A Sieve Stochastic Gradient Descent Estimator for Online Nonparametric Regression in Sobolev ellipsoids

The goal of regression is to recover an unknown underlying function that best links a set of predictors to an outcome from noisy observations. In non-parametric regression, one assumes that the regression function belongs to a pre-specified infinite dimensional function space (the hypothesis space). In the online setting, when the observations come in a stream, it is computationally-preferable to iteratively update an estimate rather than refitting an entire model repeatedly. Inspired by nonparametric sieve estimation and stochastic approximation methods, we propose a sieve stochastic gradient descent estimator (Sieve-SGD) when the hypothesis space is a Sobolev ellipsoid. We show that Sieve-SGD has rate-optimal MSE under a set of simple and direct conditions. We also show that the Sieve-SGD estimator can be constructed with low time expense, and requires almost minimal memory usage among all statistically rate-optimal estimators, under some conditions on the distribution of the predictors.