Fenchel Duals for Drifting Adversaries
This work addresses drifting regret in online convex optimization, providing a unified primal-dual framework that explains and extends prior methods, though it appears incremental as it builds on existing literature.
The paper tackles the design of online convex optimization algorithms for drifting regret, connecting existing algorithms to Online Mirror Descent and showing that bounded gradient regularizers suffice for success, with nearly optimal competitive ratios for problems with 1-lookahead and movement costs.
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.