  We consider the problem of the Zinkevich (2003)-style dynamic regret
minimization in online learning with exp-concave losses. We show that whenever
improper learning is allowed, a Strongly Adaptive online learner achieves the
dynamic regret of $\tilde O^*(n^{1/3}C_n^{2/3} \vee 1)$ where $C_n$ is the
total variation (a.k.a. path length) of the an arbitrary sequence of
comparators that may not be known to the learner ahead of time. Achieving this
rate was highly nontrivial even for squared losses in 1D where the best known
upper bound was $O(\sqrt{nC_n} \vee \log n)$ (Yuan and Lamperski, 2019). Our
new proof techniques make elegant use of the intricate structures of the primal
and dual variables imposed by the KKT conditions and could be of independent
interest. Finally, we apply our results to the classical statistical problem of
locally adaptive non-parametric regression (Mammen, 1991; Donoho and Johnstone,
1998) and obtain a stronger and more flexible algorithm that do not require any
statistical assumptions or any hyperparameter tuning.
