Daily Papers

Variational inference (VI) seeks to approximate a target distribution pi by an element of a tractable family of distributions. Of key interest in statistics and machine learning is Gaussian VI, which approximates pi by minimizing the Kullback-Leibler (KL) divergence to pi over the space of Gaussians. In this work, we develop the (Stochastic) Forward-Backward Gaussian Variational Inference (FB-GVI) algorithm to solve Gaussian VI. Our approach exploits the composite structure of the KL divergence, which can be written as the sum of a smooth term (the potential) and a non-smooth term (the entropy) over the Bures-Wasserstein (BW) space of Gaussians endowed with the Wasserstein distance. For our proposed algorithm, we obtain state-of-the-art convergence guarantees when pi is log-smooth and log-concave, as well as the first convergence guarantees to first-order stationary solutions when pi is only log-smooth.

This article considers the popular MCMC method of unadjusted Langevin Monte Carlo (LMC) and provides a non-asymptotic analysis of its sampling error in 2-Wasserstein distance. The proof is based on a refinement of mean-square analysis in Li et al. (2019), and this refined framework automates the analysis of a large class of sampling algorithms based on discretizations of contractive SDEs. Using this framework, we establish an O(d/epsilon) mixing time bound for LMC, without warm start, under the common log-smooth and log-strongly-convex conditions, plus a growth condition on the 3rd-order derivative of the potential of target measures. This bound improves the best previously known O(d/epsilon) result and is optimal (in terms of order) in both dimension d and accuracy tolerance epsilon for target measures satisfying the aforementioned assumptions. Our theoretical analysis is further validated by numerical experiments.

We revisit the problem of sampling from a target distribution that has a smooth strongly log-concave density everywhere in mathbb R^p. In this context, if no additional density information is available, the randomized midpoint discretization for the kinetic Langevin diffusion is known to be the most scalable method in high dimensions with large condition numbers. Our main result is a nonasymptotic and easy to compute upper bound on the Wasserstein-2 error of this method. To provide a more thorough explanation of our method for establishing the computable upper bound, we conduct an analysis of the midpoint discretization for the vanilla Langevin process. This analysis helps to clarify the underlying principles and provides valuable insights that we use to establish an improved upper bound for the kinetic Langevin process with the midpoint discretization. Furthermore, by applying these techniques we establish new guarantees for the kinetic Langevin process with Euler discretization, which have a better dependence on the condition number than existing upper bounds.

Score-based diffusion models, while achieving remarkable empirical performance, often suffer from low sampling speed, due to extensive function evaluations needed during the sampling phase. Despite a flurry of recent activities towards speeding up diffusion generative modeling in practice, theoretical underpinnings for acceleration techniques remain severely limited. In this paper, we design novel training-free algorithms to accelerate popular deterministic (i.e., DDIM) and stochastic (i.e., DDPM) samplers. Our accelerated deterministic sampler converges at a rate O(1/{T}^2) with T the number of steps, improving upon the O(1/T) rate for the DDIM sampler; and our accelerated stochastic sampler converges at a rate O(1/T), outperforming the rate O(1/T) for the DDPM sampler. The design of our algorithms leverages insights from higher-order approximation, and shares similar intuitions as popular high-order ODE solvers like the DPM-Solver-2. Our theory accommodates ell_2-accurate score estimates, and does not require log-concavity or smoothness on the target distribution.

The spectral smoothness properties of the low-frequency array of the Square Kilometer Array (SKA), namely SKA-Low, are an important issue for its scientific objectives to be attainable. A large array of 256 log-periodic dipole antennas, installed on top of a 42~m circular ground plane, will work as an SKA-Low station in the frequency range 50-350 MHz. In this article, the ground plane induced effects are examined in terms of antenna beam spectral characteristics, while different antenna placements are considered. Results are produced both at isolated antenna and at array level in the band 50-100 MHz, by employing an approximate method for the speeding-up of array simulations. We attempt to distinguish the ground plane effect from that of mutual coupling among antennas, which appears to be more severe at specific frequencies, using 2 figures of merit. The Discrete Fourier Transform (DFT) components of gain pattern ratios identify the fundamental spatial components of the ripple, while the Envelope Correlation Coefficient quantifies the penalty to considering an infinite ground plane.

We introduce a theoretical framework for sampling from unnormalized densities based on a smoothing scheme that uses an isotropic Gaussian kernel with a single fixed noise scale. We prove one can decompose sampling from a density (minimal assumptions made on the density) into a sequence of sampling from log-concave conditional densities via accumulation of noisy measurements with equal noise levels. Our construction is unique in that it keeps track of a history of samples, making it non-Markovian as a whole, but it is lightweight algorithmically as the history only shows up in the form of a running empirical mean of samples. Our sampling algorithm generalizes walk-jump sampling (Saremi & Hyv\"arinen, 2019). The "walk" phase becomes a (non-Markovian) chain of (log-concave) Markov chains. The "jump" from the accumulated measurements is obtained by empirical Bayes. We study our sampling algorithm quantitatively using the 2-Wasserstein metric and compare it with various Langevin MCMC algorithms. We also report a remarkable capacity of our algorithm to "tunnel" between modes of a distribution.

