  We study the problem of dynamic regret minimization in online convex
optimization, in which the objective is to minimize the difference between the
cumulative loss of an algorithm and that of an arbitrary sequence of
comparators. While the literature on this topic is very rich, a unifying
framework for the analysis and design of these algorithms is still missing. In
this paper, \emph{we show that dynamic regret minimization is equivalent to
static regret minimization in an extended decision space}. Using this simple
observation, we show that there is a frontier of lower bounds trading off
penalties due to the variance of the losses and penalties due to variability of
the comparator sequence, and provide a framework for achieving any of the
guarantees along this frontier. As a result, we prove for the first time that
adapting to the squared path-length of an arbitrary sequence of comparators to
achieve regret $R_{T}(u_{1},\dots,u_{T})\le O(\sqrt{T\sum_{t}
\|u_{t}-u_{t+1}\|^{2}})$ is impossible. However, we prove that it is possible
to adapt to a new notion of variability based on the locally-smoothed squared
path-length of the comparator sequence, and provide an algorithm guaranteeing
dynamic regret of the form $R_{T}(u_{1},\dots,u_{T})\le \tilde
O(\sqrt{T\sum_{i}\|\bar u_{i}-\bar u_{i+1}\|^{2}})$. Up to polylogarithmic
terms, the new notion of variability is never worse than the classic one
involving the path-length.
