We consider a contextual version of multi-armed bandit problem with global knapsack constraints. In each round, the outcome of pulling an arm is a scalar reward and a resource consumption vector, both dependent on the context, and the global knapsack constraints require the total consumption for each resource to be below some pre-fixed budget. The learning agent competes with an arbitrary set of context-dependent policies. This problem was introduced by Badanidiyuru et al. (2014), who gave a computationally inefficient algorithm with near-optimal regret bounds for it. We give a computationally efficient algorithm for this problem with slightly better regret bounds, by generalizing the approach of Agarwal et al. (2014) for the non-constrained version of the problem. The computational time of our algorithm scales logarithmically in the size of the policy space. This answers the main open question of Badanidiyuru et al. (2014). We also extend our results to a variant where there are no knapsack constraints but the objective is an arbitrary Lipschitz concave function of the sum of outcome vectors.
University of Maryland, College Park（马里兰大学帕克分校）；Microsoft Research NYC（微软研究院纽约市）
We unify two prominent lines of work on multi-armed bandits: bandits with knapsacks (BwK) and combinatorial semi-bandits. The former concerns limited "resources" consumed by the algorithm, e.g., limited supply in dynamic pricing. The latter allows a huge number of actions but assumes combinatorial structure and additional feedback to make the problem tractable. We define a common generalization, support it with several motivating examples, and design an algorithm for it. Our regret bounds are comparable with those for BwK and combinatorial semi- bandits.
Bandits with Knapsacks (BwK) is a general model for multi-armed bandits under supply/budget constraints. While worst-case regret bounds for BwK are well-understood, we present three results that go beyond the worst-case perspective. First, we provide upper and lower bounds which amount to a full characterization for logarithmic, instance-dependent regret rates. Second, we consider "simple regret" in BwK, which tracks algorithm's performance in a given round, and prove that it is small in all but a few rounds. Third, we provide a general "reduction" from BwK to bandits which takes advantage of …
University of Maryland（马里兰大学）；Microsoft Research（微软研究院）；Princeton University（普林斯顿大学）；UC Berkeley（加州大学伯克利分校）；Cornell University（康奈尔大学）
We propose an algorithm for tabular episodic reinforcement learning with constraints. We provide a modular analysis with strong theoretical guarantees for settings with concave rewards and convex constraints, and for settings with hard constraints (knapsacks). Most of the previous work in constrained reinforcement learning is limited to linear constraints, and the remaining work focuses on either the feasibility question or settings with a single episode. Our experiments demonstrate that the proposed algorithm significantly outperforms these approaches in existing constrained episodic environm…
Imperial College Business School（帝国理工学院商学院）；Stanford University（斯坦福大学）
In this paper, we study the bandits with knapsacks (BwK) problem and develop a primal-dual based algorithm that achieves a problem-dependent logarithmic regret bound. The BwK problem extends the multi-arm bandit (MAB) problem to model the resource consumption associated with playing each arm, and the existing BwK literature has been mainly focused on deriving asymptotically optimal distribution-free regret bounds. We first study the primal and dual linear programs underlying the BwK problem. From this primal-dual perspective, we discover symmetry between arms and knapsacks, and then propose a …
Department of Mathematics, HKUST（香港科技大学数学系）；Department of Industrial Engineering and Decision Analytics, HKUST（香港科技大学工业工程与决策分析系）；HKUST Shenzhen-Hong Kong Collaborative Innovation Research Institute（香港科技大学深港协同创新研究院）
We study the stochastic contextual bandit with knapsacks (CBwK) problem, where each action, taken upon a context, not only leads to a random reward but also costs a random resource consumption in a vector form. The challenge is to maximize the total reward without violating the budget for each resource. We study this problem under a general realizability setting where the expected reward and expected cost are functions of contexts and actions in some given general function classes $\mathcal{F}$ and $\mathcal{G}$, respectively. Existing works on CBwK are restricted to the linear function class …
