in forward/backward computations
+based on the auxiliary information. Models should now
+use the wrapped classes instead of the original ones.
+• Our implementation first constructs a computation graph
+(of a given model and loss network) dynamically by
+running a forward computation on a minibatch of input
+data. The computation graph and other information (e.g.,
+each tensor’s size) are recorded in the wrapped classes.
+• Using the auxiliary information just recorded, our
+implementation then constructs a precision assignment
+according to §4.1, uses it in gradient computations, and
+updates it after each gradient computation according
+to §4.2. We record the precision assignment also in the
+wrapped classes to automatically apply proper rounding
+functions in gradient computations.
+As no current hardware natively supports low-precision
+formats used in the experiments (e.g., 8-bit floats) and
+their operations, we simulate them with 32-bit floats
+and 32-bit operations followed by rounding functions as
+described in Eq. (1). We implement the rounding functions
+based on the QPyTorch library (Zhang et al., 2019); a few
+extensions are required though, e.g., to support exponent
+bias and signal overflows for dynamic loss scaling. We
+automatically apply these rounding functions after each
+primitive operator, by using PyTorch’s hook feature (e.g.,
+nn.Module.register_*hook).
+5.2
+Experiment Setups
+Datasets and models. As benchmarks for our experiments,
+we use the image classification task and three datasets for the
+task: CIFAR-10 and CIFAR-100 (Krizhevsky, 2009), and
+ImageNet (Russakovsky et al., 2015); we choose them since
+they have been widely used in recent works on low-precision
+training as a standard choice (Wang et al., 2018; Sakr &
+Shanbhag, 2019; Rajagopal et al., 2020; Chmiel et al., 2021).
+For the task and datasets, we use four well-known models:
+SqueezeNet (Iandola et al., 2016), ShuffleNet-v2 (Ma et al.,
+2018), MobileNet-v2 (Sandler et al., 2018), and ResNet-
+18 (He et al., 2016); they are chosen since models with
+relatively few weights, such as these, are generally known
+to be more difficult to train with low precision than those
+with more weights (Sun et al., 2019). We considered other
+tasks (e.g., language modeling) and related models (e.g.,
+RNN/transformer-based models) but did not include them
+in our experiments because substantial additional implemen-
+tation effort orthogonal to our main contributions would be
+required: these models use some PyTorch operators that do
+not support per-tensor precision assignments, so applying
+our technique to these models requires significant modifica-
+tions to PyTorch internals.
+Precision-candidate and precision assignments. For the
+experiments, we use the precision-candidate assignment
+C studied in (Sun et al., 2019), which uses 16-bit (and
+8-bit) floats for high (and low) precision; in particular,
+C(t, hi) = fp(6, 9, 0) for all (forward/backward) tensors
+t, and C(t, lo) = fp(4, 3, 4) for all forward tensors t and
+fp(5, 2, 0) otherwise. We choose this particular C because it
+uses sub-32-bit floating-point formats for both low and high
+precision and the precision assignment πop,C was shown to
+achieve accuracy comparable to 32-bit training (Sun et al.,
+2019). The three floating-point formats used in C allow
+subnormals but no infinities and NaNs, which are rounded
+to the largest or smallest representable numbers. While no
+current hardware is available for the latter two 8-bit formats,
+they will be supported natively on NVIDIA’s forthcoming
+H100 GPU (Andersch et al., 2022). Because our technique
+is parameterized by a precision-candidate assignment, it is
+easily applied to other assignments as well.
+We evaluate our technique by varying its parameter r
+(i.e., a lower bound on low-precision ratio) over deciles
+r ∈ {0, 0.1, 0.2, · · · , 1}. We write πours,r to denote the
+precision assignment chosen by our technique (described
+in §4) for a given r; e.g., πours,0 is the all-high-precision
+assignment, and πours,1 is the all-low-precision assignment
+equipped with our precision promotion technique (§4.2).
+By following (Sun et al., 2019), all precision assignments
+(including πours,r) in our experiments use high precision
+(i.e., 16 bits) for all backward weight tensors (i.e., ˆ
+dθj).
+Other setups and compute time. All experiments were
+performed on NVIDIA V100 GPUs; total compute time for
+all experiments was 1,008 GPU-days. We train all models in
+a standard way: we apply dynamic loss scaling (a standard
+technique used in low-precision floating-point training; see
+§4.2 for details) except for 32-bit training, and use standard
+settings (e.g., learning rate); see Appendix B for details.
+Due to random variations in training, we perform four runs
+of training for each configuration and report the average and
+the range of measured quantities.
+5.3
+Comparison with Existing Precision Assignments
+To compare our technique with existing precision assign-
+ments for floating-point training, we train each model
+with the following precision assignments: all-32-bit πfp32,
+uniform πunif (Micikevicius et al., 2018), operator-based
+πop (Sun et al., 2019), its variant πop′ (Kalamkar et al.,
+2019; PyTorch, 2022), and ours πours,r (see §3.1 and §5.2
+for their definitions).
+We choose πunif, πop, and πop′
+as baselines because existing precision assignments for
+floating-point training fall into one of the three assignments
+(or their variants) (see §1–2).
+
+Training with Mixed-Precision Floating-Point Assignments
+Figure 3: Results of training ShuffleNet-v2 on ImageNet with πfp32, πunif (Micikevicius et al., 2018), πop (Sun et al.,
+2019), πop′ (Kalamkar et al., 2019), and πours,r. Left: Each line shows the average training trajectory for each precision
+assignment; πours,r is colored from navy to yellow (darker for smaller r). Right: Each point shows the memory-accuracy
+tradeoff of each precision assignment; a red-dashed line shows the accuracy of πfp32; and shaded areas show the variation
+among four training runs. In the right figure, top-right points are better than bottom-left ones. Observe that there are •s
+above and to the right of
+and
+, respectively. ⋆ is missing as its y-value is too small.
