Adaptive Bound Optimization for Online Convex Optimization

TL;DR

This paper presents an adaptive online convex optimization algorithm that dynamically adjusts regularization based on observed losses, achieving optimal regret bounds and exploiting problem structure without prior knowledge.

Contribution

It introduces a novel adaptive regularization method for online convex optimization with optimal regret bounds and problem-dependent guarantees.

Findings

Achieves worst-case optimal regret bounds.

Provides better bounds for certain realistic loss functions.

Does not require prior knowledge of problem structure.

Abstract

We introduce a new online convex optimization algorithm that adaptively chooses its regularization function based on the loss functions observed so far. This is in contrast to previous algorithms that use a fixed regularization function such as L2-squared, and modify it only via a single time-dependent parameter. Our algorithm's regret bounds are worst-case optimal, and for certain realistic classes of loss functions they are much better than existing bounds. These bounds are problem-dependent, which means they can exploit the structure of the actual problem instance. Critically, however, our algorithm does not need to know this structure in advance. Rather, we prove competitive guarantees that show the algorithm provides a bound within a constant factor of the best possible bound (of a certain functional form) in hindsight.