# Estimating Population-Risk Curves Along Nonconvex Gradient Flows from the Training Sample

arXiv: https://arxiv.org/abs/2608.30261 · HTML: https://mlsanity.com/paper/2608.30261/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 9 of its 41 references that are in the map.
- First posted on arXiv 31 Aug 2026.

## Builds on (most cited first, 9 of 9)

- [A Higher-Order Swiss Army Infinitesimal Jackknife](https://mlsanity.com/paper/1907.12116/) (2019)
- [Precise gradient descent training dynamics for finite-width multi-layer neural networks](https://mlsanity.com/paper/2505.04898/) (2025)
- [Concentration inequalities for leave-one-out cross validation](https://mlsanity.com/paper/2211.02478/) (2022)
- [A Theory of Generalization in Deep Learning](https://mlsanity.com/paper/2605.01172/) (2026)
- [From Cross-Validation to SURE: Asymptotic Risk of Tuned Regularized Estimators](https://mlsanity.com/paper/2603.20388/) (2026)
- [Optimal Confidence Band for Kernel Gradient Flow Estimator](https://mlsanity.com/paper/2605.05768/) (2026)
- [A Dual Representation of Influence Functions for Linearizable Models](https://mlsanity.com/paper/2605.11239/) (2026)
- [Gradient-Flow Optimization as Dynamic Random-Effects Inference: Testing and Early Stopping with Applications to Deep Learning](https://mlsanity.com/paper/2605.27991/) (2026)
- [Statistical Inference on Gradient Flows](https://mlsanity.com/paper/2606.01257/) (2026)

## Abstract (arXiv)

We estimate the conditional population-risk curve of a realized smooth nonconvex gradient flow from the training sample. Flow approximate leave-one-out (Flow-ALO) propagates a deletion response and evaluates omitted observations at approximate deleted paths. The risk-curve error decomposes into response approximation, exact-LOO fluctuation, and deletion-to-full risk transfer. On each fixed finite horizon, bounded centered training-loss gradients, a one-sided Hessian lower bound, locally Lipschitz Hessians, and a strict tube-closure condition yield an explicit (n-1)-2 bound for the deletion-response error. Bounded evaluation-loss gradients transfer the deletion-response bound to the score without requiring the Hessian to be invertible. Direct first-order jackknife cancellation and exact-LOO concentration control deletion-to-full risk transfer and fluctuation, respectively, completing recovery of the conditional population-risk curve. For bounded smooth two-layer mean-field networks training both layers, the score-error bound is uniform in width.