No-Regret Algorithms for Unconstrained Online Convex Optimization

TL;DR

This paper introduces new no-regret algorithms for unconstrained online convex optimization that achieve near-optimal regret bounds without prior knowledge of the comparator, including constant regret for the zero comparator.

Contribution

The authors develop algorithms that attain near-optimal regret in unconstrained online convex optimization without needing prior knowledge of the comparator point.

Findings

Achieve near-optimal regret bounds in unconstrained settings

Regret with respect to x^* = 0 is constant

Prove lower bounds showing near-optimality of their guarantees

Abstract

Some of the most compelling applications of online convex optimization, including online prediction and classification, are unconstrained: the natural feasible set is R^n. Existing algorithms fail to achieve sub-linear regret in this setting unless constraints on the comparator point x^* are known in advance. We present algorithms that, without such prior knowledge, offer near-optimal regret bounds with respect to any choice of x^*. In particular, regret with respect to x^* = 0 is constant. We then prove lower bounds showing that our guarantees are near-optimal in this setting.