Training Under Challenge: Executable Certificates and Challenge-Closed Optimality for Neural Networks

2026-08-12

AI Summary

This work addresses the challenge of diagnosing why neural network training stalls—a phenomenon that may stem from convergence to global or local minima, insufficient model expressivