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

Abstract

We study high-dimensional nonparametric regression under multisource covariate shift, where $\mathrm J$ 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 $\mathsf n_{\rm eff}^{-2p/(2p+K)}$, where $p$ and $K$ denote the effective smoothness and dimension of the regression function, and the effective sample size is \begin{align} \mathsf n_{\rm eff} =\sum_{j=1}^{\mathrm J} \frac{n_j}{(1+σ_{ρ,j})σ_{ε,j}^{2}}. \nonumber \end{align} Here, $n_j$ is the sample size of source $j$, $σ_{ε,j}^{2}$ is its noise variance, and $σ_{ρ,j}$ 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.