We give an improved theoretical analysis of score-based generative modeling. Under a score estimate with small L^2 error (averaged across timesteps), we provide efficient convergence guarantees for any data distribution with second-order moment, by either employing early stopping or assuming smoothness condition on the score function of the data distribution. Our result does not rely on any log-concavity or functional inequality assumption and has a logarithmic dependence on the smoothness. In particular, we show that under only a finite second moment condition, approximating the following in reverse KL divergence in epsilon-accuracy can be done in tilde Oleft(d log (1/delta){epsilon}right) steps: 1) the variance-delta Gaussian perturbation of any data distribution; 2) data distributions with 1/delta-smooth score functions. Our analysis also provides a quantitative comparison between different discrete approximations and may guide the choice of discretization points in practice.

Stochastically Extended Adversarial (SEA) model is introduced by Sachs et al. [2022] as an interpolation between stochastic and adversarial online convex optimization. Under the smoothness condition, they demonstrate that the expected regret of optimistic follow-the-regularized-leader (FTRL) depends on the cumulative stochastic variance sigma_{1:T}^2 and the cumulative adversarial variation Sigma_{1:T}^2 for convex functions. They also provide a slightly weaker bound based on the maximal stochastic variance sigma_{max}^2 and the maximal adversarial variation Sigma_{max}^2 for strongly convex functions. Inspired by their work, we investigate the theoretical guarantees of optimistic online mirror descent (OMD) for the SEA model. For convex and smooth functions, we obtain the same O(sigma_{1:T^2}+Sigma_{1:T^2}) regret bound, without the convexity requirement of individual functions. For strongly convex and smooth functions, we establish an O(min{log (sigma_{1:T}^2+Sigma_{1:T}^2), (sigma_{max}^2 + Sigma_{max}^2) log T}) bound, better than their O((sigma_{max}^2 + Sigma_{max}^2) log T) bound. For exp-concave and smooth functions, we achieve a new O(dlog(sigma_{1:T}^2+Sigma_{1:T}^2)) bound. Owing to the OMD framework, we can further extend our result to obtain dynamic regret guarantees, which are more favorable in non-stationary online scenarios. The attained results allow us to recover excess risk bounds of the stochastic setting and regret bounds of the adversarial setting, and derive new guarantees for many intermediate scenarios.

In a differentially private sequential learning setting, agents introduce endogenous noise into their actions to maintain privacy. Applying this to a standard sequential learning model leads to different outcomes for continuous vs. binary signals. For continuous signals with a nonzero privacy budget, we introduce a novel smoothed randomized response mechanism that adapts noise based on distance to a threshold, unlike traditional randomized response, which applies uniform noise. This enables agents' actions to better reflect both private signals and observed history, accelerating asymptotic learning speed to Theta_{epsilon}(log(n)), compared to Theta(log(n)) in the non-private regime where privacy budget is infinite. Moreover, in the non-private setting, the expected stopping time for the first correct decision and the number of incorrect actions diverge, meaning early agents may make mistakes for an unreasonably long period. In contrast, under a finite privacy budget epsilon in (0,1), both remain finite, highlighting a stark contrast between private and non-private learning. Learning with continuous signals in the private regime is more efficient, as smooth randomized response enhances the log-likelihood ratio over time, improving information aggregation. Conversely, for binary signals, differential privacy noise hinders learning, as agents tend to use a constant randomized response strategy before an information cascade forms, reducing action informativeness and hampering the overall process.

Modern language models can contain billions of parameters, raising the question of whether they can generalize beyond the training data or simply regurgitate their training corpora. We provide the first non-vacuous generalization bounds for pretrained large language models (LLMs), indicating that language models are capable of discovering regularities that generalize to unseen data. In particular, we derive a compression bound that is valid for the unbounded log-likelihood loss using prediction smoothing, and we extend the bound to handle subsampling, accelerating bound computation on massive datasets. To achieve the extreme level of compression required for non-vacuous generalization bounds, we devise SubLoRA, a low-dimensional non-linear parameterization. Using this approach, we find that larger models have better generalization bounds and are more compressible than smaller models.

Understanding the gradient variance of black-box variational inference (BBVI) is a crucial step for establishing its convergence and developing algorithmic improvements. However, existing studies have yet to show that the gradient variance of BBVI satisfies the conditions used to study the convergence of stochastic gradient descent (SGD), the workhorse of BBVI. In this work, we show that BBVI satisfies a matching bound corresponding to the ABC condition used in the SGD literature when applied to smooth and quadratically-growing log-likelihoods. Our results generalize to nonlinear covariance parameterizations widely used in the practice of BBVI. Furthermore, we show that the variance of the mean-field parameterization has provably superior dimensional dependence.