From Cross-Validation to SURE: Asymptotic Risk of Tuned Regularized Estimators
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 mu...
| Authors | Karun Adusumilli, Maximilian Kasy, Ashia Wilson |
|---|---|
| Primary category | math.ST |
| Published | 20 March 2026 |
| Last updated at source | 10 July 2026 |
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