Second Order Path Variationals in Non-Stationary Online Learning
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We consider the problem of universal dynamic regret minimization under exp-concave and smooth losses. We show that appropriately designed Strongly Adaptive algorithms achieve a dynamic regret of Õ(d^2 n^1/5 C_n^2/5∨ d^2), where n is the time horizon and C_n a path variational based on second order differences of the comparator sequence. Such a path variational naturally encodes comparator sequences that are piecewise linear – a powerful family that tracks a variety of non-stationarity patterns in practice (Kim et al, 2009). The aforementioned dynamic regret rate is shown to be optimal modulo dimension dependencies and poly-logarithmic factors of n. Our proof techniques rely on analysing the KKT conditions of the offline oracle and requires several non-trivial generalizations of the ideas in Baby and Wang, 2021, where the latter work only leads to a slower dynamic regret rate of Õ(d^2.5n^1/3C_n^2/3∨ d^2.5) for the current problem.
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