Universal Online Convex Optimization with Minimax Optimal Second-Order Dynamic Regret

Abstract

We introduce an online convex optimization algorithm which utilizes projected subgradient descent with optimal adaptive learning rates. Our method provides second-order minimax-optimal dynamic regret guarantee (i.e. dependent on the sum of squared subgradient norms) for a sequence of general convex functions, which may not have strong convexity, smoothness, exp-concavity or even Lipschitz-continuity. The regret guarantee is against any comparator decision sequence with bounded path variation (i.e. sum of the distances between successive decisions). We generate the lower bound of the worst-case second-order dynamic regret by incorporating actual subgradient norms. We show that this lower bound matches with our regret guarantee within a constant factor, which makes our algorithm minimax optimal. We also derive the extension for learning in each decision coordinate individually. We demonstrate how to best preserve our regret guarantee in a truly online manner, when the bound on path variation of the comparator sequence grows in time or the feedback regarding such bound arrives partially as time goes on. We further build on our algorithm to eliminate the need of any knowledge on the comparator path variation, and provide minimax optimal second-order regret guarantees with no a priori information. Our approach can compete against all comparator sequences simultaneously (universally) in a minimax optimal manner, i.e. each regret guarantee depends on the respective comparator path variation. We discuss modifications to our approach which address complexity reductions for time, computation and memory. We further improve our results by making the regret guarantees also dependent on comparator sets' diameters in addition to the respective path variations.