Unconstrained Robust Online Convex Optimization

Abstract: This paper addresses online learning with corrupted'' feedback. Our learner is provided with potentially corrupted gradients $\tilde g_t$ instead of thetrue'' gradients $g_t$. We make no assumptions about how the corruptions arise: they could be the result of outliers, mislabeled data, or even malicious interference. We focus on the difficult ``unconstrained'' setting in which our algorithm must maintain low regret with respect to any comparison point $u \in \mathbb{R}^d$. The unconstrained setting is significantly more challenging as existing algorithms suffer extremely high regret even with very tiny amounts of corruption (which is not true in the case of a bounded domain). Our algorithms guarantee regret $ |u|G (\sqrt{T} + k) $ when $G \ge \max_t |g_t|$ is known, where $k$ is a measure of the total amount of corruption. When $G$ is unknown we incur an extra additive penalty of $(|u|^2+G^2) k$.