Simple regret for infinitely many armed bandits

TL;DR

This paper introduces a new algorithm for infinitely many armed bandits that minimizes simple regret, achieving minimax optimality depending on the distribution of near-optimal arms, with extensions for various practical scenarios.

Contribution

It proposes the first algorithm specifically designed for simple regret minimization in infinitely many armed bandits, with proven minimax optimality.

Findings

Algorithm is minimax optimal up to a constant or log factor depending on $eta$.

Extensions include unknown $eta$, small variance near-optimal arms, and unknown time horizon.

The rate of simple regret depends on the distribution parameter $eta$.

Abstract

We consider a stochastic bandit problem with infinitely many arms. In this setting, the learner has no chance of trying all the arms even once and has to dedicate its limited number of samples only to a certain number of arms. All previous algorithms for this setting were designed for minimizing the cumulative regret of the learner. In this paper, we propose an algorithm aiming at minimizing the simple regret. As in the cumulative regret setting of infinitely many armed bandits, the rate of the simple regret will depend on a parameter $\beta$ characterizing the distribution of the near-optimal arms. We prove that depending on $\beta$, our algorithm is minimax optimal either up to a multiplicative constant or up to a $\log(n)$ factor. We also provide extensions to several important cases: when $\beta$ is unknown, in a natural setting where the near-optimal arms have a small variance, and in the case of unknown time horizon.