Adaptivity and Optimality: A Universal Algorithm for Online Convex Optimization

TL;DR

This paper introduces Maler, a universal online convex optimization algorithm that adaptively achieves optimal regret bounds across various convex loss functions by running multiple algorithms in parallel and selecting the best in real-time.

Contribution

The paper presents Maler, a novel universal algorithm that attains optimal regret bounds for general convex, exponentially concave, and strongly convex functions simultaneously.

Findings

Maler achieves $O( oot{T} ext{)}$ regret for general convex functions.

Maler attains $O(d ext{log} T)$ regret for exponentially concave functions.

Maler reaches $O( ext{log} T)$ regret for strongly convex functions.

Abstract

In this paper, we study adaptive online convex optimization, and aim to design a universal algorithm that achieves optimal regret bounds for multiple common types of loss functions. Existing universal methods are limited in the sense that they are optimal for only a subclass of loss functions. To address this limitation, we propose a novel online method, namely Maler, which enjoys the optimal $O(\sqrt{T})$, $O(d\log T)$ and $O(\log T)$ regret bounds for general convex, exponentially concave, and strongly convex functions respectively. The essential idea is to run multiple types of learning algorithms with different learning rates in parallel, and utilize a meta algorithm to track the best one on the fly. Empirical results demonstrate the effectiveness of our method.