Fenchel Duals for Drifting Adversaries

TL;DR

This paper introduces a primal-dual framework connecting online mirror descent algorithms to drifting regret, providing new insights for designing algorithms with bounded gradients and nearly optimal competitive ratios for online convex optimization.

Contribution

It establishes a primal-dual approach linking mirror descent to drifting regret, explaining existing algorithms and extending to 1-lookahead and movement costs with improved bounds.

Findings

Bounded gradient regularizers are key for success.

Regularizer shifting explains previous update methods.

Achieves nearly optimal competitive ratios for 1-lookahead with movement costs.

Abstract

We describe a primal-dual framework for the design and analysis of online convex optimization algorithms for {\em drifting regret}. Existing literature shows (nearly) optimal drifting regret bounds only for the $\ell_2$ and the $\ell_1$-norms. Our work provides a connection between these algorithms and the Online Mirror Descent ($\omd$) updates; one key insight that results from our work is that in order for these algorithms to succeed, it suffices to have the gradient of the regularizer to be bounded (in an appropriate norm). For situations (like for the $\ell_1$ norm) where the vanilla regularizer does not have this property, we have to {\em shift} the regularizer to ensure this. Thus, this helps explain the various updates presented in \cite{bansal10, buchbinder12}. We also consider the online variant of the problem with 1-lookahead, and with movement costs in the $\ell_2$-norm. Our primal dual approach yields nearly optimal competitive ratios for this problem.