Columbia University（哥伦比亚大学）；Microsoft Research（微软研究院）
We consider the linear contextual bandit problem with resource consumption, in addition to reward generation. In each round, the outcome of pulling an arm is a reward as well as a vector of resource consumptions. The expected values of these outcomes depend linearly on the context of that arm. The budget/capacity constraints require that the total consumption doesn't exceed the budget for each resource. The objective is once again to maximize the total reward. This problem turns out to be a common generalization of classic linear contextual bandits (linContextual), bandits with knapsacks (BwK)…
Microsoft Research（微软研究院）；Facebook（脸书公司）；University of Maryland（马里兰大学）
We consider Bandits with Knapsacks (henceforth, BwK), a general model for multi-armed bandits under supply/budget constraints. In particular, a bandit algorithm needs to solve a well-known knapsack problem: find an optimal packing of items into a limited-size knapsack. The BwK problem is a common generalization of numerous motivating examples, which range from dynamic pricing to repeated auctions to dynamic ad allocation to network routing and scheduling. While the prior work on BwK focused on the stochastic version, we pioneer the other extreme in which the outcomes can be chosen adversariall…
The University of Sydney Business School（悉尼大学商学院）；University of Southampton（南安普顿大学）
In budget-limited multi-armed bandit (MAB) problems, the learner's actions are costly and constrained by a fixed budget. Consequently, an optimal exploitation policy may not be to pull the optimal arm repeatedly, as is the case in other variants of MAB, but rather to pull the sequence of different arms that maximises the agent's total reward within the budget. This difference from existing MABs means that new approaches to maximising the total reward are required. Given this, we develop two pulling policies, namely: (i) KUBE; and (ii) fractional KUBE. Whereas the former provides better perform…
In this paper, we consider a very general model for exploration-exploitation tradeoff which allows arbitrary concave rewards and convex constraints on the decisions across time, in addition to the customary limitation on the time horizon. This model subsumes the classic multi-armed bandit (MAB) model, and the Bandits with Knapsacks (BwK) model of Badanidiyuru et al.[2013]. We also consider an extension of this model to allow linear contexts, similar to the linear contextual extension of the MAB model. We demonstrate that a natural and simple extension of the UCB family of algorithms for MAB pr…
Bandits with knapsacks (BwK) is an influential model of sequential decision-making under uncertainty that incorporates resource consumption constraints. In each round, the decision-maker observes an outcome consisting of a reward and a vector of nonnegative resource consumptions, and the budget of each resource is decremented by its consumption. In this paper we introduce a natural generalization of the stochastic BwK problem that allows non-monotonic resource utilization. In each round, the decision-maker observes an outcome consisting of a reward and a vector of resource drifts that can be p…
We consider contextual bandits with knapsacks, with an underlying structure between rewards generated and cost vectors suffered. We do so motivated by sales with commercial discounts. At each round, given the stochastic i.i.d.\ context $\mathbf{x}_t$ and the arm picked $a_t$ (corresponding, e.g., to a discount level), a customer conversion may be obtained, in which case a reward $r(a,\mathbf{x}_t)$ is gained and vector costs $c(a_t,\mathbf{x}_t)$ are suffered (corresponding, e.g., to losses of earnings). Otherwise, in the absence of a conversion, the reward and costs are null. The reward and c…
Department of Statistics and Data Science, University of Pennsylvania（统计与数据科学系，宾夕法尼亚大学）；Department of Mathematics, The Hong Kong University of Science and Technology（数学系，香港科学与技术大学）；Department of Industrial Engineering & Decision Analytics, The Hong Kong University of Science and Technology（工业工程与决策分析系，香港科学与技术大学）
