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+Improved Algorithms for Multi-period Multi-class Packing Problems
+with Bandit Feedback
+Wonyoung Kim 1 Garud Iyengar 1 Assaf Zeevi 1
+Abstract
+We consider the linear contextual multi-class
+multi-period packing problem (LMMP) where
+the goal is to pack items such that the total vector
+of consumption is below a given budget vector
+and the total value is as large as possible. We
+consider the setting where the reward and the con-
+sumption vector associated with each action is
+a class-dependent linear function of the context,
+and the decision-maker receives bandit feedback.
+LMMP includes linear contextual bandits with
+knapsacks and online revenue management as spe-
+cial cases. We establish a new more efficient esti-
+mator which guarantees a faster convergence rate,
+and consequently, a lower regret in such problems.
+We propose a bandit policy that is a closed-form
+function of said estimated parameters. When the
+contexts are non-degenerate, the regret of the pro-
+posed policy is sublinear in the context dimen-
+sion, the number of classes, and the time hori-
+zon T when the budget grows at least as
+√
+T. We
+also resolve an open problem posed in Agrawal &
+Devanur (2016), and extend the result to a multi-
+class setting. Our numerical experiments clearly
+demonstrate that the performance of our policy is
+superior to other benchmarks in the literature.
+1. Introduction
+In the multi-period packing problem (MPP) the decision-
+maker “packs” the arrivals so that the total consumption
+across a set a resources is below a given budget vector
+and the reward is maximized. A variant of the packing
+problem, where items consume multiple resources and the
+decisions must be made sequentially with bandit feedback
+for a fixed time horizon, is known as bandits with knapsacks
+(Agrawal & Devanur, 2014a; Badanidiyuru et al., 2018;
+Immorlica et al., 2019). MPPs also arise in online revenue
+1Columbia University, New York, NY, USA. Correspondence
+to: <>.
+Preliminary work. Under review by the International Conference
+on Machine Learning (ICML). Do not distribute.
+management (Besbes & Zeevi, 2012; Ferreira et al., 2018).
+MPPs in the literature assume that all arrivals belong to a
+single class. However, in several application domains (e.g.,
+operations, healthcare, and e-commerce), the arrivals are
+heterogeneous, and personalizing decisions to each distinct
+population or class is of paramount importance. In this
+paper we consider a class of linear multi-class multi-period
+packing problems (LMMP). At each round, there is a single
+arrival that belongs to one of J classes, and the decision-
+maker observes the d-dimensional context and the cost for
+K different available actions. The outcome of selecting an
+action is a random sample of the reward and a consumption
+vector for m resources with an expected value that is a class-
+dependent linear function of the d-dimensional contexts.
+The goal of the problem is to minimize the cumulative regret
+over a time horizon T while ensuring that the total resource
+consumed is at most B.
+The LMMP problem is a generalization of several prob-
+lems including linear contextual bandits with knapsacks
+(LinCBwK) introduced by Agrawal & Devanur (2016).
+They proposed an online mirror descent-based algorithm
+that achieves ˜O(OPT/B · d
+√
+T) regret when the budget
+B for each of the m resources is Ω(
+√
+dT 3/4), where OPT
+is the reward obtained by the oracle policy. Although the
+regret bound is meaningful for B ≥ Ω(d
+√
+T), establishing
+the regret bound for smaller budget values was left as an
+open problem. Chu et al. (2011) established a regret bound
+sublinear in d for the linear contextual bandit setting, which
+is a special case of LinCBwK with no budget constraints.
+Thus, the following question remained open: “Is there an
+algorithm for LinCBwK that achieves sublinear dependence
+on d with budget B = Ω(
+√
+T)?”
+We propose a novel algorithm and an improved estimation
+strategy that settles this open problem and generalizes the
+result to the more general class of LMMP. The proposed
+algorithm achieves ˜O(OPT/B
+√
+JdT) regret with budget
+B = Ω(
+√
+JdT) under non-degenerate contexts. While re-
+gret of the existing algorithms grows linearly in the number
+of classes J, our estimator is able to pool data from differ-
+ent classes and avoids linear dependence on J. To reiterate,
+the improved regret bound results from the novel estimator
+which yields faster convergence rates.
+arXiv:2301.13791v1  [stat.ML]  31 Jan 2023
+
+Improved Algorithms for Multi-period Packing Problems with Bandit Feedback
+Our main contributions are summarized as follows:
+• We propose a new problem class – linear multi-class
+multi-period packing problems (LMMP). This problem
+generalizes a variety of problems including LinCBwK
+and online revenue management problems to the multi-
+class setting.
+• We propose a novel estimator that uses contexts for
+all actions (including the contexts in skipped rounds)
+and yields O(
+�
+Jd/n) convergence rate for J classes,
+context dimension d, and n admitted arrivals (Theorem
+4.2).
+• We propose a novel AMF (Allocate to the Maximum
+First) algorithm which achieves ˜O(OPT/B
+√
+JdT)
+regret with budget B = Ω(
+√
+JdT) where OPT is
+the reward obtained by oracle policy (Theorem 5.1).
+For the single class setting with J = 1, we improve
+the existing bound by
+√
+d and show that the bound
+is valid when B = Ω(
+√
+dT), and thus resolving an
+open problem posed in Agrawal & Devanur (2016)
+regarding LinCBwK.
+• We evaluate our proposed algorithm on a suite of syn-
+thetic experiments and demonstrate its superior perfor-
+mance.
+All proofs omitted from the front matter can be found in the
+Appendix.
+2. Related Works
+There are two streams of work that are relevant for
+LMMP. In online revenue management literature, Gallego &
+Van Ryzin (1994) introduced the dynamic pricing problem
+where the demand is a known function of price (action). Bes-
+bes & Zeevi (2009) and Besbes & Zeevi (2012) extended the
+problem under unknown demands with multiple resource
+constraints. Ferreira et al. (2018) proposed a Thompson
+sampling-based algorithm and extended it to contextual ban-
+dits with knapsacks. When the expected demand is a linear
+function of the price vector, the dynamic pricing problem
+is a special case of linear contextual bandits with knap-
+sack (LinCBwK) proposed by Agrawal & Devanur (2016).
+The LinCBwk is a common generalization of bandits with
+knapsacks (Badanidiyuru et al., 2018; Immorlica et al., 2019;
+Li et al., 2021) and online stochastic packing problems (Feld-
+man et al., 2010; Agrawal & Devanur, 2014b; Devanur et al.,
+2011). Recently, Sankararaman & Slivkins (2021) proved a
+logarithmic regret bound for LinCBwK when there exists a
+problem-dependent gap between the reward of the optimal
+action and the other actions. Instead of the gap assump-
+tion, we require non-degeneracy of the stochastic contexts
+(see Assumption 3 for a precise definition) to obtain a re-
+gret bound sublinear in d and extends to the case when the
+contexts are generated from J different class.
+Amani et al. (2019) proposed a variant of LinCBwK where
+the selected action must satisfy a single constraint with high
+probability in all rounds, i.e., LinCBwK with anytime con-
+straints. Moradipari et al. (2021) and Pacchiano et al. (2021)
+proposed a Thompson sampling-based algorithm and an
+upper confidence bound-based algorithm, respectively, for
+LinCBwK with a single anytime constraint. Liu et al. (2021)
+highlighted the difference between global and anytime con-
+straints, and proposed an pessimistic-optimistic algorithm
+for the anytime constraints. We focus on the global con-
+straints; however, we note that the extension to the anytime
+constraints is straightforward with minor modifications.
+2.1. Notation
+Let R+ denote the set of positive real numbers. For two
+real numbers a, b ∈ R, we write a ∧ b := min{a, b} and
+a ∨ b := max{a, b}. For a natural number N, let [N] =
+{1, . . . , N}.
+3. Linear Multi-period Packing Problem
+Let [J] denote the set of classes with arrival probabilities
+p = {pj}j∈[J], where pmin := minj∈[J] pj > 0. In each
+round t ∈ [T], the covariates {x(j)
+k,t ∈ [0, 1]d : k ∈ [K]}
+and costs {c(j)
+k,t ∈ [0, 1] : k ∈ [K]} are drawn from a class-
+specific distribution Fj. We assume that the class arrival
+probabilities p are known to the decision-maker; however,
+the distributions {Fj}j∈[J] are not known.
+At time t ∈ [T], the decision-maker observes an arrival of
+the form (jt, {x(jt)
+k,t , c(jt)
+k,t : k ∈ [K]}), where jt ∈ [J] is
+the arrived class. Upon observing the arrival, the decision-
+maker can either take one of K different actions or skip the
+arrival. When the arrival is skipped, the decision-maker does
+not obtain any rewards or consume any resources. When
+the decision-maker chooses an action at ∈ [K], the reward
+and consumption of the resource are given by
+E
+�
+r(jt)
+at,t
+��� Ht
+�
+=
+�
+θ(jt)
+⋆
+�⊤
+x(jt)
+at,t ∈ [−1, 1],
+E
+�
+b(jt)
+at,t
+��� Ht
+�
+=
+�
+W (jt)
+⋆
+�⊤
+x(jt)
+at,t ∈ [0, 1]m,
+for some unknown class-specific parameters θ(j)
+⋆
+∈ [0, 1]d
+and W (j)
+⋆
+∈ [0, 1]d×m. The sigma algebra Ht is generated
+by the class-specific variables {js, x(js)
+k,s , c(js)
+k,s , : s ∈ [t], k ∈
+[K]}, actions {as : s ∈ At}, consumption vectors {b(js)
+as,s :
+s ∈ At−1} and rewards {r(js)
+as,t : s ∈ At−1}, where At
+is the rounds admitted by the decision-maker until round
+t. The process terminates at the horizon T or runs out of
+
+Improved Algorithms for Multi-period Packing Problems with Bandit Feedback
+budget B ∈ Rm
++ for some resources r ∈ [m]. The problem
+reduces to LinCBwK when the number of class is J = 1
+and the costs are c(j)
+k,t = 0
+Let ρ = B/T ∈ Rm
++ denote per-period budget for m re-
+sources. Without loss of generality, one can assume that
+ρ(r) = ρ for all r ∈ [m], By rescaling W (j)
+⋆ . We assume
+that ρ is known to the decision-maker and B is possibly un-
+known at first, but known at the end of the round. This case
+happens when the total budget B is difficult to count in the
+early rounds. When ρ is not available, the decision-maker
+requires B and T to compute ρ. However, this assumption
+is more practical than in Agrawal & Devanur (2016) where
+B and OPT must be known to the decision-maker. When
+OPT is unknown, they estimate OPT with
+√
+T number
+of rounds, which requires the knowledge of T and budget
+B = Ω(
+√
+dT
+3
+4 ). Instead of estimating OPT, we use ρ to
+avoid the required budget B = Ω(
+√
+dT
+3
+4 ).
+We benchmark the performance of the decision-maker’s pol-
+icy relative to that of an oracle who knows the distributions
+{Fj : j ∈ [J]} and the parameters {θ(j)
+⋆ , W (j)
+⋆
+: j ∈ [J]},
+but does not know the arrivals {(jt, x(j)
+k,t, c(j)
+k,t) : t ∈ [T]}
+a-priori. In this case, the optimal static policy for the oracle
+{π⋆(j)
+k
+: j ∈ [J], k ∈ [K]} is the solution to the following
+optimization problem:
+max
+π(j)
+k
+J
+�
+j=1
+K
+�
+k=1
+pjπ(j)
+k E(xk,ck)∼Fj
+��
+θ(j)
+⋆
+�⊤
+xk − ck
+�
+s.t.
+j
+�
+j=1
+K
+�
+k=1
+pjπ(j)
+k Exk∼Fj
+��
+W (j)
+⋆
+�⊤
+xk
+�
+≤ ρ,
+K
+�
+k=1
+π(j)
+k
+≤ 1, ∀j ∈ [J],
+π(j)
+k
+≥ 0, ∀j ∈ [J], ∀k ∈ [K],
+(1)
+Let π⋆ denote the optimal oracle policy. Then the expected
+reward obtained by the oracle is
+OPT :=
+T
+J
+�
+j=1
+K
+�
+k=1
+pjπ⋆(j)
+k
+E(xk,ck)∼Fj
+��
+θ(j)
+⋆
+�⊤
+xk − ck
+�
+.
+Let π := {π(j)
+k,t : j ∈ [J], k ∈ [K], t ∈ [T]} denote the
+adapted (randomized) control policy of the decision-maker,
+i.e. she chooses action k ∈ [K] when the arrival at time
+t ∈ [T] belongs to class j ∈ [J]. Note that �K
+k=1 π(j)
+k,t ≤ 1
+in order to allow the decision-maker to skip an arrival and
+save the inventory for later use. Our goal is to compute a
+policy that minimizes the cumulative regret Rπ
+T defined as
+Rπ
+T := OPT − E
+� T
+�
+t=1
+Rπ
+t
+�
+,
+where Rπ
+t := �K
+k=1 π(jt)
+k,t E
+��
+θ(jt)
+⋆
+�⊤
+x(jt)
+k,t − c(jt)
+k,t
+�
+is the
+expected reward obtained by policy π at time t.
+For the LMMP problem, we assume the following regularity
+conditions on the stochastic processes.
+Assumption 1. (Sub-Gaussian and bounded errors) For
+each t ∈ [T], the error of the reward ηk,t = r(jt)
+k,t −
+�
+θ(jt)
+⋆
+�⊤
+x(jt)
+k,t is conditionally zero-mean σr-sub-Gaussian
+for a fixed constant σr ≥ 0, i.e. E [exp (vηk,t)| Ht] ≤
+exp
+�
+v2σ2
+r
+2
+�
+for all v ∈ R. For the consumption vectors,
+E
+�
+v⊤{b(jt)
+k,t − (W (jt)
+⋆
+)⊤x(jt)
+k,t }
+��� Ht
+�
+≤ exp( ∥v∥2
+2σ2
+b
+2
+) for
+all v ∈ Rm.
+Assumption 2. (Independently distributed contexts and
+costs) The set of contexts {x(j)
+k,t : k ∈ [K]} and {c(j)
+k,t :
+k ∈ [K]} are generated independently over t ∈ [T]. The
+contexts and cost in the same round and class can be corre-
+lated with each other.
+Assumption 3. (Positive definiteness of average covari-
+ances) For each t ∈ [T] and j ∈ [J], there exists α > 0,
+such that
+λmin
+�
+E
+�
+1
+K
+K
+�
+k=1
+x(j)
+k,t
+�
+x(j)
+k,t
+�⊤
+��
+≥ α.
+Assumptions 1 and 2 are standard in stochastic contex-
+tual bandits with knapsacks literature (Agrawal & Devanur,
+2016; Sankararaman & Slivkins, 2021; Sivakumar et al.,
+2022). In the multi-class case, Assumption 2 implies that all
+the contexts are drawn independently over time steps, but
+their distribution may vary depending on the class. Assump-
+tion 3 implies that the density of the covariate distribution is
+non-degenerate. Recent contextual bandit literature (without
+constraints) exploits Assumption 3 to improve the depen-
+dency of d on the regret bound (Bastani & Bayati, 2020;
+Kim et al., 2021; Bastani et al., 2021; Oh et al., 2021). The
+contexts with independent Gaussian perturbation used in
+Kannan et al. (2018); Sivakumar et al. (2020; 2022) satisfy
+the Assumption 3.
+4. Proposed Method
+In this section, we present our proposed estimator for the
+parameters {θ(j)
+⋆ , W (j)
+⋆
+: j ∈ [J]} and the proposed bandit
+policy.
+4.1. Proposed Estimator
+In sequential decision-making problems with contexts, the
+decision-maker observes the contexts for all actions, but
+the reward for only selected actions, i.e. the rewards for
+unselected actions remain missing. Kim & Paik (2019);
+
+Improved Algorithms for Multi-period Packing Problems with Bandit Feedback
+Dimakopoulou et al. (2019); Kim et al. (2021) use doubly
+robust (DR) method to handle the missing rewards for the
+linear contextual bandits. However, extensions to LinCBwK
+or LMMP problem have not explored yet.
+We adapt the DR method to the LMMP problem. For each
+n ∈ N, let τ(n) be the round when the n-th admission
+happens (recall the bandit policy allows for skipping some
+arrivals). Clearly, n ≤ τ(n) < τ(n + 1) holds. Let
+Θ⋆ :=
+�
+�
+�
+�
+θ(1)
+⋆
+...
+θ(J)
+⋆
+�
+�
+�
+� , W⋆ :=
+�
+�
+�
+�
+W (1)
+⋆
+...
+W (J)
+⋆
+�
+�
+�
+� , ˜Xk,n :=
+�
+�
+�
+�
+�
+�
+0d
+...
+x
+(jτ(n))
+k,τ(n)
+0d
+�
+�
+�
+�
+�
+�
+denote the stacked parameter vectors, and zero padded con-
+texts where x
+(jτ(n))
+k,τ(n) is located after the j − 1 of 0d vectors.
+Then the score for the ridge estimator for Θ⋆ at round τ(n)
+is:
+n
+�
+ν=1
+�
+r
+(jτ(ν))
+aτ(ν),τ(ν) − Θ⊤ ˜Xaτ(ν),ν
+�
+˜Xaτ(ν),ν
+=
+n
+�
+ν=1
+K
+�
+k=1
+I
+�
+aτ(ν) = k
+� �
+r
+(jτ(ν))
+k,τ(ν) − Θ⊤ ˜Xk,ν
+�
+˜Xk,ν,
+where Θ ∈ RJ·d. Dividing the score by the probability
+π
+(jτ(ν))
+k,τ(ν) gives the inverse probability weighted (IPW) score,
+n
+�
+ν=1
+K
+�
+k=1
+I
+�
+aτ(ν) = k
+�
+π
+(jτ(ν))
+k,τ(ν)
+�
+r
+(jτ(ν))
+k,τ(ν) − Θ⊤ ˜Xk,ν
+�
+˜Xk,ν.
+To obtain the DR score, Bang & Robins (2005); Kim et al.
+(2021) proposed to subtract the nuisance tangent space gen-
+erated by an imputed estimator ˇΘ:
+n
+�
+ν=1
+K
+�
+k=1
+I
+�
+aτ(ν) = k
+�
+π
+(jτ(ν))
+k,τ(ν)
+�
+˜X⊤
+k,ν ˇΘ − ˜X⊤
+k,νΘ
+�
+˜Xk,ν,
+from the IPW score. Then the following DR score
+n
+�
+ν=1
+K
+�
+k=1
+�
+r
+DR(jτ(ν))
+k,τ(ν)
+− ˜X⊤
+k,νΘ
+�
+˜Xk,ν,
+(2)
+is obtained where
+rDR( ˇΘ)
+k,ν
+:=I
+�
+aτ(ν)=k
+�
+π
+(jτ(ν))
+k,τ(ν)
+r
+(jτ(ν))
+k,τ(ν) +
+�
+�
+�1−I
+�
+aτ(ν)=k
+�
+π
+(jτ(ν))
+k,τ(ν)
+�
+�
+�
+˜X⊤
+k,ν ˇΘ.
+(3)
+The score (2) has a similar form with the score equation
+for the ridge estimator. The difference with the ridge es-
+timator is that it uses contexts for all actions k ∈ [K]
+with the pseudo-reward rDR( ˇΘ)
+k,ν
+which is unbiased, i.e.,
+E[rDR( ˇΘ)
+k,ν
+] = E[r
+(jτ(ν))
+k,τ(ν) ], for any given ˇΘ ∈ RJ·d. Adding
+the ℓ2 regularization norm and solving (2) leads to the DR
+estimator:
+� n
+�
+ν=1
+K
+�
+k=1
+˜Xk,ν ˜X⊤
+k,ν+IJ·d
+�−1� n
+�
+ν=1
+K
+�
+k=1
+˜Xk,νrDR( ˇΘ)
+k,τ(ν)
+�
+.
+The main advantage of the DR estimator is that it uses
+contexts from all K actions. However, in our policy, some
+π
+(jτ(ν))
+k,τ(ν) can be zero, and therefore, the pseudo-reward (3) is
+not defined. To handle this problem, we propose to introduce
+a random variable. After taking an action at round τ(ν)
+and observing the selected action aτ(ν), the decision-maker
+samples hν from the distribution:
+φk,ν:=P
+�
+hν = k| Hτ(n)
+�
+=
+�
+�
+�
+1−
+16(K−1) log( dJ
+δ )
+λmin(Fν)
+k=aτ(ν)
+16 log( dJ
+δ )
+λmin(Fν)
+k̸=aτ(ν)
+(4)
+where Fν := �ν,K
+i,k=1 ˜Xk,i ˜X⊤
+k,i+16d(K −1) log
+� dJ
+δ
+�
+IJ·d
+is the Gram matrix of contexts from ν admitted rounds
+and δ ∈ (0, 1) is the confidence level. We would like to
+emphasize that hν is sampled after observing the actions
+aτ(ν) and does not affect the policies until round τ(ν).
