Online Nonparametric Regression

TL;DR

This paper establishes optimal rates for online nonparametric regression, revealing a phase transition similar to statistical learning, and introduces a generic, potentially efficient, forecaster with proven optimality.

Contribution

It provides the first optimal rate analysis for online nonparametric regression using sequential entropy and offers a generic, efficient algorithmic framework.

Findings

Optimal rates exhibit a phase transition similar to i.i.d. case

Matching entropy conditions lead to identical rates for online and statistical learning

A generic forecaster achieving optimal rates is constructed

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.