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.

翻译：我们用Knappsacks(从这里到BwK)来考虑盗匪,这是在供应/预算限制下,多武装匪徒的一般模式。特别是,土匪算法需要解决一个众所周知的Knapack问题:找到将物品最佳包装到一个有限规模的Knapack。BwK问题是从动态定价到反复拍卖到网络路线和日程安排的动态分配等诸多激励性实例的常见概括化。虽然BwK以前的工作侧重于Stochistic版本,但我们率先探索另一个极端,即结果可以被选择为对立的极端。这比一个众所周知的Knappsack问题要困难得多:找到将物品最佳包装到有限规模的Knappsack。相反,BwK问题是将竞争比率最小化:基准奖赏与算法奖励之比。我们设计一种具有竞争性比率的算法,O(log T)相对于行动的最佳固定分配,T是时空线;我们也证明一种相对较低的约束。这个关键的概念贡献比重得多,与“经典”对后算法框架进行新的分析。我们用一个更简单的分析。我们先入一个更简单的分析,然后将它作为新的分析。我们用来分析。比较一个较简单的分析。比较一个较简单的分析。