+Sampling the random variables hν after choosing actions
+is motivated by bootstrap methods (Efron & Tibshirani,
+1994) and resampling methods (Good, 2006). To obtain the
+unbiased pseudo-rewards similar to (3), we resample the
+action with another non-zero probabilities. The probabil-
+ities {φk,ν : k ∈ [K]} is designed to control the level of
+exploration and exploitation for future rounds based on the
+ratio of confidence level to the number of admitted rounds.
+When the minimum eigenvalue of Fν is small compared
+to log(1/δ), the distribution of hν is less concentrated on
+aτ(ν) and tends to explore other actions. As ν increases, the
+probabilities {φk,ν : k ∈ [K]} concentrates on aτ(ν), and
+the decision-maker tends to exploit.
+Since we obtain non-zero probabilities {φk,ν : k ∈ [K], ν ∈
+[n]}, we define novel unbiased pseudo-rewards:
+˜rk,ν :=I (hν =k)
+φk,ν
+r
+(jτ(ν))
+k,τ(ν) +
+�
+1− I (hν =k)
+φk,ν
+�
+˜X⊤
+k,νˇΘn, (5)
+where the imputation estimator ˇΘn is an IPW estimator with
+
+Improved Algorithms for Multi-period Packing Problems with Bandit Feedback
+new probabilities:
+ˇΘt :=A−1
+n
+� �
+ν∈Ψn
+K
+�
+k=1
+I (hν =k)
+φk,ν
+˜Xk,νr
+(jτ(ν))
+k,τ(ν)
++
+�
+ν /∈Ψn
+˜Xaτ(ν),νr
+(jτ(ν))
+aτ(ν),τ(ν)
+�
+,
+An :=
+�
+ν∈Ψn
+K
+�
+k=1
+I (hν =k)
+φk,ν
+˜Xk,ν ˜X⊤
+k,ν
++
+�
+ν /∈Ψn
+˜Xaτ(ν),ν ˜X⊤
+aτ(ν),ν + IJ·d,
+Ψn :=
+�
+ν ∈ [n] : hν = aτ(ν)
+�
+.
+The set Ψn is introduced because we cannot observe
+I(hν=k)
+φk,ν
+r
+(jτ(ν))
+k,τ(ν) in case of hν ̸= aτ(ν). In other words, we
+use the pseudo-rewards in (5) only at the rounds that satisfy
+hν = aτ(ν). Then our estimator with n admitted samples is
+defined as
+�Θn :=V −1
+n
+�
+�
+�
+�
+ν∈Ψn
+K
+�
+k=1
+˜Xk,ν˜rk,ν +
+�
+ν /∈Ψn
+˜Xaτ(ν),νr
+(jτ(ν))
+aτ(ν),τ(ν)
+�
+�
+�
+Vn :=
+�
+ν∈Ψn
+K
+�
+k=1
+˜Xk,ν ˜X⊤
+k,ν +
+�
+ν /∈Ψn
+˜Xaτ(ν),ν ˜X⊤
+aτ(ν),ν +IJ·d.
+(6)
+Analogous to the construction of (6), we can also define the
+estimator for the resource consumption parameters {W (j)
+⋆
+:
+j ∈ [J]},
+�
+Wn :=V −1
+n
+� �
+ν∈Ψn
+K
+�
+k=1
+˜Xk,ν ˜b⊤
+k,ν+
+�
+ν /∈Ψn
+˜Xaτ(ν),νb
+(jτ(ν))⊤
+aτ(ν)τ(ν)
+�
+,
+(7)
+where the pseudo-consumption vectors and the imputation
+estimator are
+˜bk,ν := I (hν =k)
+φk,ν
+b
+(jτ(ν))
+aτ(ν),τ(ν)+
+�
+1− I (hν =k)
+φk,ν
+�
+ˇ
+W⊤
+n ˜Xk,ν,
+ˇ
+Wn := A−1
+n
+� �
+ν∈Ψn
+K
+�
+k=1
+I (hν =k)
+φk,ν
+˜Xk,ν
+�
+b
+(jτ(ν))
+k,ν
+�⊤
++
+�
+ν /∈Ψn
+˜Xaτ(ν),ν
+�
+b
+(jτ(ν))
+aτ(ν),τ(ν)
+�⊤
+�
+.
+The two estimators use the novel Gram matrix Vn defined
+in (6) consist of contexts from all K actions. Now, we
+present estimation error bounds normalized by the novel
+Gram matrix Vn.
+Theorem 4.1. (Self-normalized bound for the estimator)
+Suppose Assumptions 1 and 2 hold. For each t ∈ [T],
+denote nt the number of admitted arrivals until round t and
+Ψnt := {ν ∈ [nt] : hν = aτ(ν)}, where hν is defined in
+(4). Suppose Fnt := �nt
+ν=1
+�K
+k=1 ˜Xk,ν ˜X⊤
+k,ν + 16d(K −
+1) log Jd
+δ IJ·d satisfies
+λmin(Fnt)≥12Kd
+� nt
+�
+ν=1
+48(K−1) log
+� Jd
+δ
+�
+λmin(Fν)
++2 log Jd
+δ
+�
+,
+(8)
+for δ ∈ (0, 1). For each r ∈ [m] , let �
+Wnt,r and W⋆,r
+be the r-th column of �
+Wnt and W⋆, respectively. Denote
+βσ(δ) := 8
+√
+Jd + 96σ
+�
+Jd log 4
+δ. Then with probability
+at least 1 − 4(m + 1)δ,
+����Θnt − Θ∗���
+Vnt
+≤βσr(δ),
+max
+r∈[m]
+����
+Wnt,r − W⋆,r
+���
+Vnt
+≤βσb(δ).
+(9)
+The widely used self-normalized bound in Abbasi-Yadkori
+et al. (2011) uses the Gram matrix consisting of selected
+contexts only, while our bounds are normalized by Vnt
+This change in the Gram matrix enables us to develop a
+fast convergence rate. The condition (8) is required for
+the eigenvalues of the Gram matrix Fnt to be large so that
+the probability φaτ(ν),ν is large and the estimators use the
+pseudo rewards and pseudo consumption vectors for most
+of the rounds. We show in Lemma 5.3 that the condition (8)
+requires at most rounds logarithmic in T, and does not affect
+the main order of the regret bound.
+Using the novel estimators, we define the estimates for
+utility and resource consumption. Denote C(j)
+t
+:= {s ∈ [t] :
+js = j} and
+�u(j)
+k,t :=
+���C(j)
+t
+���
+−1 �
+s∈C(j)
+t
+��
+�θ(j)
+t−1
+�⊤
+x(j)
+k,s − c(j)
+k,s
+�
+,
+�b(j)
+k,t :=
+���C(j)
+t
+���
+−1 �
+s∈C(j)
+t
+�
+�
+W (j)
+t−1
+�⊤
+x(j)
+k,s.
+(10)
+The estimates (10) use the average of contexts in the same
+class to estimate the expected value over the context dis-
+tribution. In this way, the decision-maker effectively uses
+previous contexts in all rounds including the skipped rounds.
+Next, we establish the convergence rate for the estimators
+�u(j)
+k,t and �b(j)
+k,t.
+Theorem 4.2. (Convergence rate for the estimates) Suppose
+Assumptions 1-3 hold. Denote the expected utility u⋆(j)
+k
+:=
+E(xk,ck)∼Fj
+��
+θ(j)
+⋆
+�⊤
+xk − ck
+�
+and consumption b⋆(j)
+k
+:=
+Exk∼Fj
+��
+W (j)
+⋆
+�⊤
+xk
+�
+. Set γt,σ(δ) :=
+16√
+J log(JKT )
+√
+t
++
+4
+√
+2βσ(δ)
+√nt
+, where nt is the number of admitted arrivals until
+
+Improved Algorithms for Multi-period Packing Problems with Bandit Feedback
+round t and βσ(δ) is defined in Theorem 4.1. Suppose
+t ≥ 8dα−1p−1
+min log JT, δ ∈ (0, T −1) and Fnt satisfies (8).
+Then with probability at least 1 − 4(m + 1)δ − 7T −1,
+�
+�
+�
+�
+J
+�
+j=1
+pj max
+k∈[K]
+���u⋆(j)
+k
+− �u(j)
+k,t+1
+���
+2
+≤ γt,σr(δ),
+�
+�
+�
+�
+J
+�
+j=1
+pj max
+k∈[K]
+���b⋆(j)
+k
+− �b(j)
+k,t+1
+���
+2
+∞ ≤ γt,σb(δ).
+(11)
+The convergence rate of the estimates is O(
+√
+Jdn−1/2
+t
+). In
+deriving the fast rate, the novel Gram matrix Vnt plays a
+significant role. To prove Theorem 4.2, we bound the sum
+of squared maximum prediction error as follows:
+1
+nt
+�
+s∈Ψnt
+max
+k∈[K]
+��
+θ(j)
+⋆ −�θ(j)
+t
+�⊤
+x(j)
+k,s
+�2
+= 1
+nt
+�
+s∈Ψnt
+max
+k∈[K]
+�
+θ(j)
+⋆ −�θ(j)
+t
+� �
+x(j)
+k,sx(j)
+k,s
+�⊤�
+θ(j)
+⋆ −�θ(j)
+t
+�
+≤ 1
+nt
+�
+s∈Ψnt
+�
+θ(j)
+⋆ −�θ(j)
+t
+� � K
+�
+k=1
+x(j)
+k,sx(j)
+k,s
+�⊤�
+θ(j)
+⋆ −�θ(j)
+t
+�
+≤ 1
+nt
+���θ(j)
+⋆ −�θ(j)
+t
+���
+2
+Vnt
+.
+Such a bound is not available if the Gram matrix is con-
+structed using only contexts corresponding to selected ac-
+tions. In this way, we obtain faster convergence rate for the
+estimates for utility and consumption vectors.
+4.2. Proposed Algorithm
+Let (K + 1)-th action denote skipping the arrival and
+π(j)
+K+1,t := P (Skip the round t| Ht) denote the probabil-
+ity of skipping the arrival. Since the decision-maker must
+choose an action or skip the round, we have �K+1
+k=1 π(j)
+k,t = 1.
+When the decision-maker skips round t, we set x(j)
+K+1,t := 0,
+c(j)
+K+1,t := 0, and b(j)
+K+1,t := 0. In round t, the randomized
+bandit policy is given by the optimal solution of the follow-
+ing optimization problem:
+max
+π(jt)
+k,t
+K+1
+�
+k=1
+π(jt)
+k,t
+�
+�u(jt)
+k,t + γt−1,σr(δ)
+√pjt
+I (k ∈ [K])
+�
+,
+s.t.
+K+1
+�
+k=1
+π(jt)
+k,t
+�
+�b(jt)
+k,t − γt−1,σb(δ)
+√pjt
+1m
+�
+≤ ρt ∨ 0,
+K+1
+�
+k=1
+π(jt)
+k,t = 1,
+π(jt)
+k,t ≥ 0,
+∀k ∈ [K + 1],
+(12)
+Algorithm 1 Allocate to the Maximum First algorithm
+(AMF)
+INPUT: confidence lengths γθ, γb > 0, confidence level
+δ ∈ (0, 1).
+Initialize F0 := 16d(K − 1) log Jd
+δ IJ·d, ρ1 := ρ, �Θ0 :=
+0J·d, �
+W0 := 0J·d×m
+for t = 1 to T do
+Observe arrival (jt, {x(jt)
+k,t , c(jt)
+k,t }k∈[K]).
+if Ft−1 does not satisfy (8) then
+Take action at = arg maxk∈[K] ρ∥�b(jt)
+k,t ∥−1
+∞ .
+else
+Compute �u(jt)
+k,t and �b(jt)
+k,t with �θ(jt)
+t−1 and �
+W(jt)
+t−1.
+Compute ˜u(jt)
+k,t := �u(jt)
+k,t +
+γθ
+√nt and ˜b(jt)
+k,t := �b(jt)
+k,t −
+γb
+√nt 1m.
+Take action at with the policy �π(jt)
+1,t , . . . , �π(jt)
+K+1,t de-
+fined in (13).
+end if
+if at ∈ [K] then
+Observe r(jt)
+at,t and b(jt)
+at,t, then estimate �Θt and �
+Wt
+as in (6) and (7), respectively.
+Update Ft = Ft−1 + �K
+k=1 ˜Xk,t ˜X⊤
+k,t.
+end if
+Update available resource ρt+1 = ρt + ρ − b(jt)
+at,t.
+if �t
+s=1 b(js)
+as,s ≥ Tρ then
+Exit
+end if
+end for
+where ρt := tρ − �t−1
+s=1 b(js)
+as,s is the difference between
+the used resources and planned budget until round t. The
+algorithm is optimistic in that it uses upper confidence bound
+(UCB) in rewards and lower confidence bound (LCB) in
+consumption while it regulates the consumption to be less
+than tρ with ρt. In this way, the problem (12) balances
+between admitting the arrivals and saving the resources for
+later use. Next, we show that the optimal solution (12) is
+available in a closed-form.
+Lemma 4.3. (Optimal policy for bandit) Let ˜u(jt)
+k,t
+:=
+�u(jt)
+k,t
++ p−1/2
+jt
+γt−1,σr(δ)I (k ∈ [K]) and ˜b(jt)
+k,t (r)
+:=
+�b(jt)
+k,t (r) − p−1/2
+jt
+γt−1,σb(δ), for r ∈ [m]. For i ∈ [K + 1],
+let ˜u(jt)
+k⟨i⟩,t be an sequence of ordered variables of ˜u(jt)
+k,t in
+decreasing order, i.e. ˜u(jt)
+k⟨1⟩,t ≥ ˜u(jt)
+k⟨2⟩,t ≥ · · · ≥ ˜u(jt)
+k⟨K+1⟩t.
+When there is a tie between ˜u(jt)
+k⟨i⟩,t and ˜u(jt)
+k⟨i+1⟩,t, the index
+k⟨i⟩ with the higher value for
+�
+� min
+r∈[m]
+ρt(r) ∨ 0 − �i−1
+h=1 �π(jt)
+k⟨h⟩,t˜b(jt)
+k⟨h⟩,t(r)
+˜b(jt)
+k⟨h⟩,t(r)
+�
+�
+
+Improved Algorithms for Multi-period Packing Problems with Bandit Feedback
+goes first. Then the policy defined as,
+�π(jt)
+k⟨1⟩,t =
+�
+� min
+r∈[m]
+ρt(r)∨0
+˜b(jt)
+k⟨1⟩,t(r)
+�
+� ∧ 1,
+�π(jt)
+k⟨i⟩,t =
+�
+�min
+r∈[m]
+ρt(r)∨0−�i−1
+h=1�π(jt)
+k⟨h⟩,t˜b(jt)
+k⟨h⟩,t(r)
+˜b(jt)
+k⟨i⟩,t(r)
+�
+�
+∧
+�
+1 −
+i−1
+�
+h=i
+�π(jt)
+k⟨h⟩,t
+�
+, ∀i ∈ [2, K + 1],
+(13)
+is the optimal solution to (12).
+Since the objective function of (12) is linear, we can obtain
+the maximum value by permuting the objective coefficients
+in decreasing order and allocating the greatest possible prob-
+ability value in decreasing order of the objective coefficients.
+Note that �π(jt)
+k,t is automatically set to zero when the utility
+is negative. This is because of the probability of skipping
+the arrival, �π(jt)
+K+1,t = 1−�l−1
+h=1 �π(jt)
+k⟨h⟩,t, when ˜u(jt)
+K+1,t is the
+l-th largest weighted utility function and all the remaining
+probability is allocated to �π(jt)
+K+1,t. Therefor, the probabili-
+ties for actions k with ˜u(jt)
+k,t < ˜u(jt)
+K+1,t := 0 are all zero.
+Our proposed algorithm, Allocate to the Maximum First
+(AMF) is presented in Algorithm 1. The algorithm first ex-
+plores with the least consumption action until the eigenvalue
+condition for the estimator (8) holds. In each round of ex-
+ploration, the Gram matrix of all actions is added to Fnt,
+and any choice of action increases the eigenvalue of Fnt.
+Once the condition (8) holds, the algorithm solves the prob-
+lem (12) by computing the closed-form policy (13). The
+computational complexity of our algorithm is discussed in
+Appendix A.4.
+5. Regret Analysis
+In this section, we present our regret bound and regret anal-
+ysis for the AMF algorithm.
+Theorem 5.1. (Regret bound of AMF) Suppose Assumptions
+1-3 hold. Let Mα,p,T := 1152α−2p−2
+min log T + 96α−1p−1
+min
+and Cσ(δ) := 8
+√
+2(8 + 96σ
+�
+log 4
+δ ). Suppose T and
+ρ satisfies T ≥ 8dα−1p−1
+min log JdT, and ρ ≥
+�
+Jd/T.
+Setting γθ = 16√J log JKT + 4
+√
+2βσr(δ) and γb =
+16√J log JKT + 4
+√
+2βσb(δ), the regret bound of AMF is
+R�π
+T ≤
+�
+2+ OPT
+ρT
+��
+4d log JdT
+αpmin
++2dMα,p,T log Jd
+δ +15
++
+�
+96
+�
+log JKT +3Cσr∨σb(δ)
+��
+JdT log T +10mT 3δ
+�
+.
+For δ ∈ (0, m−1T −3), the regret bound is
+R�π
+T = O
+�OPT
+Tρ
+�
+JdT log mJKT log T
+�
+.
+(14)
+The regret bound (14) holds when the hyperparameter δ =
+m−1T −3, which requires the knowledge of T. However,
+in practice, selecting another value of δ does not affect the
+performance of the algorithm. We provide the discussion on
+the sensitivity to the hyperparameter choice in Section 6.3.
+Setting B = Tρ, the main term of the regret bound is
+˜O(OPT/B
+√
+JdT) for B = Ω(
+√
+JdT). The sublinear
+dependence of the regret bound on J, d, and T is a direct
+consequence of the improved ˜O(
+�
+Jd/nt) convergence rate
+for the parameter estimates. Agrawal & Devanur (2016)
+establish a regret bound Rπ
+T = ˜O(OPT/B · d
+√
+T) for the
+LinCBwK when B = Ω(
+√
+dT 3/4). Our bound for LMMP
+(which subsumes LinCBwK as a special case) is improved
+by a
+√
+d factor, and is valid under budget constraints that
+relaxed from Ω(
+√
+dT
+3
+4 ) to Ω(
+√
+dT
+1
+2 ).
+For the proof of the regret bound, we first present the lower
+bound of the reward obtained by our algorithm.
+Lemma 5.2. Let ˜u(j)
+k,t and �b(j)
+k,t be the estimates defined in
+(10). Denote �π the policy of AMF. Define the good events,
+Et :=
+�
+˜u(j)
+k,t and �b(j)
+k,t satisfies (11).
+�
+,
+Mt := {Fnt satisfies (8).} ,
+(15)
+and Gt := Et ∩ Mt−1. Let τ be the stopping time for the
+algorithm and ξ := inft∈[T ] {Mt−1 ∩ {ρt > 0}} be the
+starting time after the exploration for condition (8). Then,
+the total reward
+E
+� T
+�
+t=1
+R�π
+t
+�
+≥ OPT
+T
+E [τ − ξ]−
+�
+2+ OPT
+ρT
+� T
+�
+t=1
+P(Gc
+t )
+−2
+�
+1+ OPT
+ρT
+��
+�
+�
+�TE
+� T
+�
+t=1
+γt−1,σr∨σb(δ)2I (at ∈ [K])
+�
+.
+The lower bound consists of three main terms. The first
+term OP T
+T
+E [τ − ξ] relates to the time span for which the
+algorithm uses the optimal policy (13). The second term
+(2+ OP T
+ρT ) �T
+t=1P (Gc
+t ) is the sum of the probability of bad
+events Mc
+t−1 over which the minimum eigenvalue of the
+Gram matrix Fnt is not large enough for the fast conver-
+gence rate, and the event Ec
+t over which the estimator goes
+out of the confidence interval. And, the third term con-
+sists of the sum of confidence lengths for the reward and
+consumption.