+(a) CIFAR-10, SqueezeNet
+(b) CIFAR-100, SqueezeNet
+(c) CIFAR-100, SqueezeNet†
+(d) CIFAR-10, ShuffleNet-v2
+(e) CIFAR-100, ShuffleNet-v2
+(f) CIFAR-100, ShuffleNet-v2†
+(g) CIFAR-10, MobileNet-v2
+(h) CIFAR-100, MobileNet-v2
+(i) CIFAR-100, MobileNet-v2†
+(j) CIFAR-10, ResNet-18
+(k) CIFAR-100, ResNet-18
+(l) CIFAR-100, ResNet-18†
+Figure 4: Memory-accuracy tradeoffs of πunif (Micikevicius et al., 2018), πop (Sun et al., 2019), πop′ (Kalamkar et al.,
+2019), and πours,r for four models and their smaller variants on CIFAR-10 and CIFAR-100. The variant models have width
+multiplier 0.25 and are marked by †. Top-right points are better than bottom-left ones. In all but three plots, there are •s
+above and to the right of
+and
+, respectively; even in the three plots (g,h,k), •s have almost the same tradeoffs to
+and
+. In half of all plots, ⋆ has much smaller y-values than other points. The training trajectories for the above plots
+and the results of other smaller models are in Appendix C.1.
+
+92
+90
+88
+fp32
+op
+86
+op
+unif
+80
+ours
+78
+0.00
+0.25
+0.50
+0.75
+1.00
+low-prec. ratio72
+70
+68
+fp32
+op
+66
+op
+unif
+2
+ours
+0
+0.00
+0.25
+0.50
+0.75
+1.00
+low-prec. ratio66
+test accuracy (%)
+64
+X
+62
+fp32
+60
+op
+op
+unif
+2
+ours
+0
+0.00
+0.25
+0.50
+0.75
+1.00
+low-prec. ratio94 
+(%)
+92
+test accuracy (
+90
+fp32
+88
+op
+op
+unif
+80
+ours
+78
+0.00
+0.25
+0.50
+0.75
+1.00
+low-prec. ratio74
+(%)
+72
+test accuracy (
+70
+fp32
+68
+op
+op
+unif
+2
+ours
+0
+0.00
+0.25
+0.50
+0.75
+1.00
+low-prec. ratio68
+accuracy (%)
+66
+64
+fp32
+62
+op
+test a
+op
+unif
+40
+ours
+38
+0.00
+0.25
+0.50
+0.75
+1.00
+low-prec. ratio96
+test accuracy (%)
+94
+92
+fp32
+op
+90
+op
+unif
+76
+ours
+0.00
+0.25
+0.50
+0.75
+1.00
+low-prec. ratio80
+(%)
+78
+test accuracy (
+76
+fp32
+74
+op
+op
+unif
+2
+ours
+0
+0.00
+0.25
+0.50
+0.75
+1.00
+low-prec. ratio70
+68
+66
+fp32
+op
+64
+op
+unif
+52
+ours
+50
+0.00
+0.25
+0.50
+0.75
+1.00
+low-prec. ratio60
+test accuracy (%)
+40
+fp32
+op
+20
+op
+unif
+ours
+0
+0
+20
+40
+60
+80
+epoch66
+test accuracy (%)
+64
+62
+fp32
+op
+60
+op
+unif
+34
+ours
+32
+0.00
+0.25
+0.50
+0.75
+1.00
+low-prec. ratio94 
+(%)
+92
+test accuracy (
+90
+fp32
+88
+op
+op
+unif
+72
+ours
+70
+0.00
+0.25
+0.50
+0.75
+1.00
+low-prec. ratio72
+(%)
+70
+test accuracy (
+68
+66
+fp32
+op
+op
+unif
+2
+ours
+0
+0.00
+0.25
+0.50
+0.75
+1.00
+low-prec. ratio56
+test accuracy (%)
+54
+52
+fp32
+op
+50
+op
+unif
+2
+ours
+0
+0.00
+0.25
+0.50
+0.75
+1.00
+low-prec. ratioTraining with Mixed-Precision Floating-Point Assignments
+We train the four models mentioned in §5.2 on CIFAR-10
+and CIFAR-100, and ShuffleNet-v2 on ImageNet. We also
+train smaller variants of the four models (which are more
+difficult to train with low precision) on CIFAR-100. We
+obtain these variant models by following (Sun et al., 2019),
+i.e., by applying a well-known approach for model reduction
+that uses a parameter called the width multiplier (Howard
+et al., 2017): each variant model reduces the number of
+channels in most tensors by a width multiplier; we use three
+values {0.5, 0.25, 0.1} for the width multiplier. We train
+just one model on ImageNet due to the large amount of
+computation involved: for each model, 44 training runs (11
+choices for r and 4 runs for each choice) are required for
+πours,r and each run on ImageNet takes nearly a half day
+with 16 GPUs. We use ShuffleNet-v2 for ImageNet since
+the model shows interesting memory-accuracy tradeoffs
+when trained on the (smaller) CIFAR datasets.
+ImageNet. Figure 3 presents training results of ShuffleNet-
+v2 on ImageNet: its left graph plots the average training
+trajectory for each precision assignment, and its right graph
+shows how each precision assignment trades off between
+memory and accuracy, where memory is represented (in-
+versely) by the low-precision ratio of the assignment and ac-
+curacy is the best test accuracy of the model during training.
+Each point in the right graph shows the average accuracy
+of four runs of training, while the shaded area shows the
+variation in accuracy among those four training runs.
+Figure 3 shows three points.
+First, as the parameter r
+increases, the average accuracy drop of πours,r from πfp32
+increases (up to 5%). In contrast, πunif and πop′ have a
+much larger average accuracy drop (more than 30%), as
+some training runs diverge when πunif and πop′ are used.
+Second, the tradeoff given by πours,r is better (i.e., closer
+to Pareto-optimal) than by πop: πours,r for r ∈ {0.3, 0.4}
+has both higher accuracy and larger low-precision ratio (i.e.,
+memory reduction) than πop. In particular, πours,0.4 has
+1.6× the memory reduction of πop. Third, πours,r provides
+options that πop cannot (which has an accuracy drop of
+>1%). If we want accuracy closer to πfp32, say within 0.5%,
+we can use πours,0.2 with 2.6% more memory than πop. If
+we can tolerate a larger accuracy loss, say ≈ 3%, then we
+can use πours,0.7 with 2.9× the memory reduction of πop.
+CIFAR-10/100. Figure 4 presents the memory-accuracy
+tradeoffs of precision assignments for the four models on
+CIFAR-10 and CIFAR-100, and their smaller variants (with
+width multiplier 0.25) on CIFAR-100. The results for other
+smaller variants are similar and included in Appendix C.1.
+The conclusions from Figure 3 hold for Figure 4: πours,r
+provides a range of options by varying r and exhibits a
+better tradeoff than πunif, πop, and πop′ in almost all cases.
