Empirical Bayes linear regression in high dimensions: Method of moments and sub-linear sample complexity
We study empirical Bayes estimation of the prior in high-dimensional linear regression $\mathbf{y}=\mathbf{X}\mathbf{\beta}+\mathbf{\varepsilon}$, where the regression coefficients are drawn independently from an unknown sub-Gaussian prior. In contrast to the sequence model, the design matrix couples the latent coefficients, so that recovering the prior requires deconvolving it from both the noise and copies of itself. We introduce the \emph{Empirical Bayes Method of Moments} (EBMoM), a computationally efficient procedure for general designs that recursively estimates the prior moments through a lower-triangular system of estimating equations and runs in time $O(np^2)$. Under mild design conditions, satisfied in particular by a broad class of correlated random designs, we show that EBMoM consistently estimates a growing number of moments and hence the prior itself, provided that $n\geq p^{1-o(1)}$. A matching information-theoretic lower bound, valid for a broad class of designs, shows that this sub-linear sample complexity is optimal for nonparametric prior estimation. This improves on existing results for likelihood-based methods whose consistency requires a linear sample size $n=\Omega(p)$.