Tractable Evaluation of Stein’s Unbiased Risk Estimator with Convex Regularizers

P. Nobel, E. Candès, and S. Boyd

Manuscript, November 2022.

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.