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

We introduce an online convex optimization algorithm using projected subgradient descent with optimal adaptive learning rates, with sequential and efficient first-order updates. Our method provides a subgradient adaptive minimax optimal dynamic regret guarantee for a sequence of general convex functions with no known additional properties such as strong-convexity, smoothness, exp-concavity or even Lipschitz-continuity. The guarantee is against any comparator decision sequence with bounded "complexity", defined by the cumulative distance traveled via changes between successive decisions. We show optimality by generating a lower bound of the worst-case second-order dynamic regret, which incorporates actual subgradient norms and matches with our guarantees within a constant factor. We also derive the extension for independent learning in each decision coordinate separately. Additionally, we demonstrate how to best preserve our guarantees when the bound on total successive changes in the dynamic comparator sequence grows in time or the feedback regarding such bound arrives partially with time, both in a truly online manner. Then, as a major contribution, we examine the scenario when we receive no information regarding the successive changes, but instead, by a unique re-purposing of the expert mixture framework with novel additions, we eliminate the need of such information in, again, a truly online manner. Moreover, we show the ability to compete against all dynamic comparator sequences simultaneously (universally) with minimax optimality, where the guarantees depend on the "complexity" of each comparator separately. We also discuss potential modifications to our approach which addresses further complexity reductions for time, computation, memory, and we also further the universal competitiveness via guarantees taking into account concentrations of a comparator sequence in the decision set.