Optimal Neural Estimation for High-Dimensional Nonparametric Regression under Multisource Covariate Shift

We study high-dimensional nonparametric regression under multisource covariate shift, where J sources of training data share a common regression function but differ in their covariate distributions, sample sizes, and noise variances. Our goal is to combine these sources to estimate the regression function accurately under a target covariate distribution. We propose an oracle weighted neural estimator and establish its minimax optimality up to logarithmic factors through matching upper and lower bounds. The resulting rate is neff-2p/(2p+K), where p and K denote the effective smoothness and dimension of the regression function, and the effective sample size is {align} neff =∑ⱼ₌₁J {n_j}{(1+σρ,j)σε,j²}. {align} Here, nⱼ is the sample size of source j, σε,j² is its noise variance, and σρ,j controls the deviation between the covariate distribution of source j and the target covariate distribution. We also develop an adaptive procedure that requires no knowledge of source noise variances or covariate-shift parameters. Under suitable conditions, this procedure attains the oracle minimax rate up to logarithmic factors.