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Branch Geometry and Finite-Radius Sensitivity of Hard-ReLU Training

arXiv · AI, language, vision and robotics · article · Aug 31, 2026 · UTC

Outer-learning algorithms use infinitesimal sensitivities to propose finite changes to initialization or training parameters. For hard-ReLU training, the derivative of the finite program and the derivative of its flow limit do not by themselves specify the response at a chosen radius. We characterize the intervening regime in which the perturbation radius is proportional to the GD step. Integer event rounding then survives at leading order: smooth Euler bias shifts each discrete phase, and upstream rounding moves downstream branch boundaries. We derive the crossing indices and a uniform endpoi

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