Contextual Bandits with Packing and Covering Constraints: A Modular Lagrangian Approach via Regression

We consider contextual bandits with linear constraints (CBwLC), a variant of contextual bandits in which the algorithm consumes multiple resources subject to linear constraints on total consumption. This problem generalizes contextual bandits with knapsacks (CBwK), allowing for packing and covering constraints, as well as positive and negative resource consumption. We provide the first algorithm for CBwLC (or CBwK) that is based on regression oracles. The algorithm is simple, computationally efficient, and admits vanishing regret. It is statistically optimal for the variant of CBwK in which the algorithm must stop once some constraint is violated. Further, we provide the first vanishing-regret guarantees for CBwLC (or CBwK) that extend beyond the stochastic environment. We side-step strong impossibility results from prior work by identifying a weaker (and, arguably, fairer) benchmark to compare against. Our algorithm builds on LagrangeBwK (Immorlica et al., FOCS 2019), a Lagrangian-based technique for CBwK, and SquareCB (Foster and Rakhlin, ICML 2020), a regression-based technique for contextual bandits. Our analysis leverages the inherent modularity of both techniques

Adversarial Bandits with Knapsacks

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

Clustered Linear Contextual Bandits with Knapsacks

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