+We give a detailed comparison as follows. First, in half of
+all 12 plots, πunif shows a similar tradeoff to πours,1. But
+in the remaining half, πunif has an accuracy drop much
+larger than all other precision assignments including πours,r,
+since using πunif often makes training diverge while using,
+e.g., πours,1 does not do so. Second, in all but two plots,
+πours,r shows a strictly better tradeoff than πop: πours,r has
+noticeably larger (> 2×) memory reduction than πop while
+maintaining similar accuracy. Even in the two plots, πours,r
+has a tradeoff very close to πop. Note that in three plots,
+πop has an accuracy drop of >1% while πours,r provides
+several options that have smaller accuracy drops and more
+memory savings at the same time. Third, πours,r shows a
+strictly better (or similar) tradeoff than πop′ in all but two
+(or two) plots. Note that πop′ has accuracy smaller than πop
+in all but one plots. Also it has an accuracy drop of >1% in
+half of all plots, and sometimes makes training even diverge
+(in one plot here and three other plots in Appendix C.1).
+5.4 Ablation Study: Precision Demotion and Promotion
+Precision demotion.
+To evaluate the decision to use
+precision demotion in decreasing-size order, we train
+the four models on CIFAR-100 with πours,r, πours[inc],r
+(which demotes tensor groups in increasing-size order) and
+πours[rand],r (which demotes tensor groups in random or-
+der). For πours[rand], three different random orders are used
+in each case. The results, presented in Figure 5 (and Ap-
+pendix C.2), show that the order of precision demotion has a
+significant impact on the resulting memory-accuracy trade-
+off, and that decreasing order provides the best results in
+nearly all cases. Increasing order consistently shows the
+worst results, suggesting our intuition (given in §4.1) for
+choosing decreasing order has some basis in reality.
+Precision promotion. To understand whether precision pro-
+motion of overflowing tensors is important to our technique,
+we train ShuffleNet-v2 on ImageNet using πours[no-promo],r
+which does not promote tensors. The results, presented in
+Figure 6(a), show that several training trajectories diverge
+in early epochs and fail to recover afterwards. Figure 6(b)
+plots the top-5 tensor overflow ratios for the highlighted
+trajectory in Figure 6(a). The overflow ratios first spike
+about when divergence occurs around epoch 11. A closer
+look shows that the spike in overflow ratio occurs shortly
+before divergence, and starts first in a few tensors and then
+propagates to others. These observations indicate that an
+excessive number of overflows in a few tensors are the cause
+of the training divergence.
+Finally, Figure 6(c-d) shows that precision promotion is
+effective at preventing the divergence of training while
+sacrificing only a small amount of memory reduction. The
+figure shows ShuffleNet-v2 on ImageNet trained using our
+technique with and without precision promotion. Figure 6(c)
+shows that without precision promotion large accuracy
+drops occur due to divergence, whereas with precision pro-
+motion training converges. Figure 6(d) shows that the total
+
+Training with Mixed-Precision Floating-Point Assignments
+(a) SqueezeNet
+(b) ShuffleNet-v2
+(c) MobileNet-v2
+Figure 5: Memory-accuracy tradeoffs of πours,r, πours[inc],r, and πours[rand],r for three models on CIFAR-100. Observe
+that •s are above and to the right of other points in nearly all cases. The results of ResNet-18 are in Appendix C.2.
+(a)
+(b)
+(c)
+(d)
+Figure 6: Training ShuffleNet-v2 on ImageNet with πours,r and πours[no-promo],r. (a) Training trajectories of πours[no-promo],r
+for different r; colors denote r values (darker for smaller r). (b) Top-5 overflow ratios of tensors at each epoch, for the
+highlighted trajectory in (a); the largest ratio is blue and the fifth largest red. (c) Memory-accuracy tradeoffs of πours,r
+and πours[no-promo],r. (d) Low-precision ratio when training ends vs. when training starts, for πours,r and πours[no-promo],r.
+The results on CIFAR-10 are in Appendix C.2.
+size of tensors promoted to high precision is small for all r
+values. See Appendix C.2 for similar results for CIFAR-10.
+5.5
+Choosing the value of r
+The time and space savings of our method are most signif-
+icant when a model is regularly retrained, which commonly
+occurs when new data is periodically incorporated into
+an existing model. Assuming that new data has a similar
+distribution to existing data, we can choose a single r (a
+parameter in our method) by conducting one set of exper-
+iments where we train with πfp32 and πours,r for different r
+and then choose the r value that maximizes model aggregate
+savings while still having an acceptable drop in accuracy.
+To simulate this scenario, we create five datasets ImageNet-
+200-i (i ∈ [5]) as follows, so that each of them contains
+different but similar data: randomly select 1/5 of the classes
+in ImageNet (which has 1000 classes in total), and split the
+training data of each class evenly into five new datasets.
+For each ImageNet-200-i, we train ShuffleNet-v2 with
+πfp32 and πours,r and present the results in Figure 7. Based
+on the tradeoff results of πours,r, we can choose r = 0.4 if
+we desire an average of < 1% accuracy drop from πfp32, and
+we can choose r = 0.9 if an average ≈ 3% accuracy drop
+is tolerable. We make two more observations: the tradeoff
+result of πours,r is similar across all five datasets even
+though each dataset is different, and for each r the variance
+Figure 7:
+Memory-accuracy tradeoffs of πours,r for
+ShuffleNet-v2 on ImageNet-200-i (i ∈ [5]).
+in the accuracy of πours,r from different datasets and runs of
+training is similar to that of πfp32. Thus we expect that on
+a new but similar dataset, πours,r would have an accuracy
+drop similar to Figure 7 with acceptable variance.
+6
+LIMITATIONS AND FUTURE WORK
+Our work has the same limitation present in prior works
+on low-precision floating-point training: as 8-bit floats and
+operations are not handled natively in hardware, but rather
+simulated in software, we cannot directly measure the poten-
+tial speedup of our method, though we do expect speedups
+to be proportional to the reduction in the model aggregate.
+We leave it as future work to perform such experiments
+on future hardware (e.g., NVIDIA’s H100) that natively
+supports more low-precision formats. Another direction
+for future work is to integrate our method into systems for
+automatically optimizing deep learning computations (e.g.,
+(Jia et al., 2019; Unger et al., 2022)) to accelerate training.