+The following result bounds τ, ξ and the sum of bad events
+{Mc
+t : t ∈ [T]}.
+
+Improved Algorithms for Multi-period Packing Problems with Bandit Feedback
+Lemma 5.3. Suppose Assumptions 1-3 holds and ρ >
+�
+Jd/T Let Mα,p,T and γt,σ(δ) denote the variables
+defined in Theorem 5.1 and Theorem 4.2, respectively.
+Then, for any δ ∈ (0, 1/T 2), the starting time ξ :=
+inft∈[T ]{Mt−1 ∩ {ρt > 0}} and the stopping time τ of
+the AMF algorithm is bounded as
+E [ξ] ≤ 1+dMα,p,T log
+� Jd
+δ
+�
++T 2δ
+ρ
++ 1,
+E[T −τ]≤ 4(m + 1)Tδ + 7 + 2γ1,σb(δ)
+ρ
+,
+and for Mt defined as in (15),
+T
+�
+t=1
+P
+�
+Mc
+t−1
+�
+≤ T 2δ + dMα,p,T log
+�Jd
+δ
+�
+.
+The regret bound follows from bounding the probability of
+Ec
+t with Theorem 4.2 and showing that the sum of square
+of γt,σ(δ) is O(Jd log T). The bound holds because the
+summation of γt,σ(δ)2 = ˜O( Jd
+nt ) over the rounds that at ∈
+[K] happens is �nT
+n=1 O(Jd/n) = O(Jd log T).
+6. Numerical results
+We report the cumulative regrets for given budgets. For the
+computation of the regret, we use the following settings. For
+each round t ∈ [T] there exists the optimal action whose
+reward is 1 with consumption ρ, while the reward of other
+actions is less than 1 and the consumption is possibly greater
+than ρ. In this case, we can compute the instantaneous regret
+by subtracting the reward of a selected action from 1. Detail
+settings of the parameter and contexts are in Appendix A.1.
+6.1. Regret R�π
+T as a function of d
+Figure 1 plots log(R�π
+T ) vs. log(d) for a single-class (J =
+1) LMMP for T = {5000, 20000} and the budget B =
+√
+dT, where our ˜O( OP T
+B
+√
+JdT) regret bound implies that
+log(R�π
+T ) is constant over d. The regression line on the
+plot is nearly flat and the slope of the best fit line is 0.136
+(resp. 0.008) for T = 5000 (resp. T = 20000). The weak
+increase in T = 5000 is captured by the O(d log JdmT)
+term in our bound, which diminishes for large T.
+6.2. Comparison of AMF with OCO
+In order to compare AMF with OCO (Agrawal & Devanur,
+2016), we set the costs c(1)
+k,t = 0 and J = 1. The hyperpa-
+rameters for AMF were set to γθ = 1, γb = 1 and δ = 0.01.
+Figure 2(a) (resp. (b)) plots the cumulative regret of the
+two algorithms with budget B =
+√
+dT
+3
+4 (resp. B =
+√
+dT).
+Note that OCO requires a minimum budget B =
+√
+dT
+3
+4
+whereas AMF requires a lower minimum budget of B =
+Figure 1. Logarithm of cumulative regret of the proposed AMF
+algorithm on various dimension d when the per-period budget is
+ρ =
+�
+d/T. The gray (resp. black) line is the best fit line on the
+points when T = 5000 (resp. T = 20000).
+(a) Regret comparison with budget B =
+√
+dT 3/4
+(b) Regret comparison with budget B =
+√
+dT
+Figure 2. Regret of AMF and OCO algorithms for K = 20 and
+m = 20. The line and shade represent the average and standard
+deviation based on 20 independent experiments. Additional results
+on different K and m are in Section A.2.
+√
+dT. The regret lines cross because AMF is allowed to skip
+arrivals whereas OCO does not skip arrivals. The sudden
+bend points at the end of the round in OCO show that it
+runs out of budget and has regret = 1. In all cases, our
+algorithm performs better and the performance gap increases
+as d increases. Note that regret plot for OCO never flattens
+out for most cases, where the regret of AMF flattens as t
+increases. This is because our new estimator, that uses
+contexts from all actions with unbiased pseudo-rewards (5)
+for unselected actions, has significantly faster convergence
+rate as compared with the estimator used in OCO.
+6.3. Sensitivity Analysis
+Our proposed AMF algorithm has three hyperparameters: γθ,
+γb and δ. The choice of hyperparameters is not sensitive
+because the effect of γθ and γb diminishes fast by n−1/2
+t
+term and our policy finds the order of the utilities rather than
+their absolute values. For δ, which controls the sampling
+probabilities (4) in estimators and the exploration rounds
+in (8), it also has small effect. This is because the minimum
+eigenvalue of Fnt increases in Ω(nt)-rate and reduces the
+effect of log 1
+δ terms in (4) and (8). Therefore, our algorithm
+guarantees the similar performance for other hyperparame-
+ters than specified in Theorem 5.1. For details including the
+numerical results and specific recommendations for choos-
+ing the hyperparameters, see Appendix A.3.
+
+7.5
+7.0
+6.5
+.
+6.0
+5.5 -
+O
+5.0
+1.0
+1.5
+2.0
+2.5 
+3.0
+log d2000
+OCO
+1500
+AMF
+1000 -
+500 -
+0 
+0
+2000
+4000
+6000
+8000
+10000
+Decision points3500
+OCo
+3000
+AMF
+2500
+2000
+1500 -
+1000-
+500
+0
+0
+2000
+4000
+6000
+8000
+10000
+Decision pointsImproved Algorithms for Multi-period Packing Problems with Bandit Feedback
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+ities. Foundations and Trends® in Machine Learning, 8
+(1-2):1–230, 2015.
+
+Improved Algorithms for Multi-period Packing Problems with Bandit Feedback
+(a) Regret comparison under K = 10 and m = 10
+(b) Regret comparison under K = 20 and m = 10
+(c) Regret comparison under K = 10 and m = 20
+(d) Regret comparison under K = 20 and m = 20
+Figure 3. Regret comparison of AMF and OCO algorithms under B = dT 3/4. The line and shade represent the average and standard
+deviation based on 20 repeated experiments.
+A. Supplementary for Experiments
+A.1. Settings of Parameters and Contexts for Regret Computation
+For numerical experiments, we devise a setting where explicit regret computation is available. We set J = 1 and c(1)
+k,t = 0 for
+OCO to be compatible with the setting. For x ∈ R+, let ⌈x⌉ be the smallest integer greater than equal to x. For parameters,
+we set θ⋆ = (−1, · · · , −1, ⌈d/2⌉−1, · · · , ⌈d/2⌉−1) and
+W⋆ =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+ρ⌈d/2⌉−1
+· · ·
+ρ⌈d/2⌉−1
+...
+· · ·
+...
+ρ⌈d/2⌉−1
+· · ·
+ρ⌈d/2⌉−1
+ρ
+· · ·
+ρ
+...
+...
+...
+ρ
+· · ·
+ρ
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+,
+where the ⌈d/2⌉−1 and ρ⌈d/2⌉−1 terms are in the first ⌈d/2⌉ entries.
+For contexts, we set the optimal action as
+(0, · · · , 0, 1, · · · , 1), and for other actions, we set (U0,0.05, · · · , U0,0.05, U0,−0.05, · · · , U0,−0.05),where Ua,b the uniform
+random variable supports on [a, b]. Then we have the optimal arm with reward 1 and consumption ρ, while other arms have
+reward less than 1 and consumption more than ρ.
+A.2. Additional Results on Regret Comparison.
+Figure 3 (a)-(d) show the regret comparison of AMF and OCO on different terms of K = 10, 20, m = 10, 20, and B = dT 3/4.
+Similar to the results in Figure 2(a), our algorithm has less regret than OCO in all cases, especially at the end of the rounds.
+The crossing line occurs when our algorithm skips in the middle round when ρt < 0 while OCO does not skip until the
+inventory runs out.
+Figure 4 (a)-(d) show the regret of AMF and OCO algorithm on various K = 10, 20 and m = 10, 20 with budget B =
+√
+dT.
+Even in the smaller budget, our algorithm AMF does not run out the inventory and gains more reward than OCO. The gap of
+the performance tends to be larger than B =
+√
+dT
+3
+4 case.
+
+1000
+d=10, K=20,m=20
+OCO
+AMF
+800
+600
+400
+200 :
+0 ·
+0
+2000
+4000
+6000
+8000
+10000
+Decision points1000
+d=20, K=20, m=20
+OCO
+AMF
+800
+600
+400
+200 :
+0 
+0
+2000
+4000
+6000
+8000
+10000
+Decision points1000
+d=10,K=10, m=10
+OCO
+AMF
+800
+600
+400
+200
+0 
+0
+2000
+4000
+6000
+8000
+10000
+Decision points1000
+d=20,K=10,m=10
+OCO
+AMF
+800 
+600 -
+400 -
+200 -
+0
+0
+2000
+4000
+6000
+8000
+10000
+Decision points1000
+d=10,K=20, m=10
+OCO
+AMF
+800
+600
+400
+200 :
+0 
+0
+2000
+4000
+6000
+8000
+10000
+Decision points1000
+d=20,K=20,m=10
+OCO
+AMF
+800
+600 
+400 
+200 -
+0
+0
+2000
+4000
+6000
+8000
+10000
+Decision points1000
+d=10,K=10, m=20
+OCO
+AMF
+800
+600
+400
+200 :
+0 
+0
+2000
+4000
+6000
+8000
+10000
+Decision points1000
+d=20,K=10, m=20
+OCO
+AMF
+800
+600
+400
+200 :
+0
+0
+2000
+4000
+6000
+8000
+10000
+Decision pointsImproved Algorithms for Multi-period Packing Problems with Bandit Feedback
+(a) Regret comparison under K = 10 and m = 10
+(b) Regret comparison under K = 20 and m = 10
+(c) Regret comparison under K = 10 and m = 20
+(d) Regret comparison under K = 20 and m = 20
+Figure 4. Regret comparison of AMF and OCO algorithms under B =
+√
+dT. The line and shade represent the average and standard
+deviation based on 20 repeated experiments.
+(a) On various γθ
+(b) On various γb
+(c) On various δ
+Figure 5. The reward and inventory of AMF on various hyperparameters γθ, γb and δ. The solid (resp. dashed) line represents the reward
+(resp. inventory). The line and shade represent the average and standard deviation based on 10 repeated experiments, respectively.
+A.3. Sensitivity Analysis
+In this experiment, we present the sensitivity of our algorithm to various hyperparameters. The number of classes is J = 3
+with a uniform prior p = (1/3, 1/3, 1/3)⊤ and every d = 5 elements of K = 10 contexts are generated from the uniform
+distribution on [ kj
+KJ − 1, kj
+KJ + 1] for k ∈ [K] and j ∈ [J]. The costs are generated from the uniform distribution on
+[ k(J−j+1)−1
+KJ
+, k(J−j+1)+1
+KJ
+] for k ∈ [K] and j ∈ [J]. Each element of θ(j)
+⋆
+and W (j)
+⋆
+is generated from U0,1 and fixed
+throughout the experiment. The generated rewards and consumption vectors are not truncated to one to impose more
+variability, because our algorithm does not show apparent sensitivity on bounded rewards and consumption vectors. The
+budget is ρ = dT −1/2 with a time horizon of T = 2000.
+In our algorithm, there are three hyperparameters: (i) a confidence bound for the reward γθ, (ii) a confidence bound for
+the consumption γb and (iii) confidence level δ which affects the minimum eigenvalue condition (8). Figure 5(a) and 5(b)
+show the reward and inventory of our algorithm on various γθ ∈ {0.01, 0.1, 1} and γb ∈ {0.01, 0.1, 1}. Outside of the
+hyperparameter regions, the variability of the reward and the inventory of the algorithm are hardly visible. The algorithm
+consumes the budget earlier than previous experiments because the consumption vector is not bounded to 1. As γθ and
+γb increase the algorithm is more optimistic and admits the arrival more often, which leads to faster consumption of the
+resource. Increasing γθ has small effect on the inventory because the algorithm automatically skips when ρt < 0, i.e., when
+the consumption is too fast. However, when γb increases, the LCB of consumption is small and the algorithm uses more
+
+2500
+d=10,K=10,m=10
+OCO
+AMF
+2000
+1500 -
+1000 -
+500 -
+0
+0
+2000
+4000
+6000
+8000
+10000
+Decision points2500
+d=20,K=10,m=10
+OCO
+AMF
+2000
+1500 -
+1000 -
+500 -
+0
+0
+2000
+4000
+6000
+8000
+10000
+Decision points2500
+d=10,K=20,m=10
+OCO
+AMF
+2000
+1500 -
+1000 -
+500 -
+0
+0
+2000
+4000
+6000
+8000
+10000
+Decision points2500
+d=20,K=20,m=10
+OCO
+AMF
+2000
+1500 -
+1000 -
+500 -
+0
+0
+2000
+4000
+6000
+8000
+10000
+Decision points2500
+d=10,K=10,m=20
+OCO
+AMF
+2000
+1500 -
+1000 -
+500 -
+0
+0
+2000
+4000
+6000
+8000
+10000
+Decision points2500
+d=20, K=10,m=20
+OCO
+AMF
+2000
+1500 -
+1000 -
+500 -
+0
+0
+2000
+4000
+6000
+8000
+10000
+Decision points2500
+d=10,K=20,m=20
+OCO
+AMF
+2000
+1500 -
+1000 -
+500 -
+0
+0
+2000
+4000
+6000
+8000
+10000
+Decision points2500
+d=20, K=20, m=20
+OCO
+AMF
+2000
+1500 -
+1000 -
+500 -
+0
+0
+2000
+4000
+6000
+8000
+10000
+Decision points700
+600
+500
+400
+300
+200
+Ye=0.01
+100
+Ye=0.10
+0
+Ye=1.00
+0
+250
+500
+750
+1000
+1250
+1500
+1750
+2000
+Decision points700
+600
+500
+400
+300
+200 
+Yb=0.01
+100
+Yb=0.10
+0
+Yb=1.00
+0
+250
+500
+750
+1000
+1250
+1500
+1750
+2000
+Decision points700
+600
+500
+400
+300
+200
+6=1e-01
+§=1e-04
+100
+6=1e-07
+0
+0
+250
+500
+750
+1000
+1250
+1500
+1750
+2000
+Decision pointsImproved Algorithms for Multi-period Packing Problems with Bandit Feedback
+resource than tρ. For the specific value of the hyperparameters we recommend to use grid search on γθ × γb ∈ [0, 1]2 to
+maximize the reward.
+Figure 5(c) shows the reward and inventory of AMF on various δ ∈ {10−1, 10−4, 10−7}. When δ ≥ 10−1 (resp. δ ≤ 10−7)
+the reward and inventories are same with δ = 10−1 (resp. δ = 10−7). As δ decreases, the threshold for condition (8)
+increases and the algorithm explores more with minimum possible consumption. This results in the slower consumption
+of the resource. However, we recommend using δ = 0.1, which is greater than the specified value in Theorem 5.1 for the
+algorithm to start using its policy in earlier rounds.
+A.4. Computational Complexity of AMF
+The computational complexity of our algorithm is ˜O(d3mKT + Jd3T) where the main order occurs from updating the
+estimators and computing the eigenvalues of J symmetric positive-definite matrix Fnt. Note that Computing estimators
+does not depend on J because the algorithm updates only jt-th variables for each t ∈ [T].
+B. Missing Proofs
+B.1. Proof of Theorem 4.1
+Proof. Because the construction of �Θt and �
+Wt is the same, the bound for the �
+Wt follows immediately from the bound for
+�Θt by replacing {r
+(jτ(ν))
+aτ(ν),τ(ν) : ν ∈ [nt]} with m entries of {b
+(jτ(ν))
+aτ(ν),τ(ν) : ν ∈ [nt]}. Thus, it is sufficient to prove the bound
+for �Θt.
+Step 1. Estimation error decomposition:
+Let us fix t ∈ [T] throughout the proof. For each ν ∈ [nt] and k ∈ [K], denote
+Xk,ν := ˜Xk,ν ˜X⊤
+k,ν. Then we can write
+Vnt :=
+�
+ν∈Ψnt
+K
+�
+k=1
+Xk,ν +
+�
+ν /∈Ψnt
+Xaτ(ν),ν + IJ·d,
+Ant :=
+�
+ν∈Ψnt
+K
+�
+k=1
+I (hν = k)
+φk,ν
+Xk,ν +
+�
+ν /∈Ψnt
+Xk,ν + IJ·d.
+Denote the errors ˜ηk,ν := ˜rk,ν − ˜X⊤
+k,νΘ⋆ and ηk,ν := r
+(jτ(ν))
+k,τ(ν) − ˜X⊤
+k,νΘ⋆. By the definition of the estimator �Θnt,
+����Θnt − Θ∗���
+Vnt
+=
+������
+V −1/2
+nt
+�
+�
+�−Θ∗ +
+�
+ν∈Ψnt
+K
+�
+k=1
+˜ηk,ν ˜Xk,ν +
+�
+ν /∈Ψnt
+ηk,ν ˜Xaτ(ν),ν
+�
+�
+�
+������
+2
+≤ λmax
+�
+V −1/2
+nt
+�
+∥Θ∗∥2 +
+������
+V −1/2
+nt
+�
+�
+�
+�
+ν∈Ψnt
+K
+�
+k=1
+˜ηk,ν ˜Xk,ν +
+�
+ν /∈Ψnt
+ηk,ν ˜Xaτ(ν),ν
+�
+�
+�
+������
+2
+≤
+√
+Jd +
+������
+V −1/2
+nt
+�
+�
+�
+�
+ν∈Ψnt
+K
+�
+k=1
+˜ηk,ν ˜Xk,ν +
+�
+ν /∈Ψnt
+ηk,ν ˜Xaτ(ν),ν
+�
+�
+�
+������
+2
+,
+(16)
+where and the last inequality holds because
+���θ(j)
+⋆
+���
+2 ≤
+√
+d. Plugging in ˜rk,ν defined in (5),
+˜ηk,ν ˜Xk,ν =
+�
+1 − I (hν = k)
+φk,ν
+�
+˜Xk,ν ˜X⊤
+k,ν
+�ˇΘt − Θ∗�
++ I (hν = k)
+φk,ν
+ηk,ν ˜Xk,ν,
+
+Improved Algorithms for Multi-period Packing Problems with Bandit Feedback
+and the term �
+ν∈Ψnt
+�K
+k=1 ˜ηk,ν ˜Xk,ν is decomposed as,
+�
+ν∈Ψnt
+K
+�
+k=1
+˜ηk,ν ˜Xk,ν =
+�
+ν∈Ψnt
+K
+�
+k=1
+��
+1 − I (hν = k)
+φk,ν
+�
+Xk,ν
+�ˇΘt − Θ∗�
++ I (hν = k)
+φk,ν
+ηk,ν ˜Xk,ν
+�
+.
+(17)
+By definition of the IPW estimator ˇΘt,
+�
+ν∈Ψnt
+K
+�
+k=1
+�
+1 − I (hν = k)
+φk,ν
+�
+Xk,ν
+�ˇΘt − Θ∗�
+=
+�
+�
+�
+�
+ν∈Ψnt
+K
+�
+k=1
+�
+1 − I (hν = k)
+φk,ν
+�
+Xk,ν
+�
+�
+� A−1
+nt
+�
+�−Θ∗ +
+�
+ν∈Ψnt
+K
+�
+k=1
+I (hν = k)
+φk,ν
+ηk,ν ˜Xk,ν +
+�
+ν /∈Ψnt
+ηaτ(ν),ν ˜Xaτ(ν),ν
+�
+�
+= (Vnt − Ant) A−1
+nt
+�
+�−Θ∗ +
+�
+ν∈Ψnt
+K
+�
+k=1
+I (hν = k)
+φk,ν
+ηk,ν ˜Xk,ν +
+�
+ν /∈Ψnt
+ηaτ(ν),ν ˜Xaτ(ν),ν
+�
+�
+:= (Vnt − Ant) A−1
+nt (−Θ∗ + Snt) ,
+(18)
+where
+Snt :=
+�
+ν∈Ψnt
+K
+�
+k=1
+I (hν = k)
+φk,ν
+ηk,ν ˜Xk,ν +
+�
+ν /∈Ψnt
+ηaτ(ν),ν ˜Xaτ(ν),ν,
+then,
+����Θnt − Θ∗���
+Vnt
+≤
+(16)
+√
+Jd +
+������
+V −1/2
+nt
+�
+�
+�
+�
+ν∈Ψnt
+K
+�
+k=1
+˜ηk,ν ˜Xk,ν +
+�
+ν /∈Ψnt
+ηk,ν ˜Xaτ(ν),ν
+�
+�
+�
+������
+2
+=
+(17),(18)
+√
+Jd +
+���V −1/2
+nt
+�
+(Vnt − Ant) A−1
+nt (−Θ∗ + Snt) + Snt
+����
+2 .
