In the stochastic knapsack problem, we are given a knapsack of size B, and a set of jobs whose sizes and rewards are drawn from a known probability distribution. However, we know the actual size and reward only when the job completes. How should we schedule jobs to maximize the expected total reward? We know O(1)-approximations when we assume that (i) rewards and sizes are independent random variables, and (ii) we cannot prematurely cancel jobs. What can we say when either or both of these assumptions are changed?
The stochastic knapsack problem is of interest in its own right, but techniques developed for it are applicable to other stochastic packing problems. Indeed, ideas for this problem have been useful for budgeted learning problems, where one is given several arms which evolve in a specified stochastic fashion with each pull, and the goal is to pull the arms a total of B times to maximize the reward obtained. Much recent work on this problem focus on the case when the evolution of the arms follows a martingale, i.e., when the expected reward from the future is the same as the reward at the current state. What can we say when the rewards do not form a martingale?
In this paper, we give constant-factor approximation algorithms for the stochastic knapsack problem with correlations and/or cancellations, and also for budgeted learning problems where the martingale condition is not satisfied. Indeed, we can show that previously proposed LP relaxations have large integrality gaps. We propose new time-indexed LP relaxations, and convert the fractional solutions into distributions over strategies, and then use the LP values and the time ordering information from these strategies to devise a randomized adaptive scheduling algorithm. We hope our LP formulation and decomposition methods may provide a new way to address other correlated bandit problems with more general contexts.
Optimal regret bounds for Multi-Armed Bandit problems are now well documented. They can be classified into two categories based on the growth rate with respect to the time horizon $T$: (i) small, distribution-dependent, bounds of order of magnitude $\ln(T)$ and (ii) robust, distribution-free, bounds of order of magnitude $\sqrt{T}$. The Bandits with Knapsacks model, an extension to the framework allowing to model resource consumption, lacks this clear-cut distinction. While several algorithms have been shown to achieve asymptotically optimal distribution-free bounds on regret, there has been l…
We study the multi-armed bandit problem with arms which are Markov chains with rewards. In the finite-horizon setting, the celebrated Gittins indices do not apply, and the exact solution is intractable. We provide approximation algorithms for a more general model which includes Markov decision processes and non-unit transition times. When preemption is allowed, we provide a (1/2-eps)-approximation, along with an example showing this is tight. When preemption isn't allowed, we provide a 1/12-approximation, which improves to a 4/27-approximation when transition times are unity. Our model encompa…
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 …
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 …
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…
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…
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…
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 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…
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）
Vector norms play a fundamental role in computer science and optimization, so there is an ongoing effort to generalize existing algorithms to settings beyond $\ell_\infty$ and $\ell_p$ norms. We show that many online and bandit applications for general norms admit good algorithms as long as the norm can be approximated by a function that is ``gradient-stable'', a notion that we introduce. Roughly it says that the gradient of the function should not drastically decrease (multiplicatively) in any component as we increase the input vector. We prove that several families of norms, including all mo…
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…
One of the challenges in analyzing learning algorithms is the circular entanglement between the objective value and the stochastic noise. This is also known as the "chicken and egg" phenomenon and traditionally, there is no principled way to tackle this issue. People solve the problem by utilizing the special structure of the dynamic, and hence the analysis would be difficult to generalize. In this work, we present a streamlined three-step recipe to tackle the "chicken and egg" problem and give a general framework for analyzing stochastic dynamics in learning algorithms. Our framework composes…
