Robust Online Convex Optimization in the Presence of Outliers

TL;DR

This paper develops methods for online convex optimization that remain effective despite the presence of outliers, providing bounds on regret and exploring conditions under which outliers can be tolerated.

Contribution

It introduces the concept of robust regret for outlier-robust online convex optimization and analyzes the impact of adversarial and benign outliers on regret bounds.

Findings

Filtering extreme gradients incurs an O(k) regret overhead.

Sublinear robust regret is achievable with i.i.d. data and quantile-based outlier assumptions.

Linear outliers are manageable under specific distributional assumptions.

Abstract

We consider online convex optimization when a number k of data points are outliers that may be corrupted. We model this by introducing the notion of robust regret, which measures the regret only on rounds that are not outliers. The aim for the learner is to achieve small robust regret, without knowing where the outliers are. If the outliers are chosen adversarially, we show that a simple filtering strategy on extreme gradients incurs O(k) additive overhead compared to the usual regret bounds, and that this is unimprovable, which means that k needs to be sublinear in the number of rounds. We further ask which additional assumptions would allow for a linear number of outliers. It turns out that the usual benign cases of independently, identically distributed (i.i.d.) observations or strongly convex losses are not sufficient. However, combining i.i.d. observations with the assumption that outliers are those observations that are in an extreme quantile of the distribution, does lead to sublinear robust regret, even though the expected number of outliers is linear.