+
+72
+accuracy (%)
+70
+68
+test 
+ours[inc]
+ours[rand]
+66 
+ours
+0.00
+0.25
+0.50
+0.75
+1.00
+low-prec. ratio74-
+test accuracy (%)
+72
+70
+ours[inc]
+68
+ours[rand]
+ours
+0.00
+0.25
+0.50
+0.75
+1.00
+low-prec. ratio74
+(%)
+accuracy(
+72
+70
+test 
+ours[inc]
+ours[rand]
+68 
+ours
+0.00
+0.25
+0.50
+0.75
+1.00
+low-prec. ratio80
+60
+40
+20
+0
+20
+40
+60
+80
+epoch1.0
+0.8
+overflow ratio
+0.6
+0.4
+top-1
+top-2
+top-3
+0.2
+top-4
+top-5
+0.0
+0
+20
+40
+60
+80
+epoch80
+(%)
+60
+accuracy (
+40
+test 
+20
+ours[no-promo]
+ours
+0
+0.00
+0.25
+0.50
+0.75
+1.00
+low-prec. ratio1.0
+low-prec. ratio (end)
+0.8
+0.6
+0.2
+ours[no-promo]
+ours
+0.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+low-prec. ratio (start)64 
+62
+茶
+60
+58
+fp32
+ours
+0.00
+0.25
+0.50
+0.75
+1.00
+low-prec. ratioTraining with Mixed-Precision Floating-Point Assignments
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+A
+PROBLEM: DEFERRED PROOF
+Theorem 3.2. Problem 3.1 is NP-hard.
+Proof. We prove the NP-hardness of Problem 3.1 (the
+memory-accuracy tradeoff problem) by reducing the
+knapsack problem (which is NP-hard) to the tradeoff
+problem.
+More precisely, we prove that the knapsack
+problem can be solved in polynomial time if we assume
+an oracle for the tradeoff problem.
+Recall the knapsack problem: given n items with weights
+wi ∈ N and profits pi ∈ N (i ∈ [n]), and given a threshold
+W ∈ N, decide which items to choose such that the total
+profit of the chosen items is maximized while their total
+weight does not exceed W. That is, find α ∈ {0, 1}n that
+maximizes �
+i∈[n] αipi subject to �
+i∈[n] αiwi ≤ W. This
+problem is well-known to be NP-hard (Karp, 1972).
+Given an instance of the knapsack problem (w, p, W), we
+construct an instance of the tradeoff problem as follows:
+• Notations.
+The following construct uses a constant
+k ∈ N and floating-point formats fphi, fplo ∈ FP (one
+for high precision and the other for low precision).
+Below we will specify the conditions they should satisfy,
+and show that some k, fphi, and fplo indeed satisfy the
+conditions. We write rndhi(·) and rndlo(·) as shorthand
+for rndfphi(·) and rndfplo(·).
+• Training setups. We consider a very simple setting for
+training: the gradient descent algorithm with a learning
+rate η = 2−l (l ∈ N) is applied for just one epoch; all
+parameters are initialized to 0 and their master copies
+are represented in fphi; and the negative loss of a model
+on training data (i.e., −L(fθ(x), y) using notations to
+be described below) is used as the accuracy of the model.
+Here l ∈ N can be any natural number.
+• Model and loss networks. A model network M and
+a loss network L are given as Figure 8, where M has n
+parameter tensors θi ∈ Rwi of size wi (i ∈ [n]). For an
+input-output pair (x, y) ∈ Rn × R, M and L compute a
+predicted output fθ(x) ∈ R and a loss L(fθ(x), y) ∈ R
+as follows (assuming that no rounding functions are
+applied):
+fθ(x) =
+∆ �
+i∈[n]
+�
+j∈[wi]
+θi,jxi,
+L(fθ(x), y) =
+∆ 2−k|fθ(x) − y|.
+Roughly speaking, M is (a variant of) a linear classifier
+and L is a ℓ1-loss (scaled by 2−k).
+• Training data. Training data consists of a single input-
+output pair (x, y) ∈ Rn × R that satisfies the following:
+xi = rndlo(
+�
+pi/wi),
+y < −2−(k+l) �
+i∈[n]
+wix2
+i
+for all i ∈ [n]. Here y can take any value as long as
+it satisfies the above inequality. Note that xi can be
+different from
+�
+pi/wi since the latter value may not
+be representable in fplo.
+• Precision-candidate
+assignment.
+A
+precision-
+candidate assignment C : TS×{hi, lo} → FP is given as:
+C(t, hi) =
+∆ fphi,
+C(t, lo) =
+∆ fplo
+for all t ∈ TS.
+That is, for all tensors, fphi is used as a high-precision
+format and fphi as a low-precision format. Here fphi and
+fplo should satisfy the following:
+ehi ≥ elo,
+mhi ≥ mlo,
+(3)
+|rndlo(s) − s| < |s| · err
+for all s ∈ S1,
+(4)
+rndlo(s) = 0
+for all s ∈ S2,
+(5)
+rndhi(s) = s
+for all s ∈ S2 ∪ S3,
+(6)
+where ehi and mhi (and elo and mlo) denote the number
+of exponent bits and mantissa bits of fphi (and fplo), and
+err =
+∆ 1/(6n · maxi∈[n]pi),
+S1 =
+∆ {
+�
+pi/wi | i ∈ [n]},
+S2 =
+∆ {2−k} ∪ {2−kxi | i ∈ [n]},
+S3 =
+∆ {2−(k+l)xi | i ∈ [n]}.
+Eq. (4) says that the relative error of representing each
+s ∈ S1 in fplo should be less than err. Eq. (5) says that
+each s ∈ S2 should underflow to 0 when represented
+in fplo. Eq. (6) says that each s ∈ S2 ∪ S3 should be
+representable in fphi.
+• Low-precision ratio. A lower bound r ∈ [0, 1] on the
+low-precision ratio is given as:
+r =
+∆ max
+�
+0, 1 − 2W + 1
+size(TS)
+�
+∈ [0, 1].
+So r decreases linearly as W increases.
+We make three points on the above construction.
+• First, each part of the knapsack problem (w, p, W) is
+used in the following parts of the construction: wi is
+used mainly in the size of the parameter tensor θi; pi
+in the input xi; and W in the lower bound r.
+• Second, there exist k ∈ N and fphi, fplo ∈ FP that sat-
+isfy Eqs. (3)–(6). This can be shown as follows: first,
+by taking sufficiently many exponent and mantissa bits
+for fplo, we can make Eq. (4) satisfied; next, by taking a
+sufficiently large k, we can make Eq. (5) satisfied; finally,
+by taking sufficiently many exponent and mantissa bits
+for fphi, we can make Eq. (3) and Eq. (6) satisfied (since
+xi is representable in fplo and 2−(k+l) is a power of two).
