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

Abstract: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.

Machine Learning (stat.ML); Machine Learning (cs.LG)

arXiv:2608.30261 [stat.ML]

(or arXiv:2608.30261v1 [stat.ML] for this version)

https://doi.org/10.48550/arXiv.2608.30261 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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