Online Nonparametric Regression

Abstract: We establish optimal rates for online regression for arbitrary classes of regression functions in terms of the sequential entropy introduced in (Rakhlin, Sridharan, Tewari, 2010). The optimal rates are shown to exhibit a phase transition analogous to the i.i.d./statistical learning case, studied in (Rakhlin, Sridharan, Tsybakov 2013). In the frequently encountered situation when sequential entropy and i.i.d. empirical entropy match, our results point to the interesting phenomenon that the rates for statistical learning with squared loss and online nonparametric regression are the same. In addition to a non-algorithmic study of minimax regret, we exhibit a generic forecaster that enjoys the established optimal rates. We also provide a recipe for designing online regression algorithms that can be computationally efficient. We illustrate the techniques by deriving existing and new forecasters for the case of finite experts and for online linear regression.