Adversarially Robust Multi-Armed Bandit Algorithm with Variance-Dependent Regret Bounds

TL;DR

This paper introduces a novel best-of-both-worlds multi-armed bandit algorithm that achieves near-optimal regret bounds in both stochastic and adversarial environments, incorporating variance-dependent analysis for improved performance.

Contribution

It presents the first BOBW algorithm with gap-variance-dependent regret bounds, leveraging variance information in adversarial settings, and employs adaptive learning rates based on empirical prediction errors.

Findings

Achieves near-optimal gap-variance-dependent regret bounds.

Performs well in both stochastic and adversarial environments.

Provides data-dependent regret bounds that adapt to variance.

Abstract

This paper considers the multi-armed bandit (MAB) problem and provides a new best-of-both-worlds (BOBW) algorithm that works nearly optimally in both stochastic and adversarial settings. In stochastic settings, some existing BOBW algorithms achieve tight gap-dependent regret bounds of $O(\sum_{i: \Delta_i>0} \frac{\log T}{\Delta_i})$ for suboptimality gap $\Delta_i$ of arm $i$ and time horizon $T$. As Audibert et al. [2007] have shown, however, that the performance can be improved in stochastic environments with low-variance arms. In fact, they have provided a stochastic MAB algorithm with gap-variance-dependent regret bounds of $O(\sum_{i: \Delta_i>0} (\frac{\sigma_i^2}{\Delta_i} + 1) \log T )$ for loss variance $\sigma_i^2$ of arm $i$. In this paper, we propose the first BOBW algorithm with gap-variance-dependent bounds, showing that the variance information can be used even in the possibly adversarial environment. Further, the leading constant factor in our gap-variance dependent bound is only (almost) twice the value for the lower bound. Additionally, the proposed algorithm enjoys multiple data-dependent regret bounds in adversarial settings and works well in stochastic settings with adversarial corruptions. The proposed algorithm is based on the follow-the-regularized-leader method and employs adaptive learning rates that depend on the empirical prediction error of the loss, which leads to gap-variance-dependent regret bounds reflecting the variance of the arms.