+By triangular inequality,
+����Θt − Θ∗���
+Vnt
+≤
+√
+Jd +
+���V −1/2
+nt
+�
+(Vnt − Ant) A−1
+nt (−Θ∗ + Snt) + Snt
+����
+2
+≤
+√
+Jd +
+���V −1/2
+nt
+(Vnt − Ant) A−1
+nt (−Θ∗ + Snt)
+���
+2 + ∥Snt∥V −1
+nt
+≤
+√
+Jd +
+���
+�
+V 1/2
+nt A−1
+nt V 1/2
+nt
+− IJ·d
+� �
+−V −1/2
+nt
+Θ∗ + V −1/2
+t
+Snt
+����
+2 + ∥Snt∥V −1
+nt
+≤
+√
+Jd +
+���V 1/2
+nt A−1
+nt V 1/2
+nt
+− IJ·d
+���
+2
+���−V −1/2
+nt
+Θ∗ + V −1/2
+t
+Snt
+���
+2 + ∥Snt∥V −1
+nt
+≤
+√
+Jd +
+���V 1/2
+nt A−1
+nt V 1/2
+nt
+− IJ·d
+���
+2
+�√
+Jd + ∥Snt∥V −1
+nt
+�
++ ∥Snt∥V −1
+nt
+=
+����V 1/2
+nt A−1
+nt V 1/2
+nt
+− IJ·d
+���
+2 + 1
+� �√
+Jd + ∥Snt∥V −1
+nt
+�
+.
+(19)
+Step 2. Bounding the ∥ · ∥2 of the matrix in (19)
+We claim that
+V 1/2
+nt A−1
+nt V 1/2
+nt
+⪰ 1
+8IJ·d
+(20)
+
+Improved Algorithms for Multi-period Packing Problems with Bandit Feedback
+Define Fnt := �nt
+ν=1
+�K
+k=1 Xk,ν + 16Kd log( Jd
+δ )IJ·d. Then we have Vnt ⪯ Fnt and V 1/2
+nt A−1
+nt V 1/2
+nt
+⪰ F −1/2
+nt
+AntF −1/2
+nt
+.
+Now we decompose the matrix Ant as
+F −1/2
+nt
+AntF −1/2
+nt
+=F −1/2
+nt
+� nt
+�
+ν=1
+K
+�
+k=1
+I (hν = k)
+φk,ν
+Xk,ν + IJ·d
+�
+F −1/2
+nt
++ F −1/2
+nt
+�
+� �
+ν /∈Ψnt
+�
+Xaτ(ν),ν −
+K
+�
+k=1
+I (hν = k)
+φk,ν
+Xk,ν
+��
+� F −1/2
+nt
+.
+(21)
+For each ν ∈ [nt], the matrix �K
+k=1
+I(hν=k)
+φk,ν
+F −1/2
+nt
+Xk,νF −1/2
+nt
+symmetric positive definite and
+λmax
+�
+8 log Jd
+δ
+K
+�
+k=1
+I (hν = k)
+φk,ν
+F −1/2
+nt
+Xk,νF −1/2
+nt
+�
+≤8 log Jd
+δ
+K
+�
+k=1
+I (hν = k)
+φk,ν
+λmax
+�
+F −1
+nt
+�
+≤8 log Jd
+δ
+λmin(Fν)
+16 log
+� Jd
+δ
+�λmax
+�
+F −1
+nt
+�
+≤1
+2
+λmin(Fν)
+λmin(Fnt)
+≤1
+2.
+(22)
+With the filtration F0 := Ht and Fn := F0 ∪ {hν : ν ∈ [n]}, we use Lemma C.3 to have with probability at least 1 − δ,
+8 log Jd
+δ F −1/2
+nt
+� nt
+�
+ν=1
+K
+�
+k=1
+I (hν = k)
+φk,ν
+Xk,ν + IJ·d
+�
+F −1/2
+nt
+⪰ 8 log Jd
+δ F −1/2
+nt
+� nt
+�
+ν=1
+K
+�
+k=1
+Xk,ν + IJ·d
+�
+F −1/2
+nt
+= 4 log Jd
+δ IJ·d − log Jd
+δ IJ·d
+= 3 log Jd
+δ IJ·d,
+which implies
+F −1/2
+nt
+� nt
+�
+ν=1
+K
+�
+k=1
+I (hν = k)
+φk,ν
+Xk,ν + IJ·d
+�
+F −1/2
+nt
+⪰ 3
+8IJ·d,
+(23)
+and the first term in (21) is bounded as
+F −1/2
+nt
+AntF −1/2
+nt
+⪰ 3
+8IJ·d + F −1/2
+nt
+�
+� �
+ν /∈Ψnt
+�
+Xaτ(ν),ν −
+K
+�
+k=1
+I (hν = k)
+φk,ν
+Xk,ν
+��
+� F −1/2
+nt
+.
+(24)
+To bound the other term, observe that for ν /∈ Ψnt,
+E
+� K
+�
+k=1
+I (hν = k)
+φk,ν
+Xk,ν
+����� Ht
+�
+=
+�
+i̸=aτ(ν)
+K
+�
+k=1
+φi,ν
+I (i = k)
+φk,ν
+Xk,ν =
+�
+k̸=aτ(ν)
+Xk,ν.
+Because (22) holds for ν /∈ Ψnt, we can use Lemma C.3, to have with probability at least 1 − δ
+8 log Jd
+δ F −1/2
+nt
+�
+� �
+ν /∈Ψnt
+K
+�
+k=1
+I (hν = k)
+φk,ν
+Xk,ν
+�
+� F −1/2
+nt
+⪯ 12 log Jd
+δ F −1/2
+nt
+�
+� �
+ν /∈Ψnt
+�
+k̸=aτ(ν)
+Xk,ν
+�
+� F −1/2
+nt
++ log Jd
+δ IJ·d.
+Rearranging the terms,
+F −1/2
+nt
+�
+� �
+ν /∈Ψnt
+K
+�
+k=1
+I (hν = k)
+φk,ν
+Xk,ν
+�
+� F −1/2
+nt
+⪯ 3
+2F −1/2
+nt
+�
+� �
+ν /∈Ψnt
+�
+k̸=aτ(ν)
+Xk,ν
+�
+� F −1/2
+nt
++ 1
+8IJ·d.
+
+Improved Algorithms for Multi-period Packing Problems with Bandit Feedback
+Thus the second term in (24) is bounded as,
+F −1/2
+nt
+�
+� �
+ν /∈Ψnt
+�
+Xaτ(ν),ν −
+K
+�
+k=1
+I (hν = k)
+φk,ν
+Xk,ν
+��
+� F −1/2
+nt
+⪰ F −1/2
+nt
+�
+� �
+ν /∈Ψnt
+�
+�
+�Xaτ(ν),ν − 3
+2
+�
+ν /∈Ψt
+�
+k̸=aτ(ν)
+Xk,ν
+�
+�
+�
+�
+� F −1/2
+nt
+− 1
+8IJ·d
+⪰ −3
+2F −1/2
+nt
+�
+� �
+ν /∈Ψnt
+K
+�
+k=1
+Xk,ν
+�
+� F −1/2
+nt
+− 1
+8IJ·d
+⪰ −
+�3dK
+2
+��Ψc
+nt
+�� λmax
+�
+F −1
+nt
+�
++ 1
+8
+�
+IJ·d,
+(25)
+where the last inequality holds by λmax(Xk,ν) ≤ d. By Lemma C.3, with probability at least 1 − δ,
+1
+2
+��Ψc
+nt
+�� = 1
+2
+nt
+�
+ν=1
+I
+�
+hν ̸= aτ(ν)
+�
+≤ 3
+2
+nt
+�
+ν=1
+�
+k̸=aτ(ν)
+φk,ν + log 1
+δ ,
+which implies
+3dK
+2
+��Ψc
+nt
+�� λmax
+�
+F −1
+nt
+�
+≤
+3Kd
+2λmin(Fnt)
+�
+�
+�3
+nt
+�
+ν=1
+�
+k̸=aτ(ν)
+φk,ν + 2 log 1
+δ
+�
+�
+�
+=
+3Kd
+2λmin(Fnt)
+� nt
+�
+ν=1
+48 (K − 1) log
+� Jd
+δ
+�
+λmin(Fν)
++ 2 log 1
+δ
+�
+Because the assumption (8),
+λmin(Fnt) ≥ 12Kd
+� nt
+�
+ν=1
+48 (K − 1) log
+� Jd
+δ
+�
+λmin(Fν)
++ 2 log Jd
+δ
+�
+,
+implies
+3dK
+2
+��Ψc
+nt
+�� λmax
+�
+F −1
+nt
+�
+≤
+3Kd
+2λmin(Fnt)
+� nt
+�
+ν=1
+48 (K − 1) log
+� Jd
+δ
+�
+λmin(Fν)
++ 2 log 1
+δ
+�
+≤ 1
+8,
+(26)
+plugging in (25), with probability at least 1 − 2δ,
+F −1/2
+nt
+�
+� �
+ν /∈Ψnt
+�
+Xaτ(ν),ν −
+K
+�
+k=1
+I (hν = k)
+φk,ν
+Xk,ν
+��
+� F −1/2
+nt
+⪰ −1
+4IJ·d.
+With (24),
+F −1/2
+nt
+AntF −1/2
+nt
+⪰ 1
+8IJ·d,
+which proves (20) and the claim implies
+���V 1/2
+nt A−1
+nt V 1/2
+nt
+− IJ·d
+���
+2 ≤ 7.
+Step 3. Bounding the self-normalized vector-valued martingale Snt
+Let F0 be a sigma algebra generated by contexts
+{x(js)
+k,s : k ∈ [K], s ∈ [t]}, and Ψt. Define filtration as Fν := σ(F0 ∪ Hτ(ν+1)). Then Sν is a RJ·d-valued martingale
+
+Improved Algorithms for Multi-period Packing Problems with Bandit Feedback
+because
+E [Sν − Sν−1| Fν−1] = E
+�
+I (ν ∈ Ψnt)
+K
+�
+k=1
+I (hν = k)
+φk,ν
+ηk,ν ˜Xk,ν + I (ν /∈ Ψnt) ηaτ(ν),τ(ν) ˜Xk,ν
+����� Fν−1
+�
+= E
+�
+I (ν ∈ Ψnt)
+K
+�
+k=1
+I
+�
+aτ(ν) = k
+�
+φk,ν
+ηk,ν ˜Xk,ν + I (ν /∈ Ψnt) ηaτ(ν),τ(ν) ˜Xk,ν
+����� Fν−1
+�
+= E
+��
+I (ν ∈ Ψnt)
+φaτ(ν),ν
++ I (ν /∈ Ψnt)
+�
+ηaτ(ν),ν ˜Xk,ν
+����� Fν−1
+�
+= E
+��
+I (ν ∈ Ψnt)
+φaτ(ν),ν
++ I (ν /∈ Ψnt)
+�
+ηaτ(ν),ν ˜Xk,ν
+����� Hτ(ν)
+�
+= 0,
+where the second equality holds by definition of Ψnt and the fourth inequality holds because the distribution of {x(js)
+k,s : k ∈
+[K], s ∈ (τ(ν), t]} is independent of Hτ(ν) by Assumption 2. By Assumption 1, for any λ ∈ R,
+E
+�
+exp
+�
+λ
+�
+I (ν ∈ Ψnt)
+φaτ(ν),ν
++I (ν /∈ Ψnt)
+�
+ηk,aτ(ν)
+������ Fν−1
+�
+≤ E
+�
+�exp
+�
+�λ2σ2
+2
+�
+I (ν ∈ Ψnt)
+φaτ(ν),ν
++I (ν /∈ Ψnt)
+�2�
+�
+������
+Fν−1
+�
+�
+≤ exp
+�
+2λ2σ2
+r
+�
+,
+Thus,
+�
+I(ν∈Ψnt)
+φaτ(ν),ν + I (ν /∈ Ψnt)
+�
+ηk,aτ(ν) is 2σr-sub-Gaussian. Because
+∥Snt∥V −1
+t
+=
+������
+�
+ν∈Ψnt
+K
+�
+k=1
+I (hν = k)
+φk,ν
+ηk,ν ˜Xk,ν +
+�
+ν /∈Ψnt
+ηaτ(ν),ν ˜Xaτ(ν),ν
+������
+V −1
+nt
+=
+�����
+nt
+�
+ν=1
+�
+I (ν ∈ Ψnt)
+φaτ(ν),ν
++ I (ν /∈ Ψnt)
+�
+ηaτ(ν),ν ˜Xk,ν
+�����
+V −1
+nt
+=
+�����
+nt
+�
+ν=1
+�
+I (ν ∈ Ψnt)
+φaτ(ν),ν
++ I (ν /∈ Ψnt)
+�
+ηaτ(ν),νV −1/2
+nt
+˜Xk,ν
+�����
+2
+,
+by Lemma C.6, with probability at least 1 − δ,
+∥St∥V −1
+t
+≤
+�����
+nt
+�
+ν=1
+�
+I (ν ∈ Ψnt)
+φaτ(ν),ν
++ I (ν /∈ Ψt)
+�
+ηaτ(ν),νV −1/2
+nt
+˜Xk,ν
+�����
+2
+,
+≤12σr
+�
+�
+�
+�
+nt
+�
+ν=1
+���V −1/2
+nt
+˜Xk,ν
+���
+2
+2 log 4
+δ
+≤12σr
+�
+Jd log 4
+δ ,
+where the last inequality holds because
+nt
+�
+ν=1
+���V −1/2
+nt
+˜Xk,ν
+���
+2
+2 =
+nt
+�
+ν=1
+˜X⊤
+k,νV −1
+nt
+˜Xk,ν = Tr
+� nt
+�
+ν=1
+˜X⊤
+k,νV −1
+nt
+˜Xk,ν
+�
+= Tr
+� nt
+�
+ν=1
+˜Xk,ν ˜X⊤
+k,νV −1
+nt
+�
+≤ Tr
+�
+VntV −1
+nt
+�
+= Jd.
+With (19), the proof is completed
+
+Improved Algorithms for Multi-period Packing Problems with Bandit Feedback
+B.2. Proof of Theorem 4.2
+Proof. Similar to the proof of Theorem 4.1, the bound for consumption vector immediately follows from the bound for the
+utilities. Therefore we provide the proof for the utility bound.
+Step 1. Decomposition:
+For each k ∈ [K] and j ∈ [J],
+���u⋆(j)
+k
+− ˜u(j)
+k,t+1
+��� ≤
+�����E
+�
+x(j)
+k
+�⊤
+θ(j)
+⋆
+−
+�
+1
+�t+1
+s=1 I (js = j)
+t+1
+�
+s=1
+I (js = j)
+�
+x(j)
+k,s
+�⊤ �θ(j)
+t
+������
++
+�����E
+�
+c(j)
+k
+�
+−
+�
+1
+�t+1
+s=1 I (js = j)
+t+1
+�
+s=1
+I (js = j) c(j)
+k,s
+������
+≤
+�����E
+�
+x(j)
+k
+�⊤
+θ(j)
+⋆
+−
+�
+1
+�t+1
+s=1 I (js = j)
+t+1
+�
+s=1
+I (js = j)
+�
+x(j)
+k,s
+�⊤
+θ(j)
+⋆
+������
++
+�����E
+�
+c(j)
+k
+�
+−
+�
+1
+�t+1
+s=1 I (js = j)
+t+1
+�
+s=1
+I (js = j) c(j)
+k,s
+������
++
+�����
+1
+�t+1
+s=1 I (js = j)
+t+1
+�
+s=1
+I (js = j)
+�
+�θ(j)
+t
+− θ(j)
+⋆
+�⊤
+x(j)
+k,s
+�����
+�����
+1
+�t+1
+s=1 I (js = j)
+t+1
+�
+s=1
+I (js = j)
+�
+E
+�
+x(j)
+k
+�⊤
+θ(j)
+⋆
+−
+�
+x(j)
+k,s
+�⊤
+θ(j)
+⋆
+������
++
+�����
+1
+�t+1
+s=1 I (js = j)
+t+1
+�
+s=1
+I (js = j)
+�
+E
+�
+c(j)
+k
+�
+− c(j)
+k,s
+������
++
+�����
+1
+�t+1
+s=1 I (js = j)
+t+1
+�
+s=1
+I (js = j)
+�
+�θ(j)
+t
+− θ(j)
+⋆
+�⊤
+x(j)
+k,s
+����� .
+Taking maximum over k ∈ [K] gives the decomposition,
+max
+k∈[K]
+���u⋆(j)
+k
+− ˜u(j)
+k,t+1
+��� ≤ max
+k∈[K]
+�����
+1
+�t+1
+s=1 I (js = j)
+t+1
+�
+s=1
+I (js = j)
+�
+E
+�
+x(j)
+k
+�⊤
+θ(j)
+⋆
+−
+�
+x(j)
+k,s
+�⊤
+θ(j)
+⋆
+������
++ max
+k∈[K]
+�����
+1
+�t+1
+s=1 I (js = j)
+t+1
+�
+s=1
+I (js = j)
+�
+E
+�
+c(j)
+k
+�
+− c(j)
+k,s
+������
++ max
+k∈[K]
+�����
+1
+�t+1
+s=1 I (js = j)
+t+1
+�
+s=1
+I (js = j)
+�
+�θ(j)
+t
+− θ(j)
+⋆
+�⊤
+x(j)
+k,s
+����� .
+(27)
+Step 2.
+Bounding the difference between expectation and empirical distribution:
+The random variables
+��
+x(j)
+k,s
+�⊤
+θ(j)
+⋆
+: s ∈ [t]
+�
+and {c(j)
+k,s : s ∈ [t]} are IID by Assumption 2. Using Lemma C.1,
+�����
+1
+�t+1
+s=1 I (js = j)
+t+1
+�
+s=1
+I (js = j)
+�
+E
+�
+x(j)
+k
+�⊤
+θ(j)
+⋆
+−
+�
+x(j)
+k,s
+�⊤
+θ(j)
+⋆
+������
+=
+1
+�t+1
+s=1 I (js = j)
+�����
+t+1
+�
+s=1
+I (js = j)
+�
+E
+�
+x(j)
+k
+�⊤
+θ(j)
+⋆
+−
+�
+x(j)
+k,s
+�⊤
+θ(j)
+⋆
+������
+≤
+4
+��t+1
+s=1 I (js = j)
+�
+log JKT,
+
+Improved Algorithms for Multi-period Packing Problems with Bandit Feedback
+and
+�����
+1
+�t+1
+s=1 I (js = j)
+t+1
+�
+s=1
+I (js = j)
+�
+E
+�
+c(j)
+k
+�
+− c(j)
+k,s
+������ ≤
+4
+��t+1
+s=1 I (js = j)
+�
+log JKT
+with probability at least 1 − 4(JKT)−1. By Lemma C.3, with probability at least 1 − (JT)−1,
+t+1
+�
+s=1
+I (js = j) ≥ 1
+2pj (t + 1) − 2 log JT ≥ 1
+4pj (t + 1) ,
+(28)
+where the last inequality holds by the assumption t ≥ 8dα−1p−1
+min log JT. Summing up the probability bounds, with
+probability at least 1 − 5T −1,
+max
+k∈[K]
+�����
+1
+�t+1
+s=1 I (js = j)
+t+1
+�
+s=1
+I (js = j)
+�
+E
+�
+x(j)
+k
+�⊤
+θ(j)
+⋆
+−
+�
+x(j)
+k,s
+�⊤
+θ(j)
+⋆
+������ ≤
+4
+��t+1
+s=1 I (js = j)
+�
+log JKT
+≤
+8
+�
+pj (t + 1)
+�
+log JKT,
+max
+k∈[K]
+�����
+1
+�t+1
+s=1 I (js = j)
+t+1
+�
+s=1
+I (js = j)
+�
+E
+�
+c(j)
+k
+�
+− c(j)
+k,s
+������ ≤
+8
+�
+pj (t + 1)
+�
+log JKT.
