A Tale of Two Metrics: Simultaneous Bounds on Competitiveness and Regret

TL;DR

This paper investigates the fundamental limitations in online convex optimization, showing that no algorithm can simultaneously achieve sublinear regret and a constant competitive ratio, but provides a solution for one-dimensional cases.

Contribution

It proves the incompatibility of regret and competitiveness in general, and introduces an algorithm that balances both in one-dimensional settings.

Findings

No algorithm achieves both sublinear regret and constant competitive ratio universally.

A specialized algorithm for one-dimensional problems achieves sublinear regret with slowly growing competitive ratio.

Fundamental limits are established for the trade-off between regret and competitiveness.

Abstract

We consider algorithms for "smoothed online convex optimization" problems, a variant of the class of online convex optimization problems that is strongly related to metrical task systems. Prior literature on these problems has focused on two performance metrics: regret and the competitive ratio. There exist known algorithms with sublinear regret and known algorithms with constant competitive ratios; however, no known algorithm achieves both simultaneously. We show that this is due to a fundamental incompatibility between these two metrics - no algorithm (deterministic or randomized) can achieve sublinear regret and a constant competitive ratio, even in the case when the objective functions are linear. However, we also exhibit an algorithm that, for the important special case of one-dimensional decision spaces, provides sublinear regret while maintaining a competitive ratio that grows arbitrarily slowly.