NONPARAMETRIC STOCHASTIC APPROXIMATION WITH LARGE STEP SIZES1 BY AYMERIC DIEULEVEUT
We consider the random design least squares regression problem within the reproducing kernel Hilbert space (RKHS) framework. Given a stream of independent and identically distributed input/output data, we aim to learn a regression function within an RKHS H, even if the optimal predictor (i.e. the conditional expectation) is not in H. In a stochastic approximation framework where the estimator is updated after each observation, we show that the averaged unregularized least mean square algorithm (a form of stochastic gradient descent) given a sufficient large step size, attains optimal rates of convergence for a variety of regimes for the smoothnesses of the optimal prediction function and the functions in H. Our results apply as well in the usual finite dimensional setting of parametric least squares regression, showing adaptivity of our estimator to the spectral decay of the covariance matrix of the covariates.