Online and Distribution-Free Robustness: Regression and Contextual Bandits with Huber Contamination

TL;DR

This paper develops robust online algorithms for linear regression and contextual bandits that do not rely on distributional assumptions, achieving optimal contamination tolerance and breakdown points even under adversarial conditions.

Contribution

It introduces a novel alternating minimization algorithm that combines least-squares with convex reweighting, enabling distribution-free robustness in high-dimensional online learning.

Findings

Achieves optimal dependence on contamination level η

Reaches the optimal breakdown point

Applicable to infinite-dimensional kernel settings

Abstract

In this work we revisit two classic high-dimensional online learning problems, namely linear regression and contextual bandits, from the perspective of adversarial robustness. Existing works in algorithmic robust statistics make strong distributional assumptions that ensure that the input data is evenly spread out or comes from a nice generative model. Is it possible to achieve strong robustness guarantees even without distributional assumptions altogether, where the sequence of tasks we are asked to solve is adaptively and adversarially chosen? We answer this question in the affirmative for both linear regression and contextual bandits. In fact our algorithms succeed where conventional methods fail. In particular we show strong lower bounds against Huber regression and more generally any convex M-estimator. Our approach is based on a novel alternating minimization scheme that interleaves ordinary least-squares with a simple convex program that finds the optimal reweighting of the distribution under a spectral constraint. Our results obtain essentially optimal dependence on the contamination level $\eta$, reach the optimal breakdown point, and naturally apply to infinite dimensional settings where the feature vectors are represented implicitly via a kernel map.