+Plugging in the decomposition (27), for each j ∈ [J],
+max
+k∈[K]
+���u⋆(j)
+k
+− ˜u(j)
+k,t+1
+��� ≤
+16
+�
+pj (t + 1)
+�
+log JKT
++ max
+k∈[K]
+�����
+1
+�t+1
+s=1 I (js = j)
+t+1
+�
+s=1
+I (js = j)
+�
+�θ(j)
+t−1 − θ(j)
+⋆
+�⊤
+x(j)
+k,s
+����� .
+Taking square and summing up over j ∈ [J] gives
+J
+�
+j=1
+pj max
+k∈[K]
+���u⋆(j)
+k
+− ˜u(j)
+k,t+1
+���
+2
+≤16J log JKT
+t + 1
++
+J
+�
+j=1
+pj max
+k∈[K]
+�����
+1
+�t+1
+s=1 I (js = j)
+t+1
+�
+s=1
+I (js = j)
+�
+�θ(j)
+t
+− θ(j)
+⋆
+�⊤
+x(j)
+k,s
+�����
+2
+,
+(29)
+Step 3. Bounding the prediction error:
+By Cauchy-Schwartz inequality and (28),
+J
+�
+j=1
+max
+k∈[K] pj
+1
+��t+1
+s=1 I (js = j)
+�2
+�����
+t+1
+�
+s=1
+I (js = j)
+�
+�θ(j)
+t
+− θ(j)
+⋆
+�⊤
+x(j)
+k,s
+�����
+2
+≤
+J
+�
+j=1
+max
+k∈[K] pj
+1
+�t+1
+s=1 I (js = j)
+t+1
+�
+s=1
+I (js = j)
+��
+�θ(j)
+t
+− θ(j)
+⋆
+�⊤
+x(j)
+k,s
+�2
+≤
+J
+�
+j=1
+max
+k∈[K]
+4pj
+pj (t + 1)
+t+1
+�
+s=1
+I (js = j)
+��
+�θ(j)
+t
+− θ(j)
+⋆
+�⊤
+x(j)
+k,s
+�2
+=
+4
+(t + 1)
+J
+�
+j=1
+max
+k∈[K]
+�
+�θ(j)
+t
+− θ(j)
+⋆
+�⊤
+�t+1
+�
+s=1
+I (js = j) x(j)
+k,s
+�
+x(j)
+k,s
+�⊤
+� �
+�θ(j)
+t
+− θ(j)
+⋆
+�
+≤
+4
+(t + 1)
+J
+�
+j=1
+�
+�θ(j)
+t
+− θ(j)
+⋆
+�⊤
+�t+1
+�
+s=1
+K
+�
+k=1
+I (js = j) x(j)
+k,s
+�
+x(j)
+k,s
+�⊤
+� �
+�θ(j)
+t
+− θ(j)
+⋆
+�
+=
+4
+(t + 1)
+�
+�Θt − Θ⋆�⊤
+�t+1
+�
+s=1
+K
+�
+k=1
+˜Xk,s ˜X⊤
+k,s
+� �
+�Θt − Θ⋆�
+,
+(30)
+
+Improved Algorithms for Multi-period Packing Problems with Bandit Feedback
+where Θ⋆ := (θ(1)
+⋆ , . . . , θ(J)
+⋆
+)T ∈ RJ·d and
+˜Xk,s :=
+�
+�
+�
+�
+�
+0d
+...
+x(js)
+k,s
+0d
+�
+�
+�
+�
+� ∈ RJ·d,
+where the context x(js)
+k,s is located after js − 1 of 0d vectors. We claim that
+1
+t + 1
+t+1
+�
+s=1
+K
+�
+k=1
+˜Xk,s ˜X⊤
+k,s ⪯ 2E
+�
+˜Xk,1 ˜X⊤
+k,1
+�
+⪯
+7
+|Ψnt|
+�
+s∈Ψnt
+K
+�
+k=1
+˜Xk,s ˜X⊤
+k,s,
+(31)
+with probability at least 1 − 2T −1. The matrix Xs := �K
+k=1 ˜Xk,s ˜X⊤
+k,sis symmetric nonnegative definite which satisfies
+λmax
+�
+1
+2dK Xs
+�
+≤ 1
+2.
+By Lemma C.3, with probability at least 1 − T −1,
+1
+2Kd
+t+1
+�
+s=1
+Xs ⪯
+3
+4Kd
+t+1
+�
+s=1
+E [Xs] + (log JdT) IJ·d,
+which implies
+1
+t + 1
+t+1
+�
+s=1
+Xs ⪯
+3
+2 (t + 1)
+t+1
+�
+s=1
+E [Xs] + 2dK
+t + 1 (log JdT) IJ·d.
+(32)
+By Assumption 3, for s ∈ [t + 1],
+λmin(E [Xs]) =λmin
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+p1Exk∼F1
+��K
+k=1 xkx⊤
+k
+�
+0
+0
+0
+...
+0
+0
+0
+pJExk∼FJ
+��K
+k=1 xkx⊤
+k
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+≥λmin
+�
+�
+�
+�
+�
+�
+p1KαId
+0
+0
+0
+...
+0
+0
+0
+pJKαId
+�
+�
+�
+�
+�
+�
+≥Kpminα.
+For t ≥ 8dα−1p−1
+min log JdT ,
+t+1
+�
+s=1
+E [Xs] ⪰
+t+1
+�
+s=1
+λmin(E [Xs]) IJ·d ⪰ (t + 1) KpminαIJ·d ⪰ 4dK
+�
+log Jd
+δ
+�
+Plugging in (32) proves the first inequality of (31),
+1
+t + 1
+t+1
+�
+s=1
+Xs ⪯
+2
+(t + 1)
+t+1
+�
+s=1
+E [Xs] = 2E [X1] ,
+where the equality holds because EXs = EX1 for all s ∈ [T]. To prove the second inequality,
+E [X1] = |Ψnt|−1 �
+ν∈Ψnt
+E
+�
+Xτ(ν)
+�
+,
+
+Improved Algorithms for Multi-period Packing Problems with Bandit Feedback
+and by Lemma C.3, with probability at least 1 − T −1,
+1
+2Kd
+�
+ν∈Ψnt
+Xτ(ν) ⪰
+1
+4Kd
+�
+ν∈Ψnt
+E
+�
+Xτ(ν)
+�
+− (log JdT) IJ·d.
+Rearranging the terms,
+�
+ν∈Ψnt
+E
+�
+Xτ(ν)
+�
+⪯ 2
+�
+ν∈Ψnt
+Xτ(ν) + 4Kd (log JdT) IJ·d
+(33)
+By definition of Fnt,
+�
+ν∈Ψnt
+Xτ(ν) =Fnt −
+�
+ν /∈Ψnt
+Xτ(ν) − 16d(K − 1) log Jd
+δ IJ·d
+⪰Fnt −
+�
+Kd
+��Ψc
+nt
+�� + 16d(K − 1) log Jd
+δ
+�
+IJ·d.
+(34)
+Because the condition (8) holds, we can use (26) to have,
+Kd
+��Ψc
+nt
+�� ≤
+1
+12λmax
+�
+F −1
+nt
+� = λmin(Fnt)
+12
+,
+(35)
+and
+Fnt −
+�
+Kd
+��Ψc
+nt
+�� + 16d(K − 1) log Jd
+δ
+�
+IJ·d ⪰Fnt −
+� 1
+12λmin(Fnt) + 16d(K − 1) log Jd
+δ
+�
+IJ·d
+⪰
+�11
+12λmin(Fnt) − 16d(K − 1) log Jd
+δ
+�
+IJ·d
+⪰
+�11
+1224Kd log Jd
+δ IJ·d − 16d(K − 1) log Jd
+δ
+�
+IJ·d
+⪰
+�
+6dK log Jd
+δ
+�
+IJ·d
+⪰ {6dK log JdT} IJ·d,
+(36)
+where the third inequality holds by condition (8) and the last inequality holds by δ < T −1. Collecting the bounds (34)
+and (36)
+�
+ν∈Ψnt
+Xτ(ν) ⪰ Fnt −
+�
+Kd
+��Ψc
+nt
+�� + 16d(K − 1) log Jd
+δ
+�
+IJ·d ⪰ {6dK log JdT} IJ·d.
+Plugging in (33),
+�
+ν∈Ψnt
+E
+�
+Xτ(ν)
+�
+⪯ 2
+�
+ν∈Ψnt
+Xτ(ν) + 4Kd (log JdT) IJ·d ⪯ 7
+2
+�
+ν∈Ψnt
+Xτ(ν),
+proves the second inequality in claim (20). From (30),
+J
+�
+j=1
+max
+k∈[K] pj
+1
+��t+1
+s=1 I (js = j)
+�2
+�����
+t+1
+�
+s=1
+I (js = j)
+�
+�θ(j)
+t
+− θ(j)
+⋆
+�⊤
+x(j)
+k,s
+�����
+2
+≤
+4
+(t + 1)
+�
+�Θt − Θ⋆�⊤
+�t+1
+�
+s=1
+K
+�
+k=1
+˜Xk,s ˜X⊤
+k,s
+� �
+�Θt − Θ⋆�
+≤
+28
+|Ψnt|
+�
+�Θt − Θ⋆�⊤
+�
+�
+�
+�
+s∈Ψnt
+K
+�
+k=1
+˜Xk,s ˜X⊤
+k,s
+�
+�
+�
+�
+�Θt − Θ⋆�
+≤
+28
+|Ψnt|
+�
+�Θt − Θ⋆�⊤
+{Vnt}
+�
+�Θt − Θ⋆�
+=
+28
+|Ψnt|
+����Θt − Θ⋆���
+2
+Vnt
+(37)
+
+Improved Algorithms for Multi-period Packing Problems with Bandit Feedback
+On bounding the normalizing matrix, the novel Gram matrix Vnt plays a crucial role. To obtain an upper bound for (37), we
+need a matrix whose eigenvalue is greater than that of:
+�
+ν∈Ψnt
+Xτ(ν) =
+�
+ν∈Ψnt
+K
+�
+k=1
+˜Xk,τ(ν) ˜X⊤
+k,τ(ν),
+(38)
+However, with �
+ν∈Ψnt ˜Xaτ(ν),τ(ν) ˜X⊤
+aτ(ν),τ(ν) , a Gram matrix consist of only selected contexts, we cannot bound the
+matrix (38). Instead, by using a Gram matrix Vt, we can bound (38) as,
+�
+ν∈Ψnt
+Xτ(ν) =
+�
+ν∈Ψnt
+K
+�
+k=1
+˜Xk,τ(ν) ˜X⊤
+k,τ(ν)
+⪯
+�
+ν∈Ψnt
+K
+�
+k=1
+˜Xk,τ(ν) ˜X⊤
+k,τ(ν) +
+�
+ν /∈Ψnt
+˜Xaτ(ν),τ(ν) ˜X⊤
+aτ(ν),τ(ν)
+⪯Vnt,
+and prove the bound (37) to relate the prediction error to the self-normalized bound. From (37), by Theorem 4.1
+J
+�
+j=1
+max
+k∈[K] pj
+1
+��t+1
+s=1 I (js = j)
+�2
+�����
+t+1
+�
+s=1
+I (js = j)
+�
+�θ(j)
+t
+− θ(j)
+⋆
+�⊤
+x(j)
+k,s
+�����
+2
+≤ 28
+|Ψnt|
+����Θt − Θ⋆���
+2
+Vnt
+≤ 28
+|Ψnt|βσr(δ)2,
+with probability at least 1 − 4(m + 1)δ. Because |Ψnt| +
+��Ψc
+nt
+�� = nt and (35) holds,
+|Ψnt| ≥ nt −
+��Ψc
+nt
+�� ≥ nt − λmin(Fnt)
+12Kd
+≥ nt − ntKd
+12Kd = 11
+12nt.
+Thus,
+J
+�
+j=1
+max
+k∈[K] pj
+1
+��t+1
+s=1 I (js = j)
+�2
+�����
+t+1
+�
+s=1
+I (js = j)
+�
+�θ(j)
+t
+− θ(j)
+⋆
+�⊤
+x(j)
+k,s
+�����
+2
+≤ 28
+|Ψnt|βσr(δ)2
+≤12
+11 · 28
+nt
+βσr(δ)2
+≤32
+nt
+βσr(δ)2.
+From (29),
+�
+�
+�
+�
+J
+�
+j=1
+pj max
+k∈[K]
+���u⋆(j)
+k
+− ˜u(j)
+k,t+1
+���
+2
+≤ 16√J log JKT
+√
+t
++ 4
+√
+2βσr(δ)
+√nt
+and the proof is completed.
+B.3. Proof of Lemma 4.3
+Proof. Suppose a feasible policy ˜π(j)
+k,t for the optimization problem (1) satisfies
+K+1
+�
+k=1
+˜π(jt)
+k,t ˜u(jt)
+k,t >
+K+1
+�
+k=1
+�π(jt)
+k,t ˜u(jt)
+k,t ,
+which is equivalent to
+K+1
+�
+l=1
+˜π(jt)
+k⟨l⟩,t˜u(jt)
+k⟨l⟩,t >
+K+1
+�
+l=1
+�π(jt)
+k⟨l⟩,t˜u(jt)
+k⟨l⟩,t.
+(39)
+
+Improved Algorithms for Multi-period Packing Problems with Bandit Feedback
+Without loss of generality we assume �u(jt)
+k⟨l⟩,t ≥ 0 (Because �K+1
+l=1 ˜π(jt)
+k⟨l⟩,t = �K+1
+l=1 �π(jt)
+k⟨l⟩,t = 1, we can subtract �u(jt)
+k⟨K+1⟩,t
+on both side of (39)). By the constraints on the resources,
+˜π(jt)
+k⟨1⟩,t ≤
+�
+� min
+r∈[m]
+ρt(r)
+b(jt)
+k⟨1⟩,t(r)
+�
+� ∧ 1 = �π(jt)
+k⟨1⟩,t
+Suppose ˜π(jt)
+k⟨1⟩,t < �π(jt)
+k⟨1⟩,t. Because �K+1
+l=1 ˜π(jt)
+k⟨l⟩,t = �K+1
+l=1 �π(jt)
+k⟨l⟩,t = 1, by Lemma C.2,
+K+1
+�
+l=1
+˜π(jt)
+k⟨l⟩,t˜u(jt)
+k⟨l⟩,t ≤
+K+1
+�
+l=1
+�π(jt)
+k⟨l⟩,t˜u(jt)
+k⟨l⟩,t,
+which contradicts with (39). Thus we have ˜π(jt)
+k⟨1⟩,t = �π(jt)
+k⟨1⟩,t and
+K+1
+�
+l=2
+˜π(jt)
+k⟨l⟩,t˜u(jt)
+k⟨l⟩,t >
+K+1
+�
+l=2
+�π(jt)
+k⟨l⟩,t˜u(jt)
+k⟨l⟩,t.
+(40)
+Again, by the constraints on the resources,˜π(jt)
+k⟨2⟩,t ≤ �π(jt)
+k⟨2⟩,t. Suppose ˜π(jt)
+k⟨2⟩,t < �π(jt)
+k⟨2⟩,t. Because �K+1
+l=2 ˜π(jt)
+k⟨l⟩,t =
+�K+1
+l=2 �π(jt)
+k⟨l⟩,t, by Lemma C.2,
+K+1
+�
+l=2
+˜π(jt)
+k⟨l⟩,t˜u(jt)
+k⟨l⟩,t ≤
+K+1
+�
+l=2
+�π(jt)
+k⟨l⟩,t˜u(jt)
+k⟨l⟩,t.
+which contradicts with (40). Thus we have ˜π(jt)
+k⟨2⟩,t = �π(jt)
+k⟨2⟩,t Recursively, we have ˜π(jt)
+k⟨l⟩,t = �π(jt)
+k⟨l⟩,t for all l ∈ [K + 1]. Thus
+there exist no feasible solution ˜π(j)
+k,t such that (39) holds and the proof is completed.
+B.4. Proof of Lemma 5.2
+Proof. For each t ∈ [T], denote the good events Gt := Et ∩ Mt−1.
+Step 1. Bounds for the estimates ˜u(jt)
+k,t and ˜b(jt)
+k,t :
+For each t ∈ [T] and k ∈ [K],
+˜u(jt)
+k,t = ˜u(jt)
+k,t − u⋆(jt)
+k
++ u⋆(jt)
+k
+=γt−1,σr(δ)
+√pjt
++ �u(jt)
+k,t − u⋆(jt)
+k
++ u⋆(jt)
+k
+≥
+γt−1,σr(δ) − √pjt maxk∈[K]
+����u(jt)
+k,t − u⋆(jt)
+k
+���
+√pjt
++ u⋆(jt)
+k
+.
+Under the event Gt,
+√pjt max
+k∈[K]
+���u⋆(jt)
+k
+− �u(jt)
+k,t
+��� =
+�
+pjt max
+k∈[K]
+���u⋆(jt)
+k
+− �u(jt)
+k,t
+���
+2
+≤
+�
+�
+�
+�
+J
+�
+j=1
+pj max
+k∈[K]
+���u⋆(j)
+k
+− �u(j)
+k,t
+���
+2
+≤γt−1,σr(δ),
+which implies
+˜u(jt)
+k,t ≥ u⋆(jt)
+k
+.
+(41)
+Similarly,
+˜b(jt)
+k,t ≤ b⋆(jt)
+k
+.
+(42)
+
+Improved Algorithms for Multi-period Packing Problems with Bandit Feedback
+Another useful bound for ˜u(jt)
+k,t is
+E
+��
+t∈U
+K
+�
+k=1
+�π(jt)
+k,t
+���˜u(jt)
+k,t − u⋆(jt)
+k
+��� I (Gt)
+�
+≤ 2γt−1,σr(δ)
+�
+E [I (at ∈ [K])].
+(43)
+This bound is proved by the tower property of conditional expectation and Cauchy-Schwartz inequality,
+E
+� K
+�
+k=1
+�π(jt)
+k,t
+���˜u(jt)
+k,t − u⋆(jt)
+k
+��� I (Gt)
+�
+=E
+�
+max
+k∈[K]
+���˜u(jt)
+k,t − u⋆(jt)
+k
+���
+K
+�
+k=1
+�π(jt)
+k,t I (Gt)
+�
+=E
+�
+max
+k∈[K]
+���˜u(jt)
+k,t − u⋆(jt)
+k
+��� I (at ∈ [K]) I (Gt)
+�
+=E
+�
+�
+J
+�
+j=1
+pj max
+k∈[K]
+���˜u(j)
+k,t − u⋆(j)
+k
+��� I (at ∈ [K]) I (Gt)
+�
+�
+≤E
+�
+�
+�
+�
+�
+�
+J
+�
+j=1
+pj max
+k∈[K]
+���˜u(j)
+k,t − u⋆(j)
+k
+���
+2
+�
+�
+�
+�
+J
+�
+j=1
+pjI (at ∈ [K])I (Gt)
+�
+�
+By definition of ˜u(j)
+k,t and triangular inequality for ℓ2-norm,
+�
+�
+�
+�
+J
+�
+j=1
+pj max
+k∈[K]
+���˜u(j)
+k,t − u⋆(j)
+k
+���
+2
+I (Gt) =
+�
+�
+�
+�
+J
+�
+j=1
+pj max
+k∈[K]
+�����u(j)
+k,t − u⋆(j)
+k
++ γt−1,σr(δ)
+√pj
+����
+2
+I (Gt)
+≤
+�
+�
+�
+�
+�
+�
+J
+�
+j=1
+pj max
+k∈[K]
+����u(j)
+k,t − u⋆(j)
+k
+���
+2
++
+�
+�
+�
+�
+J
+�
+j=1
+pj
+�γt−1,σr(δ)
+√pj
+�2
+�
+� I (Gt)
+≤2γt−1,σr(δ)I (Gt)
+≤2γt−1,σr(δ).
