Scale-Free Online Learning

TL;DR

This paper introduces scale-invariant online learning algorithms that adapt to loss vector norms without prior bounds, achieving optimal regret for both bounded and unbounded decision sets.

Contribution

It presents the first adaptive algorithms for unbounded decision sets in online linear optimization that are scale-invariant and achieve optimal regret.

Findings

Algorithms are scale-invariant and adapt to loss vector norms.

First adaptive algorithms with non-vacuous regret bounds for unbounded decision sets.

Lower bounds show limitations of Mirror Descent-based scale-free algorithms.

Abstract

We design and analyze algorithms for online linear optimization that have optimal regret and at the same time do not need to know any upper or lower bounds on the norm of the loss vectors. Our algorithms are instances of the Follow the Regularized Leader (FTRL) and Mirror Descent (MD) meta-algorithms. We achieve adaptiveness to the norms of the loss vectors by scale invariance, i.e., our algorithms make exactly the same decisions if the sequence of loss vectors is multiplied by any positive constant. The algorithm based on FTRL works for any decision set, bounded or unbounded. For unbounded decisions sets, this is the first adaptive algorithm for online linear optimization with a non-vacuous regret bound. In contrast, we show lower bounds on scale-free algorithms based on MD on unbounded domains.