Prediction when fitting simple models to high-dimensional data 1

Abstract

We study linear subset regression in the context of a high-dimensional linear model. Consider y = ϑ + θz + with univariate response y and a dvector of random regressors z, and a submodel where y is regressed on a set of p explanatory variables that are given by x = Mz, for some d × p matrix M. Here, “high-dimensional” means that the number d of available explanatory variables in the overall model is much larger than the number p of variables in the submodel. In this paper, we present Pinsker-type results for prediction of y given x. In particular, we show that the mean squared prediction error of the best linear predictor of y given x is close to the mean squared prediction error of the corresponding Bayes predictor E[yx], provided only that p/log d is small. We also show that the mean squared prediction error of the (feasible) least-squares predictor computed from n independent observations of (y, x) is close to that of the Bayes predictor, provided only that both p/log d and p/n are small. Our results hold uniformly in the regression parameters and over large collections of distributions for the design variables z.