+• Third, some well-known models (e.g., ShuffleNet-v2)
+have a similar structure to M in that they apply the fol-
+lowing operations as a subroutine: split a tensor into mul-
+tiple tensors, apply some operators to each split tensor,
+and combine the resulting tensors into a single tensor.
+
+Training with Mixed-Precision Floating-Point Assignments
+⫶
+𝑥! !∈ #
+.∑!∈ # ∑$∈ %! 𝑣#&!,$.
+2() 𝑣*#&+ − 𝑦
+𝑣,,+
+𝜃+,$ ⋅ 𝑣+ .$∈ %"
+𝑣,,#
+𝜃+,$ .$∈ %"
+𝜃#,$ ⋅ 𝑣# .$∈ %#
+𝜃#,$ .$∈ %#
+𝑦
+𝑣,∈ℝ#
+𝑣+∈ℝ
+𝑣#∈ℝ
+𝜃#∈ℝ%#
+𝜃+∈ℝ%"
+𝑣*#∈ℝ%#
+𝑣#&+∈ℝ%"
+𝑣*#&+∈ℝ
+𝑣*#&*∈ℝ
+𝑦∈ℝ
+⫶
+⫶
+split
+conv
+conv
+sum
+loss
+ℳ
+ℒ
+Figure 8: The model network M and the loss network L used in the proof of Theorem 3.2.
+We now prove that the knapsack problem (w, p, W) can
+be solved in polynomial time, if an oracle to the above
+tradeoff problem is given. Suppose that π ∈ Π(C) is an
+optimal solution to the above tradeoff problem (given by
+the oracle). Define an item selection α ∈ {0, 1}n for the
+knapsack problem as:
+αi =
+∆
+�
+1
+if π(dθi) = π(dv n+i) = π(dv 2n+1) = fphi
+0
+otherwise
+for each i ∈ [n]. Note that α can be constructed from π in
+linear time. Thus, it suffices to show that α is an optimal
+solution to the knapsack problem (w, p, W), which is
+equivalent to the following two claims:
+• Claim 1: We have �
+i∈[n] αiwi ≤ W.
+• Claim 2: For any α′ ∈ {0, 1}n with �
+i∈[n] α′
+iwi ≤ W,
+we have �
+i∈[n] α′
+ipi ≤ �
+i∈[n] αipi.
+We now prove each claim as follows.
+Proof of Claim 1. If α = (0, · · · , 0), then the claim clearly
+holds. Suppose that α ̸= (0, · · · , 0). Then, we have
+1 −
+1 + 2 �
+i∈[n] αiwi
+size(TS)
+≥ ratiolo(π)
+≥ r ≥ 1 − 1 + 2W
+size(TS).
+Here the first inequality uses α ̸= (0, · · · , 0) and the defini-
+tion of α and M; the second inequality uses the fact that π is
+a valid solution to the above tradeoff problem; and the third
+inequality uses the definition of r. Hence, the claim holds.
+Proof of Claim 2. Suppose that the claim does not hold.
+Then, there exists α′ ∈ {0, 1}n such that
+�
+i∈[n]
+α′
+iwi ≤ W,
+�
+i∈[n]
+α′
+ipi >
+�
+i∈[n]
+αipi.
+Define a precision assignment π′ ∈ Π(C) as:
+π′(dv 2n+1) =
+∆ fphi,
+π′(dθi) =
+∆ π′(dv n+i) =
+∆ fphi
+for all i ∈ [n] with α′
+i = 1,
+π′(t) =
+∆ fplo
+for all other t ∈ TS.
+Then, we have ratiolo(π′) ≥ r by �
+i∈[n] α′
+iwi ≤ W and
+the definition of π′, M, and r. Hence, it suffices to show
+acc(π) < acc(π′), because this would contradict to the fact
+that π is an optimal solution.
+To show acc(π) < acc(π′), we prove the following two
+lemmas: the first lemma gives a closed form of acc(π) and
+acc(π′), and the second lemma shows that �
+i∈[n] βiwix2
+i
+is close to �
+i∈[n] βipi (where the former summation
+appears in acc(π) and acc(π′)).
+Lemma A.1. The following hold:
+acc(π) = 2−ky + 2−(2k+l) �
+i∈[n]
+αiwix2
+i ,
+acc(π′) = 2−ky + 2−(2k+l) �
+i∈[n]
+α′
+iwix2
+i .
+Proof. We prove the equation for acc(π) only, since the
+equation for acc(π′) can be proved similarly.
+First, we show that for all i ∈ [n] and j ∈ [wi],
+ˆ
+dθi,j = αi · 2−kxi.
+(7)
+Pick any i ∈ [n] and j ∈ [wi]. Note that by the definition
+of M, we have
+ˆ
+dθi,j = rndπ(dθi)
+�
+rndπ(dv n+i)(rndπ(dv 2n+1)(2−k))
+· rndvi(rndv0(xi))
+�
+= rndπ(dθi)
+�
+rndπ(dv n+i)(rndπ(dv 2n+1)(2−k)) · xi
+�
+,
+where the second equality uses Eq. (3) and that xi is
+representable in fplo. We prove Eq. (7) by case analysis
+on αi. Suppose αi = 1. Then, by the definition of αi,
+
+Training with Mixed-Precision Floating-Point Assignments
+π(dθi) = π(dv n+i) = π(dv 2n+1) = fphi. From this, we
+get the desired equation:
+ˆ
+dθi,j = rndhi
+�
+rndhi(rndhi(2−k)) · xi
+�
+= rndhi(2−k · xi) = 2−kxi,
+where the last two equalities use Eq. (6). Suppose now
+αi = 0. Then, by the definition of αi, at least one of π(dθi),
+π(dv n+i), and π(dv 2n+1) is fplo. If π(dv n+i) = fplo or
+π(dv 2n+1) = fplo, we get the desired equation:
+ˆ
+dθi,j = rndπ(dθi)
+�
+rndlo(2−k) · xi
+�
+= rndπ(dθi)(0 · xi) = 0,
+where the first equation uses Eq. (3) and Eq. (6), and the
+second equation uses Eq. (5). The remaining case is when
+π(dv n+i) = π(dv 2n+1) = fphi and π(dθi) = fplo. We get
+the desired equation in this case as well:
+ˆ
+dθi,j = rndlo
+�
+rndhi(rndhi(2−k)) · xi
+�
+= rndlo(2−k · xi) = 0,
+where the second equality uses Eq. (6), and the last equality
+uses Eq. (5). Hence, we have proved Eq. (7).
