Stochastic Linear Optimization with Adversarial Corruption

TL;DR

This paper introduces a robust algorithm for stochastic linear optimization that maintains low regret despite adversarial corruption, using ellipsoid exploration and epoch division to mitigate corruption effects.

Contribution

It extends adversarial corruption models to stochastic linear optimization, providing an algorithm with regret linearly dependent on corruption level, independent of corruption amount.

Findings

Regret increases linearly with corruption amount.

Algorithm is agnostic to the adversary's corruption level.

Uses L"owner-John's ellipsoid for exploration and epoch division for robustness.

Abstract

We extend the model of stochastic bandits with adversarial corruption (Lykouriset al., 2018) to the stochastic linear optimization problem (Dani et al., 2008). Our algorithm is agnostic to the amount of corruption chosen by the adaptive adversary. The regret of the algorithm only increases linearly in the amount of corruption. Our algorithm involves using L\"owner-John's ellipsoid for exploration and dividing time horizon into epochs with exponentially increasing size to limit the influence of corruption.