CRIMED: Lower and Upper Bounds on Regret for Bandits with Unbounded Stochastic Corruption

TL;DR

This paper introduces CRIMED, an optimal algorithm for multi-armed bandits with unbounded corruptions, providing tight regret bounds and robust median concentration results even with high corruption levels.

Contribution

We establish regret lower bounds and propose CRIMED, an asymptotically optimal algorithm that handles unbounded corruptions in bandit problems, with finite-sample guarantees and median concentration results.

Findings

CRIMED achieves the exact lower bound on regret for Gaussian bandits.

CRIMED effectively handles corruption levels up to 50%.

A new median concentration inequality under arbitrary corruptions is developed.

Abstract

We investigate the regret-minimisation problem in a multi-armed bandit setting with arbitrary corruptions. Similar to the classical setup, the agent receives rewards generated independently from the distribution of the arm chosen at each time. However, these rewards are not directly observed. Instead, with a fixed $\varepsilon\in (0,\frac{1}{2})$, the agent observes a sample from the chosen arm's distribution with probability $1-\varepsilon$, or from an arbitrary corruption distribution with probability $\varepsilon$. Importantly, we impose no assumptions on these corruption distributions, which can be unbounded. In this setting, accommodating potentially unbounded corruptions, we establish a problem-dependent lower bound on regret for a given family of arm distributions. We introduce CRIMED, an asymptotically-optimal algorithm that achieves the exact lower bound on regret for bandits with Gaussian distributions with known variance. Additionally, we provide a finite-sample analysis of CRIMED's regret performance. Notably, CRIMED can effectively handle corruptions with $\varepsilon$ values as high as $\frac{1}{2}$. Furthermore, we develop a tight concentration result for medians in the presence of arbitrary corruptions, even with $\varepsilon$ values up to $\frac{1}{2}$, which may be of independent interest. We also discuss an extension of the algorithm for handling misspecification in Gaussian model.