+Next, let θi be the i-th parameter tensor before training
+starts, and θ′
+i be the corresponding tensor after training ends
+(i ∈ [n]). Then, by the definition of the tradeoff problem
+constructed above, we have θi,j = 0 and
+θ′
+i,j = θi,j − rndhi(2−l · ˆ
+dθi,j)
+= 0 − rndhi(2−l · (αi · 2−kxi))
+= αi · (−2−(k+l)xi),
+where the second equality uses Eq. (7) and the third equality
+uses Eq. (6). Using this equation, we finally obtain the
+conclusion of this lemma:
+acc(π) = −L(fθ′(x), y)
+= −2−k���y −
+�
+i∈[n]
+�
+j∈[wi]
+θ′
+i,jxi
+���
+= −2−k���y −
+�
+i∈[n]
+�
+j∈[wi]
+αi · (−2−(k+l)xi) · xi
+���
+= −2−k���y +
+�
+i∈[n]
+αi · 2−(k+l)wix2
+i
+���
+= 2−k�
+y +
+�
+i∈[n]
+αi · 2−(k+l)wix2
+i
+�
+= 2−ky + 2−(2k+l) �
+i∈[n]
+αiwix2
+i ,
+where the first two equalities use the definition of accuracy,
+and the second last equality uses the definition of y. This
+concludes the proof of the lemma.
+■
+Lemma A.2. For any β ∈ {0, 1}n,
+���
+�
+i∈[n]
+βiwix2
+i −
+�
+i∈[n]
+βipi
+��� < 1
+2.
+Proof. We first show that for any i ∈ [n],
+|wix2
+i − pi| < 1
+2n.
+Pick any i ∈ [n]. By Eq. (4) and the definition of xi, we have
+���xi −
+� pi
+wi
+��� <
+� pi
+wi
+·
+1
+6n · maxj∈[n] pj
+≤
+� pi
+wi
+·
+1
+6npi
+.
+From this, we have
+� pi
+wi
+�
+1 −
+1
+6npi
+�
+< xi <
+� pi
+wi
+�
+1 +
+1
+6npi
+�
+,
+pi
+wi
+�
+1 −
+1
+6npi
+�2
+< x2
+i < pi
+wi
+�
+1 +
+1
+6npi
+�2
+.
+From this, we obtain the desired result:
+|wix2
+i − pi| < pi
+��
+1 +
+1
+6npi
+�2
+− 1
+�
+= pi
+�
+1
+3npi
++
+1
+(6npi)2
+�
+< pi
+�
+1
+3npi
++
+1
+6npi
+�
+= pi ·
+1
+2npi
+= 1
+2n,
+where the second inequality uses 6npi > 1 (as n, pi ∈ N).
+Using this result, we can show the conclusion as follows:
+���
+�
+i∈[n]
+βiwix2
+i −
+�
+i∈[n]
+βipi
+��� =
+���
+�
+i∈[n]
+βi(wix2
+i − pi)
+���
+≤
+�
+i∈[n]
+|βi| · |wix2
+i − pi|
+<
+�
+i∈[n]
+1
+2n = 1
+2,
+where the last inequality uses |βi| ≤ 1. This completes the
+proof of the lemma.
+■
+Using the two lemmas, we prove acc(π) < acc(π′) as fol-
+lows. First, by Lemma A.2 and �
+i∈[n] αipi < �
+i∈[n] α′
+ipi,
+we have
+�
+i∈[n]
+αiwix2
+i <
+�
+i∈[n]
+αipi + 1
+2
+≤
+�
+i∈[n]
+α′
+ipi − 1
+2 <
+�
+i∈[n]
+α′
+iwix2
+i ,
+where the second inequality comes from αi, α′
+i ∈ {0, 1}
+and pi ∈ N. From this, and by Lemma A.1, we obtain
+acc(π) < acc(π′) as desired. This concludes the proof of
+Claim 2, thereby finishing the proof of the theorem.
+
+Training with Mixed-Precision Floating-Point Assignments
+Remark A.3. In the proof of Theorem 3.2, we proved
+the NP-hardness of Problem 3.1 by making use of only a
+few limited aspects of the problem. For instance, we used
+the fact that some values representable in a high-precision
+format round to zero in a low-precision format; on the other
+hand, many other values representable in a high-precision
+format round to non-zero values in a low-precision format,
+and this indeed occurs in practical training (even more
+frequently than underflows). Also, we used a simple setting
+for training in which a gradient descent algorithm is applied
+for one epoch, training data consist of one input-output pair,
+and test data is the same as training data; on the other hand,
+in practical training, a gradient descent algorithm is applied
+for many epochs, training data consists of many input-output
+pairs, and test data is different from training data.
+Problem 3.1 is general enough so that it embraces all the
+aforementioned aspects of floating-points and training,
+including those that are not considered in the proof of
+Theorem 3.2. Since those aspects are likely to make the
+problem even more difficult, we conjecture that the problem
+would be more intractable than being NP-hard.
+B
+EXPERIMENTS: DEFERRED DETAILS
+The datasets we use have the following licenses:
+• CIFAR-10 and CIFAR-100: These datasets are under
+the MIT license.
+• ImageNet: This dataset can be used “only for non-
+commercial research and educational purposes.” For
+more details, see its webpage (Stanford Vision Lab,
+2020).
+The implementations of models we use have the following
+licenses:
+• SqueezeNet for CIFAR-10 and CIFAR-100: We adapt
+an implementation of the model in a public GitHub
+repository (Pathak, 2020), whose license information
+is not available.
+• ShuffleNet-v2,
+MobileNet-v2,
+and ResNet-18 for
+CIFAR-10 and CIFAR-100: We adapt an implementation
+of these models in a public GitHub repository (kuangliu,
+2021), which is under the MIT license.
+• ShuffleNet-v2 for ImageNet and ImageNet-200-i: We
+adapt an implementation of the model in the torchvision
+library (PyTorch, 2022b), which is under the BSD
+3-Clause license.
+The details of how we train models are as follows:
+• Four models on CIFAR-10 and CIFAR-100: We train
+the four models with a standard setup (kuangliu, 2021).
