Robust and Computationally Efficient Linear Contextual Bandits under Adversarial Corruption and Heavy-Tailed Noise

TL;DR

This paper introduces a computationally efficient algorithm for linear contextual bandits that is robust against adversarial corruption and heavy-tailed noise, achieving sublinear regret without prior knowledge of noise or corruption levels.

Contribution

It proposes a novel online mirror descent-based algorithm that handles both adversarial corruption and heavy-tailed noise efficiently, improving computational complexity and robustness over existing methods.

Findings

Reduces computational cost to O(1) per round.

Achieves regret bounds depending on noise moments and corruption.

Recovers existing guarantees when noise is finite-variance and no corruption is present.

Abstract

We study linear contextual bandits under adversarial corruption and heavy-tailed noise with finite $(1+\epsilon)$-th moments for some $\epsilon \in (0,1]$. Existing work that addresses both adversarial corruption and heavy-tailed noise relies on a finite variance (i.e., finite second-moment) assumption and suffers from computational inefficiency. We propose a computationally efficient algorithm based on online mirror descent that achieves robustness to both adversarial corruption and heavy-tailed noise. While the existing algorithm incurs $\mathcal{O}(t\log T)$ computational cost, our algorithm reduces this to $\mathcal{O}(1)$ per round. We establish an additive regret bound consisting of a term depending on the $(1+\epsilon)$-moment bound of the noise and a term depending on the total amount of corruption. In particular, when $\epsilon = 1$, our result recovers existing guarantees under finite-variance assumptions. When no corruption is present, it matches the best-known rates for linear contextual bandits with heavy-tailed noise. Moreover, the algorithm requires no prior knowledge of the noise moment bound or the total amount of corruption and still guarantees sublinear regret.