A Tale of Two Metrics: Simultaneous Bounds on Competitiveness and Regret
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.
Keywords
Cite
@article{arxiv.1508.03769,
title = {A Tale of Two Metrics: Simultaneous Bounds on Competitiveness and Regret},
author = {Lachlan L. H. Andrew and Siddharth Barman and Katrina Ligett and Minghong Lin and Adam Meyerson and Alan Roytman and Adam Wierman},
journal= {arXiv preprint arXiv:1508.03769},
year = {2015}
}