Competitive ratio versus regret minimization: achieving the best of both worlds

TL;DR

This paper develops a unified online algorithm framework that guarantees both optimal competitive ratio and low regret simultaneously, applicable to Metrical Task Systems and paging problems, extending existing algorithms to handle switching costs.

Contribution

It introduces a methodology to combine competitive ratio guarantees with regret minimization, extending regret algorithms to handle movement costs and applying this to MTS and paging problems.

Findings

Achieves simultaneous competitive ratio and regret guarantees for MTS.

Provides an efficient online algorithm for paging with combined guarantees.

Extends regret minimization algorithms to incorporate switching costs.

Abstract

We consider online algorithms under both the competitive ratio criteria and the regret minimization one. Our main goal is to build a unified methodology that would be able to guarantee both criteria simultaneously. For a general class of online algorithms, namely any Metrical Task System (MTS), we show that one can simultaneously guarantee the best known competitive ratio and a natural regret bound. For the paging problem we further show an efficient online algorithm (polynomial in the number of pages) with this guarantee. To this end, we extend an existing regret minimization algorithm (specifically, Kapralov and Panigrahy) to handle movement cost (the cost of switching between states of the online system). We then show how to use the extended regret minimization algorithm to combine multiple online algorithms. Our end result is an online algorithm that can combine a "base" online algorithm, having a guaranteed competitive ratio, with a range of online algorithms that guarantee a small regret over any interval of time. The combined algorithm guarantees both that the competitive ratio matches that of the base algorithm and a low regret over any time interval. As a by product, we obtain an expert algorithm with close to optimal regret bound on every time interval, even in the presence of switching costs. This result is of independent interest.