University of Washington（华盛顿大学）；Tsinghua University（清华大学）；Peking University（北京大学）；Princeton University（普林斯顿大学）；Microsoft Research（微软研究院）
Bandit problems with linear or concave reward have been extensively studied, but relatively few works have studied bandits with non-concave reward. This work considers a large family of bandit problems where the unknown underlying reward function is non-concave, including the low-rank generalized linear bandit problems and two-layer neural network with polynomial activation bandit problem. For the low-rank generalized linear bandit problem, we provide a minimax-optimal algorithm in the dimension, refuting both conjectures in [LMT21, JWWN19]. Our algorithms are based on a unified zeroth-order o…
University of California, San Diego（加州大学圣地亚哥分校）；University of Southampton（南安普顿大学）
This paper investigates the adversarial Bandits with Knapsack (BwK) online learning problem, where a player repeatedly chooses to perform an action, pays the corresponding cost, and receives a reward associated with the action. The player is constrained by the maximum budget $B$ that can be spent to perform actions, and the rewards and the costs of the actions are assigned by an adversary. This problem has only been studied in the restricted setting where the reward of an action is greater than the cost of the action, while we provide a solution in the general setting. Namely, we propose EXP3.…
Blind Network Revenue Management and Bandits with Knapsacks under Limited Switches
Department of Civil and Environmental Engineering, Operations Research Center, and Institute for Data, Systems, and Society, Massachusetts Institute of Technology（土木与环境工程系、运筹中心及数据、系统与社会研究所，麻省理工学院）；Department of Industrial and Enterprise Systems Engineering, Grainger College of Engineering, University of Illinois（工业与企业系统工程系、格拉inger工程学院，伊利诺伊大学）；Questrom School of Business, Boston University（乔斯特罗商学院，波士顿大学）
We consider Contextual Bandits with Concave Rewards (CBCR), a multi-objective bandit problem where the desired trade-off between the rewards is defined by a known concave objective function, and the reward vector depends on an observed stochastic context. We present the first algorithm with provably vanishing regret for CBCR without restrictions on the policy space, whereas prior works were restricted to finite policy spaces or tabular representations. Our solution is based on a geometric interpretation of CBCR algorithms as optimization algorithms over the convex set of expected rewards spann…
Google Research（谷歌研究院）；Cornell University（康奈尔大学）；Microsoft Research（微软研究院）
Multi-armed bandit problems are the predominant theoretical model of exploration-exploitation tradeoffs in learning, and they have countless applications ranging from medical trials, to communication networks, to Web search and advertising. In many of these application domains the learner may be constrained by one or more supply (or budget) limits, in addition to the customary limitation on the time horizon. The literature lacks a general model encompassing these sorts of problems. We introduce such a model, called "bandits with knapsacks", that combines aspects of stochastic integer programmi…
Quantum Algorithms for Bandits with Knapsacks with Improved Regret and Time Complexities
Center on Frontiers of Computing Studies, Peking University School of Computer Science, Peking University（前沿计算研究所以及北京大学计算机科学学院，北京大学）；Guanghua School of Management, Peking University（光华管理学院，北京大学）；Department of Management Science and Engineering, Stanford University（管理科学与工程系，斯坦福大学）
University of Bonn（波恩大学）；Princeton University（普林斯顿大学）；Institute for Advanced Study（高等研究院）
We introduce online learning with vector costs (\OLVCp) where in each time step $t \in \{1,\ldots, T\}$, we need to play an action $i \in \{1,\ldots,n\}$ that incurs an unknown vector cost in $[0,1]^{d}$. The goal of the online algorithm is to minimize the $\ell_p$ norm of the sum of its cost vectors. This captures the classical online learning setting for $d=1$, and is interesting for general $d$ because of applications like online scheduling where we want to balance the load between different machines (dimensions). We study \OLVCp in both stochastic and adversarial arrival settings, and give…
Universidad de Santiago de Chile（智利圣地亚哥大学）；Lancaster University（兰卡斯特大学）；Univ. Grenoble Alpes（格勒诺布尔阿尔卑斯大学）；CNRS（法国国家科学研究中心）；Inria（法国国家信息与自动化研究所）