+In particular, we run the (non-Nesterov) SGD optimizer
+for 200 epochs with minibatch size of 128 (over 1
+GPU), learning rate of 0.1, momentum of 0.9, weight
+decay of 5 × 10−4, and the cosine annealing scheduler
+for learning rate. For dynamic loss scaling, we use
+initial scale of 216, growth factor of 2, back-off factor
+of 0.5, and growth interval of 1 epoch, as suggested in
+PyTorch (PyTorch, 2022a).
+• ShuffleNet-v2 on ImageNet: We train the model with the
+default setup given in PyTorch’s GitHub repository (Py-
+Torch, 2022c), except that we use larger minibatch size
+and learning rate as in (Kalamkar et al., 2019; PyTorch,
+2022d; Krizhevsky, 2014; Goyal et al., 2017) to reduce
+the wall-clock time of training. In particular, we run
+the (non-Nesterov) SGD optimizer for 90 epochs with
+minibatch size of 1024 (over 16 GPUs), learning rate
+of 0.4, momentum of 0.9, weight decay of 10−4, and
+the cosine annealing scheduler for learning rate. For
+dynamic loss scale, we use initial scale of 216, growth
+factor of 2, back-off factor of 0.5, and growth interval
+of 0.5 epoch, as suggested in PyTorch (PyTorch, 2022a).
+• ShuffleNet-v2 on ImageNet-200-i: We train the model
+with the same settings for ImageNet except that we use
+the default values for minibatch size and learning rate
+given in (PyTorch, 2022c), i.e., minibatch size of 256
+(over 4 GPUs) and learning rate of 0.1.
+C
+EXPERIMENTS: DEFERRED RESULTS
+C.1
+Comparison with Existing Precision Assignments
+Figure 9 presents results omitted in Figure 4: training
+results of smaller variant models (which have width
+multiplier 0.5 or 0.1) on CIFAR-100 with πfp32, πunif,
+πop, πop′, and πours,r. The figure shows similar results
+to Figure 4: the results for the variant models with width
+multiplier 0.5 (and 0.1) are similar to those for the original
+models (and the variant models with width multiplier 0.25).
+Figures 10 and 11 show the average training trajectories
+for the configurations presented in Figures 4 and 9.
+C.2 Ablation Study: Precision Demotion and Promotion
+Figure 12 presents results omitted in Figure 5: training re-
+sults of ResNet-18 on CIFAR-100 with πours,r, πours[inc],r,
+and πours[rand],r.
+The figure shows similar results to
+Figure 5 except that it shows smaller differences in memory-
+accuracy tradeoff between the three precision assignments.
+Figure 13 presents results omitted in Figure 6: training
+results of four models on CIFAR-10 with πours,r and
+πours[no-promo],r.
+The figure shows similar results to
+Figure 6 except that the training of ResNet-18 on CIFAR-10
+does not diverge even with πours[no-promo],r for all r values.
+
+Training with Mixed-Precision Floating-Point Assignments
+(a) CIFAR-100, SqueezeNet‡
+(b) CIFAR-100, SqueezeNet¶
+(c) CIFAR-100, ShuffleNet-v2‡
+(d) CIFAR-100, ShuffleNet-v2¶
+(e) CIFAR-100, MobileNet-v2‡
+(f) CIFAR-100, MobileNet-v2¶
+(g) CIFAR-100, ResNet-18‡
+(h) CIFAR-100, ResNet-18¶
+Figure 9: Continued from Figure 4. Memory-accuracy tradeoffs of πunif (Micikevicius et al., 2018), πop (Sun et al., 2019),
+πop′ (Kalamkar et al., 2019), and πours,r for smaller variants of four models on CIFAR-100. The variant models have width
+multiplier 0.5 (marked by ‡) or 0.1 (marked by ¶). Top-right points are better than bottom-left ones. In all but one plots,
+there are •s above and to the right of
+and
+, respectively; even in the one plot (g), •s have almost the same tradeoffs to
+and
+. In three of all plots, ⋆ has much smaller y-values than other points; ⋆ is missing in (h) as its y-value is too small.
+
+66
+test accuracy (%)
+64
+62
+fp32
+60
+op
+op
+unif
+2
+ours
+0
+0.00
+0.25
+0.50
+0.75
+1.00
+low-prec. ratio44
+accuracy (%)
+40
+36 
+fp32
+32
+op
+test a
+op
+unif
+24 
+ours
+0.00
+0.25
+0.50
+0.75
+1.00
+low-prec. ratio70
+test accuracy (%)
+68
+66
+fp32
+64
+op
+op
+unif
+2
+ours
+0
+0.00
+0.25
+0.50
+0.75
+1.00
+low-prec. ratio56
+52
+X
+48
+44
+fp32
+op
+op
+unif
+24
+ours
+0.00
+0.25
+0.50
+0.75
+1.00
+low-prec. ratio72
+test accuracy (%)
+70
+68
+fp32
+66
+op
+op
+unif
+2
+ours
+0
+0.00
+0.25
+0.50
+0.75
+1.00
+low-prec. ratio56
+test accuracy (%)
+52
+48
+fp32
+44
+op
+op
+unif
+36
+ours
+32
+0.00
+0.25
+0.50
+0.75
+1.00
+low-prec. ratio76
+test accuracy (%)
+74
+72
+fp32
+op
+70
+op
+unif
+2
+ours
+0
+0.00
+0.25
+0.50
+0.75
+1.00
+low-prec. ratio60
+test accuracy (%)
+56
+52
+fp32
+op
+48
+op
+unif
+40
++
+ours
+36
+0.00
+0.25
+0.50
+0.75
+1.00
+low-prec. ratioTraining with Mixed-Precision Floating-Point Assignments
+(a) CIFAR-10, SqueezeNet
+(b) CIFAR-100, SqueezeNet
+(c) CIFAR-100, SqueezeNet†
+(d) CIFAR-10, ShuffleNet-v2
+(e) CIFAR-100, ShuffleNet-v2
+(f) CIFAR-100, ShuffleNet-v2†
+(g) CIFAR-10, MobileNet-v2
+(h) CIFAR-100, MobileNet-v2
+(i) CIFAR-100, MobileNet-v2†
+(j) CIFAR-10, ResNet-18
+(k) CIFAR-100, ResNet-18
+(l) CIFAR-100, ResNet-18†
+Figure 10: Training trajectories for the configurations shown in Figure 4. Each line shows the average training trajectory
+for each precision assignment. πours,r is colored from navy to yellow (darker for smaller r).
