Second Order Path Variationals in Non-Stationary Online Learning

We consider the problem of \emph{universal} dynamic regret minimization under exp-concave and smooth losses. We show that appropriately designed Strongly Adaptive algorithms achieve a dynamic regret of $\tilde O(d^2 n^{1/5} [\mathcal{TV}_1(w_{1:n})]^{2/5} \vee d^2)$, where $n$ is the time horizon and $\mathcal{TV}_1(w_{1:n})$ a path variational based on \emph{second order differences} of the comparator sequence. Such a path variational naturally encodes comparator sequences that are piece-wise linear -- a powerful family that tracks a variety of non-stationarity patterns in practice (Kim et al, 2009). The aforementioned dynamic regret is shown to be optimal modulo dimension dependencies and poly-logarithmic factors of $n$. To the best of our knowledge, this path variational has not been studied in the non-stochastic online learning literature before. Our proof techniques rely on analysing the KKT conditions of the offline oracle and requires several non-trivial generalizations of the ideas in Baby & Wang 2021 where the latter work only implies an $\tilde{O}(n^{1/3})$ regret for the current problem.