36 research outputs found

    Adversarial Bandits with Knapsacks

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    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 adversarially. This is a considerably harder problem, compared to both the stochastic version and the "classic" adversarial bandits, in that regret minimization is no longer feasible. Instead, the objective is to minimize the competitive ratio: the ratio of the benchmark reward to the algorithm's reward. We design an algorithm with competitive ratio O(log T) relative to the best fixed distribution over actions, where T is the time horizon; we also prove a matching lower bound. The key conceptual contribution is a new perspective on the stochastic version of the problem. We suggest a new algorithm for the stochastic version, which builds on the framework of regret minimization in repeated games and admits a substantially simpler analysis compared to prior work. We then analyze this algorithm for the adversarial version and use it as a subroutine to solve the latter.Comment: Extended abstract appeared in FOCS 201

    Approximately Stationary Bandits with Knapsacks

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    Bandits with Knapsacks (BwK), the generalization of the Bandits problem under global budget constraints, has received a lot of attention in recent years. Previous work has focused on one of the two extremes: Stochastic BwK where the rewards and consumptions of the resources of each round are sampled from an i.i.d. distribution, and Adversarial BwK where these parameters are picked by an adversary. Achievable guarantees in the two cases exhibit a massive gap: No-regret learning is achievable in the stochastic case, but in the adversarial case only competitive ratio style guarantees are achievable, where the competitive ratio depends either on the budget or on both the time and the number of resources. What makes this gap so vast is that in Adversarial BwK the guarantees get worse in the typical case when the budget is more binding. While ``best-of-both-worlds'' type algorithms are known (single algorithms that provide the best achievable guarantee in each extreme case), their bounds degrade to the adversarial case as soon as the environment is not fully stochastic. Our work aims to bridge this gap, offering guarantees for a workload that is not exactly stochastic but is also not worst-case. We define a condition, Approximately Stationary BwK, that parameterizes how close to stochastic or adversarial an instance is. Based on these parameters, we explore what is the best competitive ratio attainable in BwK. We explore two algorithms that are oblivious to the values of the parameters but guarantee competitive ratios that smoothly transition between the best possible guarantees in the two extreme cases, depending on the values of the parameters. Our guarantees offer great improvement over the adversarial guarantee, especially when the available budget is small. We also prove bounds on the achievable guarantee, showing that our results are approximately tight when the budget is small

    Adversarial blocking bandits

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    We consider a general adversarial multi-armed blocking bandit setting where each played arm can be blocked (unavailable) for some time periods and the reward per arm is given at each time period adversarially without obeying any distribution. The setting models scenarios of allocating scarce limited supplies (e.g., arms) where the supplies replenish and can be reused only after certain time periods. We first show that, in the optimization setting, when the blocking durations and rewards are known in advance, finding an optimal policy (e.g., determining which arm per round) that maximises the cumulative reward is strongly NP-hard, eliminating the possibility of a fully polynomial-time approximation scheme (FPTAS) for the problem unless P = NP. To complement our result, we show that a greedy algorithm that plays the best available arm at each round provides an approximation guarantee that depends on the blocking durations and the path variance of the rewards. In the bandit setting, when the blocking durations and rewards are not known, we design two algorithms, RGA and RGA-META, for the case of bounded duration an path variation. In particular, when the variation budget B_T is known in advance, RGA can achieve O(\sqrt{T(2\tilde{D}+K)B_{T}}) dynamic approximate regret. On the other hand, when B_T is not known, we show that the dynamic approximate regret of RGA-META is at most O((K+\tilde{D})^{1/4}\tilde{B}^{1/2}T^{3/4}) where \tilde{B} is the maximal path variation budget within each batch of RGA-META (which is provably in order of o(\sqrt{T}). We also prove that if either the variation budget or the maximal blocking duration is unbounded, the approximate regret will be at least Theta(T). We also show that the regret upper bound of RGA is tight if the blocking durations are bounded above by an order of O(1)

    Clustered Linear Contextual Bandits with Knapsacks

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    In this work, we study clustered contextual bandits where rewards and resource consumption are the outcomes of cluster-specific linear models. The arms are divided in clusters, with the cluster memberships being unknown to an algorithm. Pulling an arm in a time period results in a reward and in consumption for each one of multiple resources, and with the total consumption of any resource exceeding a constraint implying the termination of the algorithm. Thus, maximizing the total reward requires learning not only models about the reward and the resource consumption, but also cluster memberships. We provide an algorithm that achieves regret sublinear in the number of time periods, without requiring access to all of the arms. In particular, we show that it suffices to perform clustering only once to a randomly selected subset of the arms. To achieve this result, we provide a sophisticated combination of techniques from the literature of econometrics and of bandits with constraints

    Towards Thompson Sampling for Complex Bayesian Reasoning

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    Paper III, IV, and VI are not available as a part of the dissertation due to the copyright.Thompson Sampling (TS) is a state-of-art algorithm for bandit problems set in a Bayesian framework. Both the theoretical foundation and the empirical efficiency of TS is wellexplored for plain bandit problems. However, the Bayesian underpinning of TS means that TS could potentially be applied to other, more complex, problems as well, beyond the bandit problem, if suitable Bayesian structures can be found. The objective of this thesis is the development and analysis of TS-based schemes for more complex optimization problems, founded on Bayesian reasoning. We address several complex optimization problems where the previous state-of-art relies on a relatively myopic perspective on the problem. These includes stochastic searching on the line, the Goore game, the knapsack problem, travel time estimation, and equipartitioning. Instead of employing Bayesian reasoning to obtain a solution, they rely on carefully engineered rules. In all brevity, we recast each of these optimization problems in a Bayesian framework, introducing dedicated TS based solution schemes. For all of the addressed problems, the results show that besides being more effective, the TS based approaches we introduce are also capable of solving more adverse versions of the problems, such as dealing with stochastic liars.publishedVersio
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