+Then by Jensen’s inequality,
+E
+� K
+�
+k=1
+�π(jt)
+k,t
+���˜u(jt)
+k,t − u⋆(jt)
+k
+��� I (Gt)
+�
+≤E
+�
+�
+�
+�
+�
+�
+J
+�
+j=1
+pj max
+k∈[K]
+���˜u(j)
+k,t − u⋆(j)
+k
+���
+2
+�
+�
+�
+�
+J
+�
+j=1
+pjI (at ∈ [K])I (Gt)
+�
+�
+≤2γt−1,σr(δ)E
+�
+�
+�
+�
+�
+�
+J
+�
+j=1
+pjI (at ∈ [K])
+�
+�
+≤2γt−1,σr(δ)
+�
+�
+�
+�
+�E
+�
+�
+J
+�
+j=1
+pjI (at ∈ [K])
+�
+�
+=2γt−1,σr(δ)
+�
+E [I (at ∈ [K])],
+which proves (43). Similarly,
+E
+� K
+�
+k=1
+�π(jt)
+k,t
+���˜b(jt)
+k,t − b⋆(jt)
+k
+���
+∞ I (Gt)
+�
+≤ 2γt−1,σb(δ)
+�
+E [I (at ∈ [K])]
+(44)
+Step 2. Reward decomposition:
+Let τ be the stopping time of the algorithm and let U := {t ∈ [τ] : ρt > 0}. Then for
+t /∈ U, the allocated resource is ρt ∨ 0 = 0 and the algorithm skips the round. Thus,
+E
+� T
+�
+t=1
+R�π
+t
+�
+=E
+��
+t∈U
+R�π
+t
+�
+.
+
+Improved Algorithms for Multi-period Packing Problems with Bandit Feedback
+Then, the reward is decomposed as
+E
+��
+t∈U
+R�π
+t
+�
+=E
+��
+t∈U
+R�π
+t I (Gt)
+�
++ E
+��
+t∈U
+R�π
+t I (Gc
+t )
+�
+≥E
+��
+t∈U
+K
+�
+k=1
+�π(jt)
+k,t u⋆(jt)
+k
+I (Gt)
+�
+−
+T
+�
+t=1
+P (Gc
+t )
+≥E
+��
+t∈U
+K
+�
+k=1
+�π(jt)
+k,t ˜u(jt)
+k,t I (Gt)
+�
+− E
+��
+t∈U
+K
+�
+k=1
+�π(jt)
+k,t
+���˜u(jt)
+k,t − u⋆(jt)
+k
+��� I (Gt)
+�
+−
+T
+�
+t=1
+P (Gc
+t )
+≥E
+��
+t∈U
+K
+�
+k=1
+�π(jt)
+k,t ˜u(jt)
+k,t I (Gt)
+�
+−
+T
+�
+t=1
+E
+� K
+�
+k=1
+�π(jt)
+k,t
+���˜u(jt)
+k,t − u⋆(jt)
+k
+��� I (Gt)
+�
+−
+T
+�
+t=1
+P (Gc
+t ) .
+By the bound (43),
+T
+�
+t=1
+E
+� K
+�
+k=1
+�π(jt)
+k,t
+���˜u(jt)
+k,t − u⋆(jt)
+k
+��� I (Gt)
+�
+≤2
+T
+�
+t=1
+γt−1,σr(δ)
+�
+E [I (at ∈ [K])]
+≤2
+�
+�
+�
+�T
+T
+�
+t=1
+γt−1,σr(δ)2E [I (at ∈ [K])]
+=2
+�
+�
+�
+�TE
+� T
+�
+t=1
+γt−1,σr(δ)2I (at ∈ [K])
+�
+where the last ineqaulity holds by Cauchy-Schwartz inequality. Thus, the reward is decomposed as
+E
+� T
+�
+t=1
+R�π
+t
+�
+=E
+��
+t∈U
+R�π
+t
+�
+≥E
+��
+t∈U
+K
+�
+k=1
+�π(jt)
+k,t ˜u(jt)
+k,t I (Gt)
+�
+− 2
+�
+�
+�
+�TE
+� T
+�
+t=1
+γt−1,σr(δ)2I (at ∈ [K])
+�
+−
+T
+�
+t=1
+P (Gc
+t )
+(45)
+Step 3. A lower bound for ρt:
+Denote u1 < u2 < . . . < u|U| the indexes in U. For s /∈ U, we have ρs = 0m and
+b(js)
+as,s = 0m. Thus for ν ∈ [|U| − 1],
+ρuν+1 = uν+1ρ −
+uν+1−1
+�
+s=1
+b(js)
+as,s = uν+1ρ −
+uν
+�
+s=1
+b(js)
+as,s.
+(46)
+By the resource constrain at round uν,
+K
+�
+k=1
+�π(juν )
+k,uν ˜b(juν )
+k,uν ≤uνρ −
+uν−1
+�
+s=1
+b(js)
+as,s
+=uνρ + b(juν )
+auν ,uν −
+uν
+�
+s=1
+b(js)
+as,s.
+Plugging in (46),
+ρuν+1 ≥ (uν+1 − uν) ρ − b(juν )
+auν ,uν +
+K
+�
+k=1
+�π(juν )
+k,uν ˜b(juν )
+k,uν
+≥ (uν+1 − uν) ρ − b(juν )
+auν ,uν +
+K
+�
+k=1
+�π(juν )
+k,uν b⋆(juν )
+k
++
+K
+�
+k=1
+�π(juν )
+k,uν
+�
+˜b(juν )
+k,uν − b⋆(juν )
+k
+�
+.
+
+Improved Algorithms for Multi-period Packing Problems with Bandit Feedback
+Taking conditional expectation on both sides gives
+E
+�
+ρuν+1
+�� juν+1
+�
+≥E
+�
+uν+1 − uν| juν+1
+�
+ρ + E
+�
+−b(juν )
+auν ,uν +
+K
+�
+k=1
+�π(juν )
+k,uν b⋆(juν )
+k
+����� juν+1
+�
++ E
+� K
+�
+k=1
+�π(juν )
+k,uν
+�
+˜b(juν )
+k,uν − b⋆(juν )
+k
+������ juν+1
+�
+=E
+�
+uν+1 − uν| juν+1
+�
+ρ + E
+� K
+�
+k=1
+�π(juν )
+k,uν
+�
+˜b(juν )
+k,uν − b⋆(juν )
+k
+��
+≥E
+�
+uν+1 − uν| juν+1
+�
+ρ + E
+� K
+�
+k=1
+�π(juν )
+k,uν
+�
+˜b(juν )
+k,uν − b⋆(juν )
+k
+�
+I (Guν)
+�
+− P
+�
+Gc
+uν
+�
+1m,
+where the equality holds by Assumption 1 and
+E
+��
+−b(juν )
+auν ,uν +
+K
+�
+k=1
+�π(juν )
+k,uν b⋆(juν )
+k
+��
+=E
+��
+−b⋆(juν )
+auν
++
+K
+�
+k=1
+�π(juν )
+k,uν b⋆(juν )
+k
+��
+=E
+��
+−
+K
+�
+k=1
+�π(juν )
+k,uν b⋆(juν )
+k
++
+K
+�
+k=1
+�π(juν )
+k,uν b⋆(juν )
+k
+��
+=0.
+For the last term, by the bound (44),
+E
+� K
+�
+k=1
+�π(juν )
+k,uν
+�
+˜b(juν )
+k,uν − b⋆(juν )
+k
+�
+I (Guν)
+�
+≥ − E
+� K
+�
+k=1
+�π(juν )
+k,uν
+���˜b(juν )
+k,uν − b⋆(juν )
+k
+���
+∞ I (Guν)
+�
+1m
+≥ − E
+�
+2γuν−1,σb(δ)
+�
+E [I (auν ∈ [K])| uν]
+�
+1m.
+Thus we obtain a lower bound,
+E
+�
+ρuν+1
+�� juν+1
+�
+≥E
+�
+uν+1 − uν| juν+1
+�
+ρ − P
+�
+Gc
+uν
+�
+1m
+− 2E
+�
+γuν−1,σb(δ)
+�
+E [I (auν ∈ [K])| uν]
+�
+1m.
+(47)
+Step 4. An upper bound for OPT
+In the optimization problem (1), all constraints are linear with respect to the variable
+and there exist a feasible solution. Thus the problem satisfies the Slater’s condition and strong duality (Boyd et al., 2004).
+Then,
+OPT
+T
+= max
+π(j)
+k
+min
+λ∈Rm
++
+min
+µ(j)≥0 min
+ν(j)
+k
+≥0
+L
+�
+π(j)
+k , λ, µ(j), ν(j)
+k
+�
+,
+where L is the Lagrangian function:
+L
+�
+π(j)
+k , λ, µ(j), ν(j)
+k
+�
+:=
+J
+�
+j=1
+K
+�
+k=1
+pjπ(j)
+k u⋆(j)
+k
++
+�
+�ρ −
+J
+�
+j=1
+K
+�
+k=1
+pjπ(j)
+k b⋆(j)
+k
+�
+�
+⊤
+λ
++
+J
+�
+j=1
+µ(j)
+�
+1 −
+K
+�
+k=1
+π(j)
+k,1
+�
++
+J
+�
+j=1
+K
+�
+k=1
+ν(j)
+k π(j)
+k,1.
+
+Improved Algorithms for Multi-period Packing Problems with Bandit Feedback
+Minimizing over µ(j) and ν(j)
+k
+gives
+min
+µ(j)
+t
+≥0
+min
+ν(j)
+k,t≥0
+L
+�
+π(j)
+k , λ, µ(j), ν(j)
+k
+�
+=
+�
+�
+�
+�J
+j=1
+�K
+k=1 pjπ(j)
+k u⋆(j)
+k
++
+�
+ρ − �J
+j=1
+�K
+k=1 pjπ(j)
+k b⋆(j)
+k
+�⊤
+λ
+�K
+k=1 π(j)
+k
+≤ 1, π(j)
+k
+≥ 0
+−∞
+o.w.
+,
+which implies
+OPT
+T
+= max
+π(j)
+k
+min
+λ∈Rm
++
+min
+µ(j)
+t
+≥0
+min
+ν(j)
+k,t≥0
+L
+�
+π(j)
+k , λ, µ(j), ν(j)
+k
+�
+≤
+max
+�K
+k=1 π(j)
+k
+≤1,π(j)
+k
+≥0
+min
+λ∈Rm
++
+J
+�
+j=1
+K
+�
+k=1
+pjπ(j)
+k u⋆(j)
+k
++
+�
+�ρ −
+J
+�
+j=1
+K
+�
+k=1
+pjπ(j)
+k b⋆(j)
+k
+�
+�
+⊤
+λ
+=
+max
+�K
+k=1 π(j)
+k
+≤1,π(j)
+k
+≥0
+min
+λ∈Rm
++
+J
+�
+j=1
+pj
+�
+�
+�
+K
+�
+k=1
+π(j)
+k u⋆(j)
+k
++
+�
+ρ −
+K
+�
+k=1
+π(j)
+k b⋆(j)
+k
+�⊤
+λ
+�
+�
+�
+≤ min
+λ∈Rm
++
+max
+�K
+k=1 π(j)
+k
+≤1,π(j)
+k
+≥0
+J
+�
+j=1
+pj
+�
+�
+�
+K
+�
+k=1
+π(j)
+k u⋆(j)
+k
++
+�
+ρ −
+K
+�
+k=1
+π(j)
+k b⋆(j)
+k
+�⊤
+λ
+�
+�
+�
+≤ min
+λ∈Rm
++
+J
+�
+j=1
+pj
+max
+�K
+k=1 π(j)
+k
+≤1,π(j)
+k
+≥0
+�
+�
+�
+K
+�
+k=1
+π(j)
+k u⋆(j)
+k
++
+�
+ρ −
+K
+�
+k=1
+π(j)
+k b⋆(j)
+k
+�⊤
+λ
+�
+�
+� .
+Let {¯π(j)
+k
+: j ∈ [J], k ∈ [K]} be the maximizer. If ρ − �K
+k=1 ¯π(j)
+k b⋆(j)
+k
+is negative for some element and j ∈ [J], then the
+optimal value becomes −∞. Thus
+OPT
+T
+≤ min
+λ∈Rm
++
+J
+�
+j=1
+pj
+max
+�K
+k=1 π(j)
+k
+≤1,π(j)
+k
+≥0
+�
+�
+�
+K
+�
+k=1
+π(j)
+k u⋆(j)
+k
++
+�
+ρ −
+K
+�
+k=1
+π(j)
+k b⋆(j)
+k
+�⊤
+λ
+�
+�
+�
+= min
+λ∈Rm
++
+J
+�
+j=1
+pj
+max
+�K
+k=1 π(j)
+k
+≤1,π(j)
+k
+≥0,ρ−�K
+k=1 π(j)
+k
+b⋆(j)
+k
+≥0
+�
+�
+�
+K
+�
+k=1
+π(j)
+k u⋆(j)
+k
++
+�
+ρ −
+K
+�
+k=1
+π(j)
+k b⋆(j)
+k
+�⊤
+λ
+�
+�
+�
+=
+J
+�
+j=1
+pj
+max
+�K
+k=1 π(j)
+k
+≤1,π(j)
+k
+≥0,ρ−�K
+k=1 π(j)
+k
+b⋆(j)
+k
+≥0
+� K
+�
+k=1
+π(j)
+k u⋆(j)
+k
+�
+.
+For each j ∈ [J] and v ∈ Rm
++, let ˜π(j)
+k,v be the solution to the optimization problem:
+max
+π(j)
+k,v
+K
+�
+k=1
+π(j)
+k,vu⋆(j)
+k
+s.t.
+K
+�
+k=1
+π(j)
+k,vb⋆(j)
+k
+≤ v.
+(48)
+Then,
+OPT
+T
+≤
+J
+�
+j=1
+pj
+max
+�K
+k=1 π(j)
+k
+≤1,π(j)
+k
+≥0,ρ−�K
+k=1 π(j)
+k
+b⋆(j)
+k
+≥0
+� K
+�
+k=1
+π(j)
+k u⋆(j)
+k
+�
+=
+J
+�
+j=1
+pj
+K
+�
+k=1
+˜π(j)
+k,ρu⋆(j)
+k
+.
+
+Improved Algorithms for Multi-period Packing Problems with Bandit Feedback
+For each ν ∈ [|U| − 1],
+E
+�
+(uν+1 − uν) OPT
+T
+�
+≤E
+�
+�(uν+1 − uν)
+J
+�
+j=1
+pj
+K
+�
+k=1
+˜π(j)
+k,ρu⋆(j)
+k
+�
+�
+=E
+�
+(uν+1 − uν)
+K
+�
+k=1
+˜π
+(juν+1)
+k,ρ
+u
+⋆(juν+1)
+k
+�
+In (48), all constraints are linear with respect to the variable and there exist a feasible solution. Thus the problem satisfies
+the Slater’s condition and strong duality (Boyd et al., 2004). The dual problem of (48) is
+min
+λ(j)
+v ∈Rm
++
+v⊤λ(j)
+v
+s.t.
+�
+b⋆(j)
+k
+�⊤
+λ(j)
+v
+≥ u⋆(j)
+k
+,
+∀k ∈ [K].
+(49)
+Let ˜λ(j)
+v
+be the solution to (49). By strong duality, for each ν ∈ [|U| − 1],
+E
+�
+(uν+1 − uν)
+K
+�
+k=1
+˜π
+(juν+1)
+k,ρ
+u
+⋆(juν+1)
+k
+�
+= E
+�
+(uν+1 − uν) ρ⊤˜λ
+(juν+1)
+ρ
+�
+= E
+�
+E
+�
+(uν+1 − uν) ρ| juν+1
+�⊤ ˜λ
+(juν+1)
+ρ
+�
+= E
+��
+P
+�
+Gc
+uν
+�
++ 2E
+��
+E [I (auν ∈ [K])| uν]γuν−1,λ(σb)
+��
+1⊤
+m˜λ
+(juν+1)
+ρ
+�
++ E
+��
+E
+�
+(uν+1 − uν) ρ| juν+1
+�
+− P
+�
+Gc
+uν
+�
+− 2E
+��
+E [I (auν ∈ [K])| uν]γuν−1,σb(δ)
+��
+1⊤
+m˜λ
+(juν+1)
+ρ
+�
+.
+(50)
+For the first term, we observe the dual problem of (1),
+min
+λ∈Rm
++
+ρ1⊤
+mλ
+s.t.λ⊤b⋆(j)
+k
+≥ u⋆(j)
+k
+, ∀j ∈ [J], ∀k ∈ [K].
+(51)
+Comparing to the dual problem (49), when v = ρ1m and j = juν+1, (51) has more constraints than (49) with same objective
+function. Denote λ⋆ be the solution to (51). Then,
+ρ1⊤
+m˜λ
+(juν+1)
+ρ1m
+≤ ρ1⊤
+mλ⋆ = OPT
+T
+,
+where the last equality holds by strong duality for the oracle problem (1). Thus the first term in (50) is bounded as
+E
+��
+P
+�
+Gc
+uν
+�
++ 2E
+��
+E [I (auν ∈ [K])| uν]γuν−1,λ(σb)
+��
+1⊤
+m˜λ
+(juν+1)
+ρ1m
+�
+≤
+�
+P
+�
+Gc
+uν
+�
++ 2E
+��
+E [I (auν ∈ [K])| uν]γuν−1,λ(σb)
+�� OPT
+ρT .
+For the second term in (50), we observe that ˜λ(juν +1)
+E[ ρuν+1|juν+1] is a feasible solution to (49) when v = ρ1m and j = juν+1.
+Thus
+E
+��
+E
+�
+(uν+1 − uν) ρ| juν+1
+�
+−2E
+��
+E [I (auν ∈[K])| uν]γuν−1,λ(σb)
+�
+−P
+�
+Gc
+uν
+��
+1⊤
+m˜λ
+(juν+1)
+ρ1m
+�
+≤E
+���
+E
+�
+(uν+1 − uν) ρ| juν+1
+�
+−2E
+��
+E [I (auν ∈[K])| uν]γuν−1,λ(σb)
+�
+−P
+�
+Gc
+uν
+��
+∨ 0
+�
+1⊤
+m˜λ
+(juν+1)
+ρ1m
+�
+≤E
+���
+E
+�
+(uν+1 − uν) ρ| juν+1
+�
+−2E
+��
+E [I (auν ∈[K])| uν]γuν−1,λ(σb)
+�
+−P
+�
+Gc
+uν
+��
+∨ 0
+�
+1⊤
+m˜λ(juν +1)
+E[ ρuν+1|juν+1]
+�
+≤E
+��
+E
+�
+ρuν+1
+�� juν+1
+�
+∨ 0m
+�⊤ ˜λ(juν +1)
+E[ ρuν+1|juν+1]
+�
+,
+
+Improved Algorithms for Multi-period Packing Problems with Bandit Feedback
+where the last inequality holds by (47). Because uν+1 ∈ U, we have ρuν+1 > 0 and
+E
+��
+E
+�
+ρuν+1
+�� juν+1
+�
+∨ 0m
+�⊤ ˜λ(juν +1)
+E[ ρuν+1|juν+1]
+�
+≤E
+�
+E
+�
+ρuν+1 ∨ 0m
+�� juν+1
+�⊤ ˜λ(juν +1)
+E[ ρuν+1|juν+1]
+�
+=E
+�
+E
+�
+ρuν+1
+�� juν+1
+�⊤ ˜λ(juν +1)
+E[ ρuν+1|juν+1]
+�
+.
+Collecting the bounds, we have
+E
+�
+(uν+1 − uν) OPT
+T
+�
+≤E
+�
+E
+�
+ρuν+1
+�� juν+1
+�⊤ ˜λ(juν +1)
+E[ ρuν+1|juν+1]
+�
++
+�
+2E
+��
+E [I (auν ∈[K])| u��]γuν−1,σb(δ)
+�
++ P
+�
+Gc
+uν
+�� OPT
+ρT .
+Similar to Step 4, by strong duality,
+E
+�
+ρuν+1
+�� juν+1
+�⊤ ˜λ(juν +1)
+E[ ρuν+1|juν+1]
+=
+max
+�K
+k=1 π
+(juν +1)
+k
+≤1,π
+(juν +1)
+k
+≥0
+min
+λ∈Rm
++
+K
+�
+k=1
+π(juν +1)
+k
+u⋆(juν +1)
+k
++
+�
+E
+�
+ρuν+1
+�� juν+1
+�
+−
+K
+�
+k=1
+π(juν +1)
+k
+b⋆(juν +1)
+k
+�⊤
+λ
+≤ min
+λ∈Rm
++
+max
+�K
+k=1 π
+(juν +1)
+k
+≤1,π
+(juν +1)
+k
+≥0
+K
+�
+k=1
+π(juν +1)
+k
+u⋆(juν +1)
+k
++
+�
+E
+�
+ρuν+1
+�� juν+1
+�
+−
+K
+�
+k=1
+π(juν +1)
+k
+b⋆(juν +1)
+k
+�⊤
+λ
+≤ min
+λ∈Rm
++
+E
+�
+�
+max
+�K
+k=1 π
+(juν +1)
+k
+≤1,π
+(juν +1)
+k
+≥0
+K
+�
+k=1
+π(juν +1)
+k
+u⋆(juν +1)
+k
++
+�
+ρuν+1 −
+K
+�
+k=1
+π(juν +1)
+k
+b⋆(juν +1)
+k
+�⊤
+λ
+������
+juν+1
+�
+�
+≤ E
+�
+�
+max
+�K
+k=1 π
+(juν +1)
+k
+≤1,π
+(juν +1)
+k
+≥0,ρuν+1−�K
+k=1 π
+(juν +1)
+k
+b
+⋆(juν +1)
+k
+≥0
+K
+�
+k=1
+π(juν +1)
+k
+u⋆(juν +1)
+k
+������
+juν+1
+�
+�
+=
+K
+�
+k=1
+˜π
+(juν+1)
+k,ρuν+1 u⋆(juν +1)
+k
+.
