Fully Unconstrained Online Learning

TL;DR

This paper introduces an online learning algorithm that achieves near-optimal regret bounds without prior knowledge of key parameters, effectively handling all scenarios where sublinear regret is possible.

Contribution

It presents a fully unconstrained online learning algorithm that attains optimal regret bounds without knowing the Lipschitz constant or comparison norm beforehand.

Findings

Achieves regret close to the best known bounds without prior parameter knowledge.

Matches the optimal regret bounds in all sublinear regret scenarios.

Handles arbitrary comparison points with no prior constraints.

Abstract

We provide an online learning algorithm that obtains regret $G\|w_\star\|\sqrt{T\log(\|w_\star\|G\sqrt{T})} + \|w_\star\|^2 + G^2$ on $G$-Lipschitz convex losses for any comparison point $w_\star$ without knowing either $G$ or $\|w_\star\|$. Importantly, this matches the optimal bound $G\|w_\star\|\sqrt{T}$ available with such knowledge (up to logarithmic factors), unless either $\|w_\star\|$ or $G$ is so large that even $G\|w_\star\|\sqrt{T}$ is roughly linear in $T$. Thus, it matches the optimal bound in all cases in which one can achieve sublinear regret, which arguably most "interesting" scenarios.