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We consider the framework of non-stationary Online Convex Optimization wherea learner seeks to control its dynamic regret against an arbitrary sequence ofcomparators. When the loss functions are strongly convex or exp-concave, wedemonstrate that Strongly Adaptive (SA) algorithms can be viewed as aprincipled way of controlling dynamic regret in terms of path variation $V_T$of the comparator sequence. Specifically, we show that SA algorithms enjoy$\tilde O(\sqrt{TV_T} \vee \log T)$ and $\tilde O(\sqrt{dTV_T} \vee d\log T)$dynamic regret for strongly convex and exp-concave losses respectively withoutapriori knowledge of $V_T$. The versatility of the principled approach isfurther demonstrated by the novel results in the setting of learning againstbounded linear predictors and online regression with Gaussian kernels.  Under a related setting, the second component of the paper addresses an openquestion posed by Zhdanov and Kalnishkan (2010) that concerns online kernelregression with squared error losses. We derive a new lower bound on a certainpenalized regret which establishes the near minimax optimality of online KernelRidge Regression (KRR). Our lower bound can be viewed as an RKHS extension tothe lower bound derived in Vovk (2001) for online linear regression in finitedimensions.

Dynamic Regret for Strongly Adaptive Methods and Optimality of Online KRR