Precise Error Analysis of Regularized $M$ -Estimators in High Dimensions

A popular approach for estimating an unknown signal <inline-formula> <tex-math notation="LaTeX">$ \mathbf {x}_{0}\in \mathbb {R} ^{n}$ </tex-math></inline-formula> from noisy, linear measurements <inline-formula> <tex-math notation="LaTeX">$ \mathbf {y}= \mathbf {A} \mathbf {x} _{0}+ \mathbf {z}\in \mathbb {R}^{m}$ </tex-math></inline-formula> is via solving a so called <italic>regularized</italic> <inline-formula> <tex-math notation="LaTeX">$M$ </tex-math></inline-formula>-estimator: <inline-formula> <tex-math notation="LaTeX">$\hat{\mathbf {x}} :=\arg \min _ \mathbf {x} \mathcal {L} (\mathbf {y}- \mathbf {A} \mathbf {x})+\lambda f(\mathbf {x})$ </tex-math></inline-formula>. Here, <inline-formula> <tex-math notation="LaTeX">$ \mathcal {L}$ </tex-math></inline-formula> is a convex loss function, <inline-formula> <tex-math notation="LaTeX">$f$ </tex-math></inline-formula> is a convex (typically, non-smooth) regularizer, and <inline-formula> <tex-math notation="LaTeX">$\lambda > 0$ </tex-math></inline-formula> is a regularizer parameter. We analyze the squared error performance <inline-formula> <tex-math notation="LaTeX">$\|\hat{\mathbf {x}} - \mathbf {x}_{0}\|_{2}^{2}$ </tex-math></inline-formula> of such estimators in the <italic>high-dimensional proportional regime</italic> where <inline-formula> <tex-math notation="LaTeX">$m,n\rightarrow \infty $ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$m/n\rightarrow \delta $ </tex-math></inline-formula>. The design matrix <inline-formula> <tex-math notation="LaTeX">$ \mathbf {A}$ </tex-math></inline-formula> is assumed to have entries iid Gaussian; only minimal and rather mild regularity conditions are imposed on the loss function, the regularizer, and on the noise and signal distributions. We show that the squared error converges in probability to a nontrivial limit that is given as the solution to a minimax convex-concave optimization problem on four scalar optimization variables. We identify a new summary parameter, termed the <italic>expected Moreau envelope</italic> to play a central role in the error characterization. The <italic>precise</italic> nature of the results permits an accurate performance comparison between different instances of regularized <inline-formula> <tex-math notation="LaTeX">$M$ </tex-math></inline-formula>-estimators and allows to optimally tune the involved parameters (such as the regularizer parameter and the number of measurements). The key ingredient of our proof is the <italic>convex Gaussian min-max theorem</italic> which is a tight and strengthened version of a classical Gaussian comparison inequality that was proved by Gordon in 1988.

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