This paper examines the long-run behavior of learning with bandit feedback in non-cooperative concave games. The bandit framework accounts for extremely low-information environments where the agents may not even know they are playing a game; as such, the agents' most sensible choice in this setting would be to employ a no-regret learning algorithm. In general, this does not mean that the players' behavior stabilizes in the long run: no-regret learning may lead to cycles, even with perfect gradient information. However, if a standard monotonicity condition is satisfied, our analysis shows that …
Otto von Guericke University Magdeburg（马格德堡奥托·冯·格里克大学）
We investigate the problem dependent regime in the stochastic Thresholding Bandit problem (TBP) under several shape constraints. In the TBP, the objective of the learner is to output, at the end of a sequential game, the set of arms whose means are above a given threshold. The vanilla, unstructured, case is already well studied in the literature. Taking $K$ as the number of arms, we consider the case where (i) the sequence of arm's means $(μ_k)_{k=1}^K$ is monotonically increasing (MTBP) and (ii) the case where $(μ_k)_{k=1}^K$ is concave (CTBP). We consider both cases in the problem dependent …
The linear submodular bandit problem was proposed to simultaneously address diversified retrieval and online learning in a recommender system. If there is no uncertainty, this problem is equivalent to a submodular maximization problem under a cardinality constraint. However, in some situations, recommendation lists should satisfy additional constraints such as budget constraints, other than a cardinality constraint. Thus, motivated by diversified retrieval considering budget constraints, we introduce a submodular bandit problem under the intersection of $l$ knapsacks and a $k$-system constrain…
We study contextual bandits with ancillary constraints on resources, which are common in real-world applications such as choosing ads or dynamic pricing of items. We design the first algorithm for solving these problems that handles constrained resources other than time, and improves over a trivial reduction to the non-contextual case. We consider very general settings for both contextual bandits (arbitrary policy sets, e.g. Dudik et al. (UAI'11)) and bandits with resource constraints (bandits with knapsacks, Badanidiyuru et al. (FOCS'13)), and prove a regret guarantee with near-optimal statis…
Carnegie Mellon University（卡内基梅隆大学）；University of Pennsylvania（宾夕法尼亚大学）
The goal of data-driven algorithm design is to obtain high-performing algorithms for specific application domains using machine learning and data. Across many fields in AI, science, and engineering, practitioners will often fix a family of parameterized algorithms and then optimize those parameters to obtain good performance on example instances from the application domain. In the online setting, we must choose algorithm parameters for each instance as they arrive, and our goal is to be competitive with the best fixed algorithm in hindsight. There are two major challenges in online data-driven…
City University of Hong Kong（香港城市大学）；University of Toronto（多伦多大学）；Fudan University（复旦大学）
In contextual continuum-armed bandits, the contexts $x$ and the arms $y$ are both continuous and drawn from high-dimensional spaces. The payoff function to learn $f(x,y)$ does not have a particular parametric form. The literature has shown that for Lipschitz-continuous functions, the optimal regret is $\tilde{O}(T^{\frac{d_x+d_y+1}{d_x+d_y+2}})$, where $d_x$ and $d_y$ are the dimensions of contexts and arms, and thus suffers from the curse of dimensionality. We develop an algorithm that achieves regret $\tilde{O}(T^{\frac{d_x+1}{d_x+2}})$ when $f$ is globally concave in $y$. The global concavi…
Penn State University（宾夕法尼亚州立大学）；University at Buffalo（布法罗大学）
We propose a minimax concave penalized multi-armed bandit algorithm under generalized linear model (G-MCP-Bandit) for a decision-maker facing high-dimensional data in an online learning and decision-making process. We demonstrate that the G-MCP-Bandit algorithm asymptotically achieves the optimal cumulative regret in the sample size dimension T , O(log T), and further attains a tight bound in the covariate dimension d, O(log d). In addition, we develop a linear approximation method, the 2-step weighted Lasso procedure, to identify the MCP estimator for the G-MCP-Bandit algorithm under non-iid …
