Pure Exploration in Bandits with Linear Constraints

TL;DR

This paper studies the problem of identifying the best policy in a constrained multi-armed bandit setting, introducing optimal algorithms and analyzing how linear constraints affect problem complexity.

Contribution

It characterizes the geometry of constrained bandit problems via an information-theoretic lower bound and proposes two asymptotically optimal algorithms based on different approaches.

Findings

Algorithms match the lower bound asymptotically

Constraints significantly impact the problem's hardness

Empirical results validate theoretical bounds

Abstract

We address the problem of identifying the optimal policy with a fixed confidence level in a multi-armed bandit setup, when \emph{the arms are subject to linear constraints}. Unlike the standard best-arm identification problem which is well studied, the optimal policy in this case may not be deterministic and could mix between several arms. This changes the geometry of the problem which we characterize via an information-theoretic lower bound. We introduce two asymptotically optimal algorithms for this setting, one based on the Track-and-Stop method and the other based on a game-theoretic approach. Both these algorithms try to track an optimal allocation based on the lower bound and computed by a weighted projection onto the boundary of a normal cone. Finally, we provide empirical results that validate our bounds and visualize how constraints change the hardness of the problem.