Locally-Adaptive Nonparametric Online Learning
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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 regret 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 whose performance can provably take advantage of these local patterns are hardly known. We fill this gap introducing efficient online algorithms (based on a single versatile master algorithm) that adapt to: (1) local Lipschitzness of the competitor function, (2) local metric dimension of the instance sequence, (3) local performance of the predictor across different regions of the instance space. Extending previous approaches, we design algorithms that dynamically grow hierarchical packings of the instance space, and 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 scaling with quantities associated with all three types of local regularities. 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.
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