Fenchel Duals for Drifting Adversaries

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

Online optimization and regret guarantees for non-additive long-term constraints

We consider online optimization in the 1-lookahead setting, where the objective does not decompose additively over the rounds of the online game. The resulting formulation enables us to deal with non-stationary and/or long-term constraints , which arise, for example, in online display advertising problems. We propose an on-line primal-dual algorithm for which we obtain dynamic cumulative regret guarantees. They depend on the convexity and the smoothness of the non-additive penalty, as well as terms capturing the smoothness with which the residuals of the non-stationary and long-term constraints vary over the rounds. We conduct experiments on synthetic data to illustrate the benefits of the non-additive penalty and show vanishing regret convergence on live traffic data collected by a display advertising platform in production