Data-Dependent Regret Bounds for Constrained MABs

TL;DR

This paper develops data-dependent regret bounds for constrained multi-armed bandits, showing they can be significantly tighter than classical bounds and are fundamental to understanding the problem's complexity.

Contribution

It introduces the first data-dependent regret bounds for constrained MABs with adversarial losses and stochastic constraints, including a new algorithm and lower bounds.

Findings

Regret bounds with two data-dependent terms capturing constraint difficulty and learning complexity

Lower bounds confirming the fundamental nature of these two terms

Novel results in soft constraints settings that may be of independent interest

Abstract

This paper initiates the study of data-dependent regret bounds in constrained MAB settings. These bounds depend on the sequence of losses that characterize the problem instance. Thus, they can be much smaller than classical $\widetilde{\mathcal{O}}(\sqrt{T})$ regret bounds, while being equivalent to them in the worst case. Despite this, data-dependent regret bounds have been completely overlooked in constrained MAB settings. The goal of this paper is to answer the following question: Can data-dependent regret bounds be derived in the presence of constraints? We answer this question affirmatively in constrained MABs with adversarial losses and stochastic constraints. Specifically, our main focus is on the most challenging and natural settings with hard constraints, where the learner must ensure that the constraints are always satisfied with high probability. We design an algorithm with a regret bound consisting of two data-dependent terms. The first term captures the difficulty of satisfying the constraints, while the second one encodes the complexity of learning independently of the presence of constraints. We also prove a lower bound showing that these two terms are not artifacts of our specific approach and analysis, but rather the fundamental components that inherently characterize the complexities of the problem. Finally, in designing our algorithm, we also derive some novel results in the related (and easier) soft constraints settings, which may be of independent interest.