We consider a budgeted combinatorial multi-armed bandit setting where, in every round, the algorithm selects a super-arm consisting of one or more arms. The goal is to minimize the total expected regret after all rounds within a limited budget. Existing techniques in this literature either fix the budget per round or fix the number of arms pulled in each round. Our setting is more general where based on the remaining budget and remaining number of rounds, the algorithm can decide how many arms to be pulled in each round. First, we propose CBwK-Greedy-UCB algorithm, which uses a greedy techniqu…
Google Research（谷歌研究）；University of Maryland（马里兰大学）；Microsoft Research（微软研究院）；Harvard University（哈佛大学）
We consider online variations of the Pandora's box problem (Weitzman. 1979), a standard model for understanding issues related to the cost of acquiring information for decision-making. Our problem generalizes both the classic Pandora's box problem and the prophet inequality framework. Boxes are presented online, each with a random value and cost drew jointly from some known distribution. Pandora chooses online whether to open each box given its cost, and then chooses irrevocably whether to keep the revealed prize or pass on it. We aim for approximation algorithms against adversaries that can c…
Imperial College London（伦敦帝国理工学院）；NYU Stern School of Business（纽约大学斯特恩商学院）
In this paper, we study the problem of bandits with knapsacks (BwK) in a non-stationary environment. The BwK problem generalizes the multi-arm bandit (MAB) problem to model the resource consumption associated with playing each arm. At each time, the decision maker/player chooses to play an arm, and s/he will receive a reward and consume certain amount of resource from each of the multiple resource types. The objective is to maximize the cumulative reward over a finite horizon subject to some knapsack constraints on the resources. Existing works study the BwK problem under either a stochastic o…
Otto von Guericke University Magdeburg（马格德堡奥托·冯·格里克大学）；Centre Inria Lille - Nord Europe（里尔-北欧欧洲INRIA中心）
We investigate the stochastic Thresholding Bandit problem (TBP) under several shape constraints. On top of (i) the vanilla, unstructured TBP, we consider the case where (ii) the sequence of arm's means $(μ_k)_k$ is monotonically increasing MTBP, (iii) the case where $(μ_k)_k$ is unimodal UTBP and (iv) the case where $(μ_k)_k$ is concave CTBP. In the TBP problem the aim is to output, at the end of the sequential game, the set of arms whose means are above a given threshold. The regret is the highest gap between a misclassified arm and the threshold. In the fixed budget setting, we provide probl…
Technical University of Berlin（柏林工业大学）；University of California at Los Angeles（加州大学洛杉矶分校）；University of Cambridge（剑桥大学）
The ever-increasing demand for media streaming together with limited backhaul capacity renders developing efficient file-delivery methods imperative. One such method is femto-caching, which, despite its great potential, imposes several challenges such as efficient resource management. We study a resource allocation problem for joint caching and transmission in small cell networks, where the system operates in two consecutive phases: (i) cache placement, and (ii) joint file- and transmit power selection followed by broadcasting. We define the utility of every small base station in terms of the …
First- and Second-Order Bounds for Adversarial Linear Contextual Bandits
Vrije Universiteit Amsterdam（阿姆斯特丹自由大学）；University of Amsterdam（阿姆斯特丹大学）；Universitat Pompeu Fabra（庞培法布拉大学）；Massachusetts Institute of Technology（麻省理工学院）
Continuum-armed Bandit Optimization with Batch Pairwise Comparison Oracles
Department of Information Systems and Intelligence Business, Xi’an Jiaotong University, Xi’an 710049, China（西安交通大学信息系统与智能商业系）；Stern School of Business, New York University, New York, NY 10012, USA（纽约大学斯特恩商学院）；Naveen Jindal School of Management, University of Texas at Dallas, Richardson, TX 75080, USA（德克萨斯大学达拉斯分校奈维恩·金达管理学院）
Department of CSE, POSTECH, Pohang, Republic of Korea（POSTECH 电子工程系）；Graduate School of AI, POSTECH, Pohang, Republic of Korea（POSTECH 人工智能研究生院）；Microsoft Research, Beijing, China（微软研究院）；Qiuzhen College, Tsinghua University, Beijing, China（清华大学求真学院）
