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

We consider 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 present a new algorithm that is simple, computationally efficient, and admits vanishing regret. It is statistically optimal for CBwK when an algorithm must stop once some constraint is violated. 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.