Fenchel Duals for Drifting Adversaries

We describe a primal-dual framework for the design and analysis of online convex optimization algorithms for drifting regret. Existing literature shows (nearly) optimal drifting regret bounds only for the l2 and the l1-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 l1 norm) where the vanilla regularizer does not have this property, we have to shift the regularizer to ensure this. Thus, this helps explain the various updates presented in [3, 10]. We also consider the online variant of the problem with 1-lookahead, and with movement costs in the l2-norm. Our primal dual approach yields nearly optimal competitive ratios for this problem.