Optimal Dynamic Regret in Proper Online Learning with Strongly Convex Losses and Beyond

We study the framework of universal dynamic regret minimization with strongly convex losses. We answer an open problem in (Baby and Wang, 2021) by showing that in a proper learning setup, Strongly Adaptive algorithms can achieve the near optimal dynamic regret of (O) over tilde (d(1/3)n(1/3) TV[u(1:n)](2/3) V d) against any comparator sequence u(1); : : :; u(n) simultaneously, where n is the time horizon and TV[u1:n] is the Total Variation of comparator. These results are facilitated by exploiting a number of new structures imposed by the KKT conditions that were not considered in (Baby and Wang, 2021) which also lead to other improvements over their results such as: (a) handling non-smooth losses and (b) improving the dimension dependence on regret. Further, we also derive near optimal dynamic regret rates for the special case of proper online learning with exp-concave losses and an L1 constrained decision set.