+Thus we have
+E
+�
+(uν+1 − uν) OPT
+T
+�
+≤E
+� K
+�
+k=1
+˜π
+(juν+1)
+k,ρuν+1 u⋆(juν +1)
+k
+�
++
+�
+2E
+��
+E [I (auν ∈[K])| uν]γuν−1,σb(δ)
+�
++ P
+�
+Gc
+uν
+�� OPT
+ρT .
+Under the event Guν+1, the policy ˜π
+(juν+1)
+k,ρuν+1 is a feasible solution to the bandit problem (12),
+E
+� K
+�
+k=1
+˜π
+(juν+1)
+k,ρuν+1 u⋆(juν +1)
+k
+�
+≤E
+� K
+�
+k=1
+˜π
+(juν+1)
+k,ρuν+1 u⋆(juν +1)
+k
+I
+�
+Guν+1
+�
+�
++ P
+�
+Gc
+uν+1
+�
+≤E
+� K
+�
+k=1
+˜π
+(juν+1)
+k,ρuν+1 ˜u
+(juν+1)
+k,uν+1 I
+�
+Guν+1
+�
+�
++ P
+�
+Gc
+uν+1
+�
+≤E
+� K
+�
+k=1
+�π
+(juν+1)
+k,uν+1 ˜u
+(juν+1)
+k,uν+1 I
+�
+Guν+1
+�
+�
++ P
+�
+Gc
+uν+1
+�
+.
+
+Improved Algorithms for Multi-period Packing Problems with Bandit Feedback
+Thus, for each ν ∈ [|U| − 1],
+E
+�
+(uν+1 − uν) OPT
+T
+�
+≤E
+� K
+�
+k=1
+�π
+(juν+1)
+k,uν+1 ˜u
+(juν+1)
+k,uν+1 I
+�
+Guν+1
+�
+�
++ P
+�
+Gc
+uν+1
+�
++
+�
+2E
+��
+E [I (auν ∈[K])| uν]γuν−1,σb(δ)
+�
++ P
+�
+Gc
+uν
+�� OPT
+ρT .
+Summing up over ν,
+E
+�
+�
+|U|−1
+�
+ν=1
+(uν+1 − uν) OPT
+T
+�
+� ≤E
+��
+t∈U
+K
+�
+k=1
+�π(jt)
+k,t ˜u(jt)
+k,t I (Gt)
+�
++
+�
+1 + OPT
+ρT
+�
+T
+�
+t=1
+P (Gc
+t )
++
+�
+�
+|U|−1
+�
+ν=1
+2E
+��
+E [I (auν ∈[K])| uν]γuν−1,σb(δ)
+�
+�
+� OPT
+ρT
+≤E
+��
+t∈U
+K
+�
+k=1
+�π(jt)
+k,t ˜u(jt)
+k,t I (Gt)
+�
++
+�
+1 + OPT
+ρT
+�
+T
+�
+t=1
+P (Gc
+t )
++ 2
+� T
+�
+t=1
+E
+��
+E [I (at ∈[K])]γt−1,σb(δ)
+��
+OPT
+ρT
+≤E
+��
+t∈U
+K
+�
+k=1
+�π(jt)
+k,t ˜u(jt)
+k,t I (Gt)
+�
++
+�
+1 + OPT
+ρT
+�
+T
+�
+t=1
+P (Gc
+t )
++ 2
+�
+�
+�
+�TE
+� T
+�
+t=1
+γt−1,σb(δ)2I (at ∈ [K])
+�
+OPT
+ρT ,
+where the last inequality holds by Cauchy-Schwartz inequality, By (45),
+E
+�
+�
+|U|−1
+�
+ν=1
+(uν+1 − uν) OPT
+T
+�
+� ≤E
+��
+t∈U
+K
+�
+k=1
+�π(jt)
+k,t ˜u(jt)
+k,t I (Gt)
+�
++
+�
+1 + OPT
+ρT
+�
+T
+�
+t=1
+P (Gc
+t )
++ 2
+�
+�
+�
+�TE
+� T
+�
+t=1
+γt−1,σb(δ)2I (at ∈ [K])
+�
+OPT
+ρT
+≤E
+� T
+�
+t=1
+R�π
+t
+�
++
+�
+2 + OPT
+ρT
+�
+T
+�
+t=1
+P (Gc
+t )
++ 2
+�
+�
+�
+�TE
+� T
+�
+t=1
+γt−1,σb(δ)2I (at ∈ [K])
+�
+OPT
+ρT
+2
+�
+�
+�
+�TE
+� T
+�
+t=1
+γt−1,σr(δ)2I (at ∈ [K])
+�
+≤E
+� T
+�
+t=1
+R�π
+t
+�
++
+�
+2 + OPT
+ρT
+�
+T
+�
+t=1
+P (Gc
+t )
+2
+�
+1 + OPT
+ρT
+� �
+�
+�
+�TE
+� T
+�
+t=1
+γt−1,σb∨σr(δ)2I (at ∈ [K])
+�
+
+Improved Algorithms for Multi-period Packing Problems with Bandit Feedback
+Because the last choice of the algorithm happens at round τ, we have ρτ > 0 and u|U| = τ. And by definition, u1 = ξ. Thus
+E
+�
+�
+|U|−1
+�
+ν=1
+(uν+1 − uν) OPT
+T
+�
+� = E
+��
+u|U| − u1
+� OPT
+T
+�
+= OPT
+T
+E [τ − ξ] .
+Rearranging the terms
+E
+� T
+�
+t=1
+R�π
+t
+�
+≥ E
+�
+�
+|U|−1
+�
+ν=1
+(uν+1 − uν) OPT
+T
+�
+� −
+�
+2 + OPT
+ρT
+�
+T
+�
+t=1
+P (Gc
+t )
+− 2
+�
+1 + OPT
+ρT
+� �
+�
+�
+�TE
+� T
+�
+t=1
+γt−1,σb∨σr(δ)2I (at ∈ [K])
+�
+≥ OPT
+T
+E [τ − ξ] −
+�
+2 + OPT
+ρT
+�
+T
+�
+t=1
+P (Gc
+t )
+− 2
+�
+1 + OPT
+ρT
+� �
+�
+�
+�TE
+� T
+�
+t=1
+γt−1,σb∨σr(δ)2I (at ∈ [K])
+�
+,
+completes the proof.
+B.5. Proof of Lemma 5.3
+Proof. Let us fix δ ∈ (0, T −2) throughout the proof.
+Step 1. Bounding the minimum eigenvalue of {Fν : ν ∈ [nT ]}:
+By Lemma C.3, with probability at least 1 − Tδ,
+1
+2KdFν =
+1
+2Kd
+ν
+�
+u=1
+˜Xk,τ(u) ˜X⊤
+k,τ(u) + 8K − 1
+K
+log Jd
+δ
+⪰
+1
+4Kd
+ν
+�
+u=1
+E
+��
+˜Xk,τ(u) ˜X⊤
+k,τ(u)
+��� Hτ(u)−1
+�
++ 8K − 1
+K
+log Jd
+δ − log Jd
+δ
+⪰
+1
+4Kd
+ν
+�
+u=1
+E
+�
+˜Xk,τ(u) ˜X⊤
+k,τ(u)
+��� Hτ(u)−1
+�
+,
+for all ν ∈ [nT ]. By Assumption 2 and 3,
+λmin
+�
+E
+�
+˜Xk,τ(u) ˜X⊤
+k,τ(u)
+��� Hτ(u)−1
+��
+=λmin
+�
+�
+�
+�
+�
+p1EXk∼F1
+��K
+k=1 XkX⊤
+k
+�
+0
+0
+0
+...
+0
+0
+0
+pJExk∼FJ
+��K
+k=1 XkX⊤
+k
+�
+�
+�
+�
+�
+�
+≥pmin min
+j∈[J] λmin
+�
+EXk∼Fj
+� K
+�
+k=1
+XkX⊤
+k
+��
+≥pminKα.
+Thus, with probability at least 1 − Tδ,
+λmin(Fν) ≥1
+2λmin
+� ν
+�
+u=1
+E
+�
+˜Xk,u ˜X⊤
+k,u
+��� Hu−1
+��
+≥1
+2
+ν
+�
+u=1
+λmin
+�
+E
+�
+˜Xk,u ˜X⊤
+k,u
+��� Hu−1
+��
+≥pminKαν
+2
+,
+
+Improved Algorithms for Multi-period Packing Problems with Bandit Feedback
+for all ν ∈ [nT ].
+Step 2. Bounding the probability of Mt:
+Under the event proved in Step 1, the event Mt is implied by
+pminKαnt
+2
+≥ 12Kd
+� nt
+�
+ν=1
+96 (K − 1) log
+� Jd
+δ
+�
+αKpminν
++ 2 log Jd
+δ
+�
+,
+(52)
+for all t ∈ [T]. The left hand side is bounded as
+12Kd
+� nt
+�
+ν=1
+96 (K − 1) log
+� Jd
+δ
+�
+αKpminν
++ 2 log Jd
+δ
+�
+≤12 · 96Kd log
+� Jd
+δ
+�
+log nt
+αpmin
++ 24Kd log Jd
+δ
+≤12 · 96Kd log
+� Jd
+δ
+�
+log T
+αpmin
++ 24Kd log Jd
+δ .
+Plugging in (52) and rearranging the terms,
+nt ≥ 96d log
+�Jd
+δ
+� �24 log T
+α2p2
+min
++
+1
+αpmin
+�
+,
+implies the event Mt for all t ∈ [T] with probability at least 1 − Tδ. In other words,
+P (Mc
+t) ≤ P
+�
+nt < dMα,p,T log
+�Jd
+δ
+��
++ Tδ,
+for all t ∈ [T], where Mα,p,T := 96
+�
+24 log T
+α2p2
+min +
+1
+αpmin
+�
+.
+Step 3. Bounding ξ:
+Let ˜t = inft∈[T ]{Mt happens} be the first round that Mt happens. After round ˜t, the algorithm
+skips the rounds until ρt > 0 holds and then pulls an action according to the policy. Thus, for the round ξ − 1,
+(ξ − 1) ρ −
+ξ−2
+�
+s=1
+b(js)
+as,s = (ξ − 1) ρ −
+˜t
+�
+s=1
+b(js)
+as,s ≤ 0.
+Rearraging the terms, and taking expectation,
+E [ξ] ≤ 1 + ρ−1E
+�
+�
+˜t
+�
+s=1
+b(js)
+as,s
+�
+� ≤ 1 + ρ−1E
+�˜t
+�
+.
+(53)
+Now we need an upper bound for ˜t. For t ∈ [˜t − 1], the event Mt does not happen and the algorithm admits the arrival for
+t ∈ [˜t]. Thus, nt = t for all t ∈ [˜t]. For t = ˜t − 1, the event M˜t−1 does not happen and
+λmin
+�
+Fn˜t−1
+�
+≤ 12Kd
+�n˜t−1
+�
+ν=1
+48 (K − 1) log
+� Jd
+δ
+�
+λmin(Fν)
++ 2 log Jd
+δ
+�
+.
+By the fact proved in Step 1, with probability at least 1 − Tδ,
+pminKαn˜t−1
+2
+≤ 12Kd
+�n˜t−1
+�
+ν=1
+96 (K − 1) log
+� Jd
+δ
+�
+pminKαν
++ 2 log Jd
+δ
+�
+.
+Plugging in n˜t−1 = ˜t − 1 and rearranging the terms,
+˜t − 1 ≤ 24d
+pminα
+�
+�
+�
+˜t−1
+�
+ν=1
+96 (K − 1) log
+� Jd
+δ
+�
+pminKαν
++ 2 log Jd
+δ
+�
+�
+�
+≤ 24d
+pminα
+�96 log T
+pminα + 2 log Jd
+δ
+�
+.
+
+Improved Algorithms for Multi-period Packing Problems with Bandit Feedback
+Then with probability at least 1 − Tδ,
+˜t ≤1 + 96d log
+�Jd
+δ
+� �24 log T
+α2p2
+min
++
+1
+αpmin
+�
+:=1 + Mα,p,T d log
+�Jd
+δ
+�
+.
+Thus,
+E
+�˜t
+�
+=E
+�
+˜tI
+�
+˜t < 1 + Mα,p,T d log
+�Jd
+δ
+���
++ E
+�
+˜tI
+�
+˜t ≥ 1 + Mα,p,T d log
+�Jd
+δ
+���
+≤1 + dMα,p,T log
+�Jd
+δ
+�
++ TP
+�
+˜t ≥ 1 + Mα,p,T d log
+�Jd
+δ
+��
+≤1 + dMα,p,T log
+�Jd
+δ
+�
++ T 2δ
+Plugging in (53),
+E [ξ] ≤1 + ρ−1E
+�˜t
+�
+≤1 + 1 + dMα,p,T log
+� Jd
+δ
+�
++ T 2δ
+ρ
+.
+Step 4. Proving the lower bound for τ:
+Let τ be the stopping time of the algorithm. Because the algorithm admits
+arrival at round τ, we have ρτ > 0. From the resource constraint in the bandit problem (12),
+K
+�
+k=1
+�π(jτ )
+k,τ
+�
+�b(jτ )
+k,τ − γτ−1,σb(δ)
+√pjτ
+1m
+�
+:=
+K
+�
+k=1
+�π(jτ )
+k,τ ˜b(jτ )
+k,τ :≤ τρ −
+τ−1
+�
+s=1
+b(js)
+as,s
+Because algorithm stops at round τ, there exists an r ∈ [m] such that �τ
+s=1 b(js)
+as,s(r) ≥ Tρ(r). Rearranging the terms,
+τρ ≥
+τ−1
+�
+s=1
+b(js)
+as,s(r) +
+K
+�
+k=1
+�π(jτ )
+k,τ ˜b(jτ )
+k,τ (r)
+≥Tρ − b(jτ )
+aτ ,τ(r) +
+K
+�
+k=1
+�π(jτ )
+k,τ ˜b(jτ )
+k,τ (r)
+=Tρ − b(jτ )
+aτ ,τ(r) +
+K
+�
+k=1
+�π(jτ )
+k,τ b⋆(jτ )
+k,τ
+(r) +
+K
+�
+k=1
+�π(jτ )
+k,τ
+�
+˜b(jτ )
+k,τ (r) − b⋆(jτ )
+k,τ
+(r)
+�
+≥Tρ − b(jτ )
+aτ ,τ(r) +
+K
+�
+k=1
+�π(jτ )
+k,τ b⋆(jτ )
+k
+(r) −
+K
+�
+k=1
+�π(jτ )
+k,τ
+���˜b(jτ )
+k,τ − b⋆(jτ )
+k
+���
+∞ .
+Taking expectation on both side,
+E [τρ] ≥Tρ + E
+�
+−b(jτ )
+aτ ,τ(r) +
+K
+�
+k=1
+�π(jτ )
+k,τ b⋆(jτ )
+k,τ
+(r)
+�
+− E
+� K
+�
+k=1
+�π(jτ )
+k,τ
+���˜b(jτ )
+k,τ − b⋆(jτ )
+k
+���
+∞
+�
+=Tρ − E
+� K
+�
+k=1
+�π(jτ )
+k,τ
+���˜b(jτ )
+k,τ − b⋆(jτ )
+k
+���
+∞
+�
+=Tρ − E
+� K
+�
+k=1
+�π(jτ )
+k,τ
+���˜b(jτ )
+k,τ − b⋆(jτ )
+k
+���
+∞ I (Eτ ∩ Mτ−1)
+�
+− E
+� K
+�
+k=1
+�π(jτ )
+k,τ
+���˜b(jτ )
+k,τ − b⋆(jτ )
+k
+���
+∞ I
+�
+Ec
+τ ∪ Mc
+τ−1
+�
+�
+≥Tρ − 2E
+�
+γτ−1,σb(δ)
+�
+E [I (at ∈ [K])| τ]
+�
+− E
+� K
+�
+k=1
+�π(jτ )
+k,τ
+���˜b(jτ )
+k,τ − b⋆(jτ )
+k
+���
+∞ I
+�
+Ec
+τ ∪ Mc
+τ−1
+�
+�
+,
+(54)
+
+Improved Algorithms for Multi-period Packing Problems with Bandit Feedback
+where the last inequality holds by (44).
+Because ˜b(jτ )
+k,τ (r) ≤ Tρ almost surely,
+E
+� K
+�
+k=1
+�π(jτ )
+k,τ
+���˜b(jτ )
+k,τ − b⋆(jτ )
+k
+���
+∞ I
+�
+Ec
+τ ∪ Mc
+τ−1
+�
+�
+≤TρP
+�
+Ec
+τ ∪ Mc
+τ−1
+�
+=TρP (Ec
+τ)
+≤Tρ
+�
+4(m + 1)δ + 7T −1�
+=7ρ + 4(m + 1)Tδ,
+where the equality holds because the algorithm takes action according to the policy at round τ and the last inequality holds
+by Theorem 4.2. from (54),
+E [τρ] ≥Tρ − 7ρ + 4(m + 1)Tρδ − 2E
+�
+γτ−1,σb(δ)
+�
+E [I (at ∈ [K])| τ]
+�
+≥Tρ − 7ρ + 4(m + 1)Tρδ − 2γ1,σb(δ)
+Rearranging the terms,
+E [T − τ] ≤4(m + 1)Tδ + 7 + 2γτ−1,σb(δ)
+ρ
+.
+Step 5. Proving a bound for the sum of probabilities
+Because the algorithm admits the arrival when Mt−1 does not
+happen,
+Mc
+t−1 = Mc
+t−1 ∩ {at ∈ [K]} .
+Then
+P
+�
+Mc
+t−1
+�
+=P
+�
+Mc
+t−1 ∩ {at ∈ [K]}
+�
+=P
+�
+Mc
+t−1 ∩ {at ∈ [K]} ∩
+�
+nt−1 ≥ Mα,p,T d log
+�Jd
+δ
+���
++ P
+�
+Mc
+t−1 ∩ {at ∈ [K]} ∩
+�
+nt−1 < Mα,p,T d log
+�Jd
+δ
+���
+≤P
+�
+Mc
+t−1 ∩
+�
+nt−1 ≥ Mα,p,T d log
+�Jd
+δ
+���
++ P
+�
+{at ∈ [K]} ∩
+�
+nt−1 < Mα,p,T d log
+�Jd
+δ
+���
+≤Tδ + P
+�
+{at ∈ [K]} ∩
+�
+nt−1 < Mα,p,T d log
+�Jd
+δ
+���
+,
+where the last inequality holds by the fact proved in Step 2. Summing over t ∈ [T],
+T
+�
+t=1
+P
+�
+Mc
+t−1
+�
+≤T 2δ +
+T
+�
+t=1
+P
+�
+{at ∈ [K]} ∩
+�
+nt−1 < Mα,p,T d log
+�Jd
+δ
+���
+=T 2δ + E
+� T
+�
+t=1
+I (at ∈ [K]) I
+�
+nt−1 < Mα,p,T d log
+�Jd
+δ
+���
+.
+Set µ := Mα,p,T d log
+� Jd
+δ
+�
+and suppose
+T
+�
+t=1
+I (at ∈ [K]) I (nt−1 < µ) > µ.
