Online Linear Optimization with Many Hints

TL;DR

This paper introduces an online linear optimization algorithm that leverages multiple hints per round, achieving logarithmic regret when hints correlate with costs, extending prior single-hint approaches.

Contribution

It develops a method to combine multiple hint-based algorithms, enabling improved regret bounds in online linear optimization with many hints.

Findings

Achieves logarithmic regret with multiple hints when a convex combination correlates with costs.

Extends previous work from a single hint to many hints.

Provides a novel technique for combining multiple OLO algorithms.

Abstract

We study an online linear optimization (OLO) problem in which the learner is provided access to $K$ "hint" vectors in each round prior to making a decision. In this setting, we devise an algorithm that obtains logarithmic regret whenever there exists a convex combination of the $K$ hints that has positive correlation with the cost vectors. This significantly extends prior work that considered only the case $K=1$. To accomplish this, we develop a way to combine many arbitrary OLO algorithms to obtain regret only a logarithmically worse factor than the minimum regret of the original algorithms in hindsight; this result is of independent interest.