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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 $ε$-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.