+
+100
+(%)
+90
+test accuracy (
+80
+fp32
+op
+70
+op'
+unif
+ours
+60
+0
+50
+100
+150
+200
+epoch70
+accuracy (%)
+60
+fp32
+50
+op
+test a
+op'
+unif
+40
+ours
+0
+50
+100
+150
+200
+epoch60
+test accuracy (%)
+50
+40
+fp32
+op
+op'
+30
+unif
+ours
+20
+0
+50
+100
+150
+200
+epoch100
+(%)
+90
+test accuracy (
+80
+fp32
+op
+op'
+70
+unif
+ours
+60
+0
+50
+100
+150
+200
+epoch80
+test accuracy (%)
+70
+60
+fp32
+op
+50
+op'
+unif
+ours
+40
+0
+50
+100
+150
+200
+epoch70
+test accuracy (%)
+60
+50
+fp32
+op
+40
+op'
+unif
+ours
+30
+0
+50
+100
+150
+200
+epoch100
+(%)
+90
+test accuracy (
+80
+fp32
+op
+op'
+70
+unif
+ours
+60
+0
+50
+100
+150
+200
+epoch80
+test accuracy (%)
+70
+60
+fp32
+op
+50
+op'
+unif
+ours
+40
+0
+50
+100
+150
+200
+epoch70
+test accuracy (%)
+60
+50
+fp32
+op
+40
+op'
+unif
+ours
+30
+0
+50
+100
+150
+200
+epoch100
+(%)
+90
+test accuracy (
+80
+fp32
+op
+70
+op'
+unif
+ours
+60
+0
+50
+100
+150
+200
+epoch80
+accuracy (%)
+70
+fp32
+60
+op
+test a
+,do
+unif
+50
+ours
+0
+50
+100
+150
+200
+epoch70
+accuracy (%)
+60
+fp32
+50
+op
+test a
+op'
+unif
+40
+ours
+0
+50
+100
+150
+200
+epochTraining with Mixed-Precision Floating-Point Assignments
+(a) CIFAR-100, SqueezeNet‡
+(b) CIFAR-100, SqueezeNet¶
+(c) CIFAR-100, ShuffleNet-v2‡
+(d) CIFAR-100, ShuffleNet-v2¶
+(e) CIFAR-100, MobileNet-v2‡
+(f) CIFAR-100, MobileNet-v2¶
+(g) CIFAR-100, ResNet-18‡
+(h) CIFAR-100, ResNet-18¶
+Figure 11: Training trajectories for the configurations shown in Figure 9. Each line shows the average training trajectory
+for each precision assignment. πours,r is colored from navy to yellow (darker for smaller r).
+
+70
+test accuracy (%)
+60
+50
+fp32
+op
+40
+op'
+unif
+ours
+30
+0
+50
+100
+150
+200
+epoch50
+test accuracy (%)
+40
+30
+fp32
+op
+20
+op'
+unif
+ours
+10
+0
+50
+100
+150
+200
+epoch70
+accuracy (%)
+60
+fp32
+50
+op
+test 
+op'
+unif
+40
+ours
+0
+50
+100
+150
+200
+epoch60
+50
+40
+fp32
+op
+30
+op
+unif
+ours
+20
+0
+50
+100
+150
+200
+epoch70
+accuracy (%)
+60
+fp32
+50
+op
+test a
+op'
+unif
+40
+ours
+0
+50
+100
+150
+200
+epoch60
+50
+40
+fp32
+op
+30
+op'
+unif
+ours
+20
+0
+50
+100
+150
+200
+epoch80
+70
+60
+fp32
+op
+50
+,do
+unif
+ours
+40
+0
+50
+100
+150
+200
+epoch70
+test accuracy (%)
+60
+50
+fp32
+op
+40
+op'
+unif
+ours
+30
+0
+50
+100
+150
+200
+epochTraining with Mixed-Precision Floating-Point Assignments
+(a) ResNet-18
+Figure 12: Continued from Figure 5. Memory-accuracy tradeoffs of πours,r, πours[inc],r, and πours[rand],r for ResNet-18
+on CIFAR-100. Observe that •s are above and to the right of other points in nearly all cases.
+(a) CIFAR-10, SqueezeNet
+(b) CIFAR-10, ShuffleNet-v2
+(c) CIFAR-10, MobileNet-v2
+(d) CIFAR-10, ResNet-18
+Figure 13: Continued from Figure 6. Training four models on CIFAR-10 with πours,r and πours[no-promo],r. Column 1:
+Training trajectories of πours[no-promo],r for different r; colors denote r values (darker for smaller r). Column 2: Top-5
+overflow ratios of tensors at each epoch, for the highlighted trajectory in (a); the largest ratio is blue and the fifth largest
+red. Column 3: Memory-accuracy tradeoffs of πours,r and πours[no-promo],r. Column 4: Low-precision ratio when training
+ends vs. when training starts, for πours,r and πours[no-promo],r.
+
+82
+(%)
+80
+test accuracy (
+78
+76
+ours[inc]
+ours[rand]
+ours
+0.00
+0.25
+0.50
+0.75
+1.00
+low-prec. ratio100
+test accuracy (%)
+80
+60
+40
+20
+0
+0
+50
+100
+150
+200
+epoch1.0
+0.8
+overflow ratio
+0.6
+0.4
+top-1
+top-2
+top-3
+0.2
+top-4
+top-5
+0.0
+0
+50
+100
+150
+200
+epoch100
+(%)
+80
+accuracy (
+60
+40
+test 
+20
+ours[no-promo]
+ours
+0
+0.00
+0.25
+0.50
+0.75
+1.00
+low-prec. ratio1.0
+ratio (end)
+0.8
+0.6
+low-prec.
+0.2
+ours[no-promo]
+ours
+0.0
+0.0
+0.2
+0.4
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+low-prec. ratio (start)Training with Mixed-Precision Floating-Point Assignments
+REFERENCES (FOR APPENDIX)
+Goyal, P., Dollár, P., Girshick, R. B., Noordhuis, P.,
+Wesolowski, L., Kyrola, A., Tulloch, A., Jia, Y., and He,
+K. Accurate, Large Minibatch SGD: Training ImageNet
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+madaka, N., Huang, J., Yuen, H., Yang, J., Park, J.,
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+pytorch_Squeezenet, 2020.
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+

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