  We investigate online convex optimization in non-stationary environments and
choose the dynamic regret as the performance measure, defined as the difference
between cumulative loss incurred by the online algorithm and that of any
feasible comparator sequence. Let $T$ be the time horizon and $P_T$ be the
path-length that essentially reflects the non-stationarity of environments, the
state-of-the-art dynamic regret is $\mathcal{O}(\sqrt{T(1+P_T)})$. Although
this bound is proved to be minimax optimal for convex functions, in this paper,
we demonstrate that it is possible to further enhance the dynamic regret by
exploiting the smoothness condition. Specifically, we propose novel online
algorithms that are capable of leveraging smoothness and replace the dependence
on $T$ in the dynamic regret by problem-dependent quantities: the variation in
gradients of loss functions, the cumulative loss of the comparator sequence,
and the minimum of the previous two terms. These quantities are at most
$\mathcal{O}(T)$ while could be much smaller in benign environments. Therefore,
our results are adaptive to the intrinsic difficulty of the problem, since the
bounds are tighter than existing results for easy problems and meanwhile
guarantee the same rate in the worst case.
