Bandits with Knapsacks (BwK), the generalization of the Multi-Armed Bandits under budget constraints, has received a lot of attention in recent years. It has numerous applications, including dynamic pricing, repeated auctions, etc. Previous work has focused on one of the two extremes: Stochastic BwK where the rewards and consumptions of the resources each round are sampled from an i.i.d. distribution, and Adversarial BwK where these values are picked by an adversary. Achievable guarantees in the two cases exhibit a massive gap: No-regret learning is achievable in Stochastic BwK, but in Adversarial BwK, only competitive ratio style guarantees are achievable, where the competitive ratio depends on the budget. 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 (algorithms that provide the best achievable guarantee in both extreme cases), their guarantees 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.

翻译：带背包的赌博机（BwK）是预算约束下多臂赌博机的推广，近年来受到了广泛关注。该模型具有众多应用，包括动态定价、重复拍卖等。以往的研究聚焦于两个极端情况：随机BwK，其中每轮的奖励和资源消耗从独立同分布（i.i.d.）中采样；对抗性BwK，其中这些值由对手选择。这两种情况的可实现保证存在巨大差距：在随机BwK中可实现无遗憾学习，但在对抗性BwK中，仅能实现竞争比类型保证，且竞争比取决于预算。造成这一巨大差距的原因在于，在对抗性BwK中，当预算更为紧张（典型情况）时，保证性能会恶化。尽管已知“两全其美”型算法（即在两种极端情况下都能提供最佳可实现保证的算法），一旦环境不完全随机，其保证性能就会退化为对抗性情况。本文旨在弥合这一差距，为既非完全随机也非最坏情况的工作负载提供保证。我们定义了一个条件——近似平稳BwK，该条件通过参数刻画实例接近随机或对抗的程度。基于这些参数，我们探索了BwK中可达到的最佳竞争比。我们研究了两种算法，它们对参数值无感知，但能保证竞争比根据参数值在两种极端情况的最佳可实现保证之间平滑过渡。我们的保证相比对抗性保证有显著改进，尤其在可用预算较小的情况下。我们还证明了可实现保证的下界，表明当预算较小时，我们的结果近似紧致。