An Online Projection Estimator for Nonparametric Regression in Reproducing Kernel Hilbert Spaces

The goal of nonparametric regression is to recover an underlying regression function from noisy observations, under the assumption that the regression function belongs to a pre-specified infinite dimensional function space. In the online setting, when the observations come in a stream, it is generally computationally infeasible to refit the whole model repeatedly. There are as of yet no methods that are both computationally efficient and statistically rate-optimal. In this paper, we propose an estimator for online nonparametric regression. Notably, our estimator is an empirical risk minimizer (ERM) in a deterministic linear space, which is quite different from existing methods using random features and functional stochastic gradient. Our theoretical analysis shows that this estimator obtains rate-optimal generalization error when the regression function is known to live in a reproducing kernel Hilbert space. We also show, theoretically and empirically, that the computational expense of our estimator is much lower than other rate-optimal estimators proposed for this online setting.