# From Cross-Validation to SURE: Asymptotic Risk of Tuned Regularized Estimators

arXiv: https://arxiv.org/abs/2603.20388 · HTML: https://mlsanity.com/paper/2603.20388/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 20 Mar 2026.

## Built on by (most cited first, 1 of 1)

- [Estimating Population-Risk Curves Along Nonconvex Gradient Flows from the Training Sample](https://mlsanity.com/paper/2608.30261/) (2026)

## Abstract (arXiv)

We derive the asymptotic risk function of regularized empirical risk minimization (ERM) estimators tuned by n-fold cross-validation (CV). The out-of-sample prediction loss of such estimators converges in distribution to the squared-error loss (risk function) of shrinkage estimators in the normal means model, tuned by Stein's unbiased risk estimate (SURE). This risk function provides a more fine-grained picture of predictive performance than uniform bounds on worst-case regret, which are common in learning theory: it quantifies how risk varies with the true parameter. As key intermediate steps, we show that (i) n-fold CV converges uniformly to SURE, and (ii) while SURE typically has multiple local minima, its global minimum is generically well separated. Well-separation ensures that uniform convergence of CV to SURE translates into convergence of the tuning parameter chosen by CV to that chosen by SURE.