+(55)
+
+Improved Algorithms for Multi-period Packing Problems with Bandit Feedback
+Let τ(1) < τ(2) < · · · < τ(|A|) be the ordered admitted round in A := {t ∈ [T] : at ∈ [K]}. By definition, nτ(ν) = ν
+for ν ∈ [|A|]. By (55), the event{at ∈ [K]} happens at least µ + 1 times over the horizon [T] and |A| > µ. For any
+ν ∈ (µ, |A|],the number of admitted round is nτ(ν) > µ and
+T −1
+�
+t=ε
+I (nt−1 < µ) I (at ∈ [K]) =
+|A|
+�
+ν=1
+I
+�
+nτ(ν)−1 < µ
+�
+I
+�
+aτ(ν) ∈ [K]
+�
+≤
+|A|
+�
+ν=1
+I
+�
+nτ(ν)−1 < µ
+�
+I
+�
+nτ(ν) = nτ(ν)−1 + 1
+�
+=
+|A|
+�
+ν=1
+I
+�
+nτ(ν)−1 < µ
+�
+I
+�
+ν = nτ(ν)−1 + 1
+�
+≤
+|A|
+�
+ν=1
+I (ν − 1 < µ) ,
+=
+|A|
+�
+ν=1
+I (ν < µ + 1)
+=µ,
+which contradicts with (55). Thus
+E
+� T
+�
+t=1
+I (at ∈ [K]) I
+�
+nt−1 < Mα,p,T d log
+�Jd
+δ
+���
+≤ µ := Mα,p,T d log
+�Jd
+δ
+�
+,
+which proves,
+T
+�
+t=1
+P
+�
+Mc
+t−1
+�
+≤ T 2δ + Mα,p,T d log
+�Jd
+δ
+�
+.
+B.6. Proof of Theorem 5.1
+Proof. From Lemma 5.2, rearranging the terms,
+R�π
+T :=OPT − E
+� T
+�
+t=1
+R�π
+t
+�
+≤OPT
+T
+{T − E [τ − ξ]}
++
+�
+2 + OPT
+ρT
+�
+T
+�
+t=1
+P
+�
+Mc
+t−1 ∪ Ec
+t
+�
++ 2
+�
+�
+�
+�TE
+� T
+�
+t=1
+γt−1,σr(δ)2I (at ∈ [K])
+�
++ 2
+�
+�
+�
+�TE
+� T
+�
+t=1
+γt−1,σb(δ)2I (at ∈ [K])
+�
+OPT
+ρT .
+
+Improved Algorithms for Multi-period Packing Problems with Bandit Feedback
+By Lemma 5.3,
+E [ξ] ≤ 1 + 1+dMα,p,T log
+� Jd
+δ
+�
++T 2δ
+ρ
+,
+E [T − τ] ≤ 4(m + 1)Tδ + 7 + 2γτ−1,σb(δ)
+ρ
+.
+By definition of γt,σ(δ),
+E [T − τ] ≤4(m + 1)Tδ + 7 + 32√J log JKT + 8
+√
+2βσb(δ)
+ρ
+=4(m + 1)Tδ + 7 + 32√J log JKT + Cσ(δ)
+√
+Jd
+ρ
+.
+This implies
+OPT
+T
+{T − E [τ − ξ]}
+≤ OPT
+Tρ
+�
+ρ + 4(m + 1)Tδ + 8 + 32
+�
+J log
+�JK
+δ
+�
++Cσb(δ)
+√
+Jd + dMα,p,T log
+�Jd
+δ
+�
++T 2δ
+�
+≤ OPT
+Tρ
+�
+ρ + 8 +
+�
+5mT + T 2�
+δ + 32
+�
+J log
+�JK
+δ
+�
++Cσb(δ)
+√
+Jd + dMα,p,T log
+�Jd
+δ
+��
+.
+[Step 3. Bounding the sum of probability] Because T ≥ 8dα−1p−1
+min log JdT, by Theorem 4.2 and Lemma 5.3,
+T
+�
+t=1
+P
+�
+Mc
+t−1 ∪ Ec
+t
+�
+=
+T
+�
+t=1
+�
+P
+�
+Mc
+t−1
+�
++ P (Mt−1 ∩ Ec
+t )
+�
+≤T 3δ + dMα,p,T log
+�Jd
+δ
+�
++
+T
+�
+t=1
+P (Mt−1 ∩ Ec
+t )
+≤T 3δ + dMα,p,T log
+�Jd
+δ
+�
++ 8dα−1p−1
+min log JdT
++
+T
+�
+t=8dα−1p−1
+min log JdT
+P (Mt−1 ∩ Ec
+t )
+≤T 3δ + dMα,p,T log
+�Jd
+δ
+�
++ 8dα−1p−1
+min log JdT + 4(m + 1)Tδ + 7.
+By definition of γt,σ(δ) and βσ(δ),
+E
+� T
+�
+t=1
+γt−1,σr(δ)2I (at ∈ [K])
+�
+=E
+�
+�
+T
+�
+t=1
+�
+16√J log JKT
+√t − 1
++ 4
+√
+2βσr(δ)
+√nt−1
+�2
+I (at ∈ [K])
+�
+�
+≤E
+� T
+�
+t=1
+�
+16√J log JKT + 4
+√
+2βσr(δ)
+�2
+nt−1
+I (at ∈ [K])
+�
+≤E
+� T
+�
+t=1
+�
+16√J log JKT + 4
+√
+2βσr(δ)
+�2
+nt−1
+I (nt = nt−1 + 1)
+�
+≤
+�
+16
+�
+J log JKT + 4
+√
+2βσr(δ)
+�2
+log T,
+
+Improved Algorithms for Multi-period Packing Problems with Bandit Feedback
+where the first inequality holds by nt ≤ t almost surely. Thus by definition of βσ(δ) := 8
+√
+Jd + 96σ
+�
+Jd log 4
+δ ,
+2
+�
+�
+�
+�TE
+� T
+�
+t=1
+γt−1,σr(δ)2I (at ∈ [K])
+�
+≤
+�
+32
+�
+J log JKT + 4
+√
+6βσr(δ)
+� �
+T log T
+≤
+�
+32
+�
+J log JKT + Cσr(δ)
+√
+Jd
+� �
+T log T,
+where Cσ(δ) := 8
+√
+2 ·
+�
+8 + 96σ
+�
+log 4
+δ
+�
+. Similarly,
+2
+�
+�
+�
+�TE
+� T
+�
+t=1
+γt−1,σb(δ)2I (at ∈ [K])
+�
+OPT
+ρT
+≤
+�
+32
+�
+J log JKT + Cσb(δ)
+√
+Jd
+� �
+T log T OPT
+ρT
+Collecting the bounds,
+R�π
+T ≤ OPT
+Tρ
+�
+ρ + 8 +
+�
+5mT + T 2�
+δ + 32
+�
+J log JKT +Cσb(δ)
+√
+Jd + dMα,p,T log
+�Jd
+δ
+��
++
+�
+2 + OPT
+ρT
+� �
+T 3δ + dMα,p,T log
+�Jd
+δ
+�
++ 4dα−1p−1
+min log JdT + 4(m + 1)Tδ + 7
+�
++ 2
+�
+1 + OPT
+ρT
+� �
+32
+�
+J log JKT + Cσb∨σr(δ)
+√
+Jd
+� �
+T log T
+≤
+�
+2 + OPT
+ρT
+� � �
+96
+�
+J log JKT + 3Cσr∨σr(δ)
+√
+Jd
+� �
+T log T + 2dMα,p,T log
+�Jd
+δ
+�
++ 4dα−1p−1
+min log JdT + 15 + 10mT 3δ
+�
+,
+Plugging in δ = m−1T −3 proves (14).
+C. Technical lemmas
+Lemma C.1. (Azuma-Hoeffding’s inequality) Azuma (1967) If a super-martingale (Yt; t ≥ 0) corresponding to filtration
+Ft, satisfies |Yt − Yt−1| ≤ ct for some constant ct, for all t = 1, . . . , T, then for any a ≥ 0,
+P (YT − Y0 ≥ a) ≤ e
+−
+a2
+2 �T
+t=1 c2
+t .
+Thus with probability at least 1 − δ,
+YT − Y0 ≤
+�
+�
+�
+�2 log 1
+δ
+T
+�
+t=1
+c2
+t.
+Lemma C.2. For a sequence u1 ≥ u2 ≥ · · · ≥ un ≥ 0 and nonnegative real sequences {pi}i∈[n] and {qi}i∈[n] such that
+�n
+i=1 pi = �n
+i=1 qi, if p1 > q1 then
+n
+�
+i=1
+piui ≥
+n
+�
+i=1
+qiui.
+Proof. When n = 1, p1u1 ≥ q1u1, for any u1 ≥ 0. Suppose for any sequence u1 ≥ u2 ≥ · · · ≥ un−1 ≥ 0 and nonnegative
+real sequences {pi}i∈[n−1] and {qi}i∈[n−1] such that �n−1
+i=1 pi = �n−1
+i=1 qi,
+p1 > q1 =⇒
+n−1
+�
+i=1
+piui ≥
+n−1
+�
+i=1
+qiui.
+
+Improved Algorithms for Multi-period Packing Problems with Bandit Feedback
+For a sequence u1 ≥ u2 ≥ · · · ≥ un ≥ 0 and nonnegative real sequences {pi}i∈[n] and {qi}i∈[n] such that �n
+i=1 pi =
+�n
+i=1 qi, and p1 > q1, there exist k ∈ [n]\{1} such that pk < qk. In case of k = n, define a sequence
+˜qi = qi,
+∀i ∈ [n − 2]
+˜qn−1 = qn−1 − pn + qn ≥ 0.
+Then �n−1
+i=1 ˜qi = �n−1
+i=1 pi and
+n
+�
+i=1
+piui =
+n−1
+�
+i=1
+piui + pnun
+≥
+n−1
+�
+i=1
+˜qiui + pnun
+=
+n−1
+�
+i=1
+qiui + (−pn + qn) un−1 + pnun
+≥
+n−1
+�
+i=1
+qiui + (−pn + qn) un + pnun
+=
+n
+�
+i=1
+qiui.
+In case of k ̸= n, denote a sequence
+˜qi = qi,
+∀i ∈ [n − 1]\{k}
+˜qk = qk − pk + qn.
+Then �n−1
+i=1 ˜qi = �
+j̸=k pi and
+n
+�
+i=1
+piui =
+�
+i̸=k
+piui + pkuk
+≥
+n−1
+�
+i=1
+˜qiui + pkuk
+≥
+n−1
+�
+i=1
+qiui − pkuk + qnuk + pkuk
+=
+n−1
+�
+i=1
+qkuk + qnuk
+≥
+n
+�
+i=1
+qkuk.
+By induction, the proof is complete.
+Lemma C.3. Let {Xτ : τ ∈ [t]} be a Rd×d-valued stochastic process adapted to the filtration {Fτ : τ ∈ [t]}, i.e., Xτ
+is Fτ-measurable for τ ∈ [t]. Suppose Xτ is a positive definite symmetric matrices such thatλmax(Xτ) ≤ 1
+2.Then with
+probability at least 1 − δ,
+t
+�
+τ=1
+Xτ ⪰ 1
+2
+t
+�
+τ=1
+E [Xτ| Fτ−1] − log d
+δ Id.
+
+Improved Algorithms for Multi-period Packing Problems with Bandit Feedback
+In addition, with probability at least 1 − δ,
+t
+�
+τ=1
+Xτ ⪯ 3
+2
+t
+�
+τ=1
+E [Xτ| Fτ−1] + log d
+δ Id.
+Proof. This proof is an adapted version of Lemma 12.2 in Lattimore & Szepesv´ari (2020) for matrix stochastic process
+using the argument of Tropp (2012). For the lower bound, It is sufficient to prove that
+λmax
+�
+−
+t
+�
+τ=1
+Xτ + 1
+2
+t
+�
+τ=1
+E [Xτ| Fτ−1]
+�
+≤ log d
+δ ,
+with probability at least 1 − δ. By the spectral mapping theorem,
+exp
+�
+λmax
+�
+−
+t
+�
+τ=1
+Xτ + 1
+2
+t
+�
+τ=1
+E [Xτ| Fτ−1]
+��
+≤λmax
+�
+exp
+�
+−
+t
+�
+τ=1
+Xτ + 1
+2
+t
+�
+τ=1
+E [Xτ| Fτ−1]
+��
+≤Tr
+�
+exp
+�
+−
+t
+�
+τ=1
+Xτ + 1
+2
+t
+�
+τ=1
+E [Xτ| Fτ−1]
+��
+.
+Taking expectation on both side gives,
+E exp
+�
+λmax
+�
+−
+t
+�
+τ=1
+Xτ + 1
+2
+t
+�
+τ=1
+E [Xτ| Fτ−1]
+��
+≤ETr
+�
+exp
+�
+−
+t
+�
+τ=1
+Xτ + 1
+2
+t
+�
+τ=1
+E [Xτ| Fτ−1]
+��
+=ETr
+�
+E
+�
+exp
+�
+−
+t−1
+�
+τ=1
+Xτ + 1
+2
+t
+�
+τ=1
+E [Xτ| Fτ−1] + log exp (−Xt)
+������ Ft−1
+��
+≤ETr
+�
+exp
+�
+−
+t−1
+�
+τ=1
+Xτ + 1
+2
+t
+�
+τ=1
+E [Xτ| Fτ−1] + log E [exp (−Xt)| Ft−1]
+��
+.
+The last inequality holds due to Lieb’s theorem Tropp (2015). Because ex ≤ 1+ 1
+2xfor all x ∈ [−1/2, 0], and the eigenvalue
+of −Xt lies in [−1/2, 0], we have
+E [exp (−Xt)| Ft−1] ⪯ I − 1
+2E [Xt| Ft−1] ⪯ exp
+�
+−1
+2E [Xt| Ft−1]
+�
+,
+by the spectral mapping theorem. Thus we have
+E exp
+�
+λmax
+�
+−
+t
+�
+τ=1
+Xτ + 1
+2
+t
+�
+τ=1
+E [Xτ| Fτ−1]
+��
+≤ ETr
+�
+exp
+�
+−
+t−1
+�
+τ=1
+Xτ + 1
+2
+t
+�
+τ=1
+E [Xτ| Fτ−1] + log exp
+�
+−1
+2E [Xt| Ft−1]
+���
+= ETr
+�
+exp
+�
+−
+t−1
+�
+τ=1
+Xτ + 1
+2
+t
+�
+τ=1
+E [Xτ| Fτ−1] − 1
+2E [Xt| Ft−1]
+��
+= ETr
+�
+exp
+�
+−
+t−1
+�
+τ=1
+Xτ + 1
+2
+t−1
+�
+τ=1
+E [Xτ| Fτ−1]
+��
+≤ ...
+≤ ETr (exp (O)) = d
+
+Improved Algorithms for Multi-period Packing Problems with Bandit Feedback
+Now my Markov’s inequality,
+P
+�
+λmax
+�
+−
+t
+�
+τ=1
+Xτ + 1
+2
+t
+�
+τ=1
+E [Xτ| Fτ−1]
+�
+> log d
+δ
+�
+≤ E exp
+�
+λmax
+�
+−
+t
+�
+τ=1
+Xτ + 1
+2
+t
+�
+τ=1
+E [Xτ| Fτ−1]
+��
+δ
+d
+≤ δ.
+For the upper bound, we prove
+λmax
+�
+t
+�
+τ=1
+Xτ − 3
+2
+t
+�
+τ=1
+E [Xτ| Fτ−1]
+�
+≤ log d
+δ ,
+in a similar way using the fact that ex ≤ 1 + (3/2)x on x ∈ [0, 1/2].
+Lemma C.4. Suppose a random variable X satisfies E[X] = 0, and let Y be an σ-sub-Gaussian random variable. If
+|X| ≤ |Y | almost surely, then X is 6σ-sub-Gaussian.
+Proof. Because |X| ≤ |Y |
+E
+� X2
+6σ2
+�
+≤E
+� Y 2
+6σ2
+�
+.
+=1 + E
+�� ∞
+0
+I (|Y | ≥ x) x
+3σ2 e
+x2
+6σ2 dx
+�
+≤1 +
+� ∞
+0
+P (|Y | ≥ x) x
+3σ2 e
+x2
+6σ2 dx.
+Because
+P (|Y | ≥ x) =P (Y ≥ x) + P (−Y ≤ x)
+≤2e− x2
+2σ2 ,
+we have
+E
+� X2
+6σ2
+�
+≤1 +
+� ∞
+0
+2x
+3σ2 e− x2
+3σ2 dx
+≤2.
+Now for any λ ∈ R,
+E [exp (λX)] =E
+� ∞
+�
+n=0
+(λX)n
+n!
+�
+=1 + E
+� ∞
+�
+n=2
+(λX)n
+n!
+�
+≤1 + E
+�
+λ2X2
+2
+∞
+�
+n=2
+|λX|n−2
+(n − 2)!
+�
+≤1 + λ2
+2 E
+�
+X2 exp (|λX|)
+�
+.
+
+Improved Algorithms for Multi-period Packing Problems with Bandit Feedback
+Because 6σ2λ2 +
+X2
+12σ2 ≥ |λX| ,
+E [exp (λX)] ≤1 + λ2
+2 exp
+�
+6σ2λ2�
+E
+�
+X2 exp
+� X2
+12σ2
+��
+=1 + 6σ2λ2 exp
+�
+6σ2λ2�
+E
+� X2
+12σ2 exp
+� X2
+12σ2
+��
+≤1 + 6σ2λ2 exp
+�
+6σ2λ2�
+E
+�
+exp
+� X2
+6σ2
+��
+≤1 + 12σ2λ2 exp
+�
+6σ2λ2�
+≤
+�
+1 + 12σ2λ2�
+exp
+�
+6σ2λ2�
+≤ exp
+�36
+2 σ2λ2
+�
+.
+Thus X is 6σ-sub-Gaussian.
+Lemma C.5. (Lee et al., 2016, Lemma 2.3) Let {Nt} be a martingale on a Hilbert space (H, ∥·∥H). Then there exists a
+R2-valued martingale {Pt} such that for any time t ≥ 0, ∥Pt∥2 = ∥Nt∥H and ∥Pt+1 − Pt∥2 = ∥Nt+1 − Nt∥H.
+Lemma C.6. (A dimension-free bound for vector-valued martingales.) Let {Fs}t
+s=0 be a filtration and {ηs}t
+s=1 be a
+real-valued stochastic process such that ηs is Fτ-measurable. Let {Xs}t
+s=1 be an Rd-valued stochastic process where Xs
+is F0-measurable. Assume that {ηs}t
+s=1 are σ-sub-Gaussian as in Assumption 1. Then with probability at least 1 − δ,
+�����
+t
+�
+s=1
+ηsXs
+�����
+2
+≤ 12σ
+�
+�
+�
+�
+t
+�
+s=1
+∥Xs∥2
+2
+�
+log 4t2
+δ .
+(56)
+Proof. Fix a t ≥ 1. For each s = 1, . . . , t, we have E [ηs| Fs−1] = 0 and Xs is F0-measurable. Thus the stochastic process,
+� u
+�
+s=1
+ηsXs
+�t
+u=1
+(57)
+is a (Rd, ∥·∥2)-martingale. Since (Rd, ∥·∥2) is a Hilbert space, by Lemma C.5, there exists an R2-martingale {Mu}t
+u=1
+such that
+�����
+u
+�
+s=1
+ηsXs
+�����
+2
+= ∥Mu∥2 , ∥ηuXu∥2 = ∥Mu − Mu−1∥2 ,
+(58)
+and M0 = 0. Set Mu = (M1(u), M2(u))⊤. Then for each i = 1, 2, and u ≥ 2,
+|Mi(u) − Mi(u − 1)| ≤ ∥Mu − Mu−1∥2
+= ∥ηuXu∥2
+= |ηu| ∥Xu∥2 ,
+almost surely. By Lemma C.4, Mi(u) − Mi(u − 1) is 6σ-sub-Gaussian. By Lemma C.1, for x > 0,
+P (|Mi(t)| > x) =P
+������
+t
+�
+u=1
+Mi(u) − Mi(u − 1)
+����� > x
+�
+≤2 exp
+�
+−
+x2
+72tσ2 �t
+s=1 ∥Xs∥2
+2
+�
+,
+for each i = 1, 2. Thus, with probability 1 − δ/2,
+Mi(t)2 ≤ 72
+�
+t
+�
+s=1
+∥Xs∥2
+2
+�
+σ2 log 4
+δ .
+
+Improved Algorithms for Multi-period Packing Problems with Bandit Feedback
+In summary, with probability at least 1 − δ/2,
+�����
+t
+�
+τ=1
+ηsXs
+�����
+2
+=
+�
+M1(t)2 + M2(t)2 ≤ 6σ
+�
+�
+�
+�
+t
+�
+s=1
+∥Xs∥2
+2
+�
+2 log 4t2
+δ .
+