Tractable Evaluation of Stein's Unbiased Risk Estimate with Convex Regularizers

TL;DR

This paper introduces scalable methods for efficiently evaluating Stein's Unbiased Risk Estimate (SURE) for convex regularized estimators, enabling practical risk assessment in large-scale Gaussian mean estimation problems.

Contribution

It develops new techniques to compute SURE analytically and efficiently for a broad class of convex regularizers, overcoming scalability issues of previous methods.

Findings

Methods enable SURE evaluation for large-scale problems

Analytical formulas derived for specific convex regularizers

Scalable algorithms improve risk estimation accuracy

Abstract

Stein's unbiased risk estimate (SURE) gives an unbiased estimate of the $\ell_2$ risk of any estimator of the mean of a Gaussian random vector. We focus here on the case when the estimator minimizes a quadratic loss term plus a convex regularizer. For these estimators SURE can be evaluated analytically for a few special cases, and generically using recently developed general purpose methods for differentiating through convex optimization problems; these generic methods however do not scale to large problems. In this paper we describe methods for evaluating SURE that handle a wide class of estimators, and also scale to large problem sizes.