  This work focuses on the setting of dynamic regret in the context of online
learning with full information. In particular, we analyze regret bounds with
respect to the temporal variability of the loss functions. By assuming that the
sequence of loss functions does not vary much with time, we show that it is
possible to incur improved regret bounds compared to existing results. The key
to our approach is to use the loss function (and not its gradient) during the
optimization process. Building on recent advances in the analysis of Implicit
algorithms, we propose an adaptation of the Implicit version of Online Mirror
Descent to the dynamic setting. Our proposed algorithm is adaptive not only to
the temporal variability of the loss functions, but also to the path length of
the sequence of comparators when an upper bound is known. Furthermore, our
analysis reveals that our results are tight and cannot be improved without
further assumptions. Next, we show how our algorithm can be applied to the
setting of learning with expert advice or to settings with composite loss
functions. Finally, when an upper bound to the path-length is not fixed
beforehand we show how to combine a greedy strategy with existing
strongly-adaptive algorithms to compete optimally against different sequences
of comparators simultaneously.
