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The goal of regression is to recover an unknown underlying function that bestlinks a set of predictors to an outcome from noisy observations. Innonparametric regression, one assumes that the regression function belongs to apre-specified infinite-dimensional function space (the hypothesis space). Inthe online setting, when the observations come in a stream, it iscomputationally-preferable to iteratively update an estimate rather thanrefitting an entire model repeatedly. Inspired by nonparametric sieveestimation and stochastic approximation methods, we propose a sieve stochasticgradient descent estimator (Sieve-SGD) when the hypothesis space is a Sobolevellipsoid. We show that Sieve-SGD has rate-optimal mean squared error (MSE)under a set of simple and direct conditions. The proposed estimator can beconstructed with a low computational (time and space) expense: We also formallyshow that Sieve-SGD requires almost minimal memory usage among allstatistically rate-optimal estimators.

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