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We study Smoothed Online Convex Optimization, a version of online convexoptimization where the learner incurs a penalty for changing her actionsbetween rounds. Given a $\Omega(\sqrt{d})$ lower bound on the competitive ratioof any online algorithm, where $d$ is the dimension of the action space, we askunder what conditions this bound can be beaten. We introduce a novelalgorithmic framework for this problem, Online Balanced Descent (OBD), whichworks by iteratively projecting the previous point onto a carefully chosenlevel set of the current cost function so as to balance the switching costs andhitting costs. We demonstrate the generality of the OBD framework by showinghow, with different choices of "balance," OBD can improve upon state-of-the-artperformance guarantees for both competitive ratio and regret, in particular,OBD is the first algorithm to achieve a dimension-free competitive ratio, $3 +O(1/\alpha)$, for locally polyhedral costs, where $\alpha$ measures the"steepness" of the costs. We also prove bounds on the dynamic regret of OBDwhen the balance is performed in the dual space that are dimension-free andimply that OBD has sublinear static regret.

Smoothed Online Convex Optimization in High Dimensions via Online Balanced Descent