From Optimality to Robustness: Dirichlet Sampling Strategies in Stochastic Bandits

TL;DR

This paper introduces a Dirichlet Sampling algorithm for stochastic bandits that maintains near-optimal regret under various distribution assumptions, including model misspecification, and demonstrates its effectiveness through theoretical analysis and experiments.

Contribution

It proposes a robust Dirichlet Sampling strategy that adapts to different distribution classes and provides theoretical regret guarantees under model misspecification.

Findings

Achieves optimal regret for bounded distributions.

Ensures logarithmic regret for semi-bounded distributions.

Provides robustness to unbounded distributions with slight regret trade-offs.

Abstract

The stochastic multi-arm bandit problem has been extensively studied under standard assumptions on the arm's distribution (e.g bounded with known support, exponential family, etc). These assumptions are suitable for many real-world problems but sometimes they require knowledge (on tails for instance) that may not be precisely accessible to the practitioner, raising the question of the robustness of bandit algorithms to model misspecification. In this paper we study a generic Dirichlet Sampling (DS) algorithm, based on pairwise comparisons of empirical indices computed with re-sampling of the arms' observations and a data-dependent exploration bonus. We show that different variants of this strategy achieve provably optimal regret guarantees when the distributions are bounded and logarithmic regret for semi-bounded distributions with a mild quantile condition. We also show that a simple tuning achieve robustness with respect to a large class of unbounded distributions, at the cost of slightly worse than logarithmic asymptotic regret. We finally provide numerical experiments showing the merits of DS in a decision-making problem on synthetic agriculture data.