Artificial Constraints and Lipschitz Hints for Unconstrained Online Learning

TL;DR

This paper introduces algorithms for online convex optimization that achieve regret bounds without prior knowledge of the Lipschitz constant or comparison norm, improving over previous exponential penalties with polynomial bounds.

Contribution

The authors develop new algorithms with regret bounds that do not require prior knowledge of key parameters, and they show these bounds are nearly optimal with polynomial dependence.

Findings

Regret bounds without prior knowledge of G or ||u||

Polynomial penalty bounds in all parameters

Optimal adaptation to unknown ||u||

Abstract

We provide algorithms that guarantee regret $R_T(u)\le \tilde O(G\|u\|^3 + G(\|u\|+1)\sqrt{T})$ or $R_T(u)\le \tilde O(G\|u\|^3T^{1/3} + GT^{1/3}+ G\|u\|\sqrt{T})$ for online convex optimization with $G$-Lipschitz losses for any comparison point $u$ without prior knowledge of either $G$ or $\|u\|$. Previous algorithms dispense with the $O(\|u\|^3)$ term at the expense of knowledge of one or both of these parameters, while a lower bound shows that some additional penalty term over $G\|u\|\sqrt{T}$ is necessary. Previous penalties were exponential while our bounds are polynomial in all quantities. Further, given a known bound $\|u\|\le D$, our same techniques allow us to design algorithms that adapt optimally to the unknown value of $\|u\|$ without requiring knowledge of $G$.