  One of the main strengths of online algorithms is their ability to adapt to
arbitrary data sequences. This is especially important in nonparametric
settings, where performance is measured against rich classes of comparator
functions that are able to fit complex environments. Although such hard
comparators and complex environments may exhibit local regularities, efficient
algorithms, which can provably take advantage of these local patterns, are
hardly known. We fill this gap by introducing efficient online algorithms
(based on a single versatile master algorithm) each adapting to one of the
following regularities: (i) local Lipschitzness of the competitor function,
(ii) local metric dimension of the instance sequence, (iii) local performance
of the predictor across different regions of the instance space. Extending
previous approaches, we design algorithms that dynamically grow hierarchical
$\epsilon$-nets on the instance space whose prunings correspond to different
"locality profiles" for the problem at hand. Using a technique based on tree
experts, we simultaneously and efficiently compete against all such prunings,
and prove regret bounds each scaling with a quantity associated with a
different type of local regularity. When competing against "simple" locality
profiles, our technique delivers regret bounds that are significantly better
than those proven using the previous approach. On the other hand, the time
dependence of our bounds is not worse than that obtained by ignoring any local
regularities.
