Optimal Dynamic Regret in Proper Online Learning with Strongly Convex Losses and Beyond
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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 Õ(d^1/3 n^1/3TV[u_1:n]^2/3∨ d) against any comparator sequence u_1,…,u_n simultaneously, where n is the time horizon and TV[u_1: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 L_∞ constrained decision set.
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