Parameter-free Mirror Descent

TL;DR

This paper introduces a new parameter-free mirror descent framework for online optimization in unbounded domains, achieving optimal dynamic regret and surpassing traditional strategies like Follow-the-Regularized-Leader.

Contribution

It presents the first unconstrained online linear optimization algorithm with optimal dynamic regret and develops new parameter-free implicit and scale-free algorithms.

Findings

Achieves optimal dynamic regret in unbounded domains.

Shows traditional strategies like Follow-the-Regularized-Leader are insufficient.

Provides simplified, improved unconstrained scale-free algorithms.

Abstract

We develop a modified online mirror descent framework that is suitable for building adaptive and parameter-free algorithms in unbounded domains. We leverage this technique to develop the first unconstrained online linear optimization algorithm achieving an optimal dynamic regret bound, and we further demonstrate that natural strategies based on Follow-the-Regularized-Leader are unable to achieve similar results. We also apply our mirror descent framework to build new parameter-free implicit updates, as well as a simplified and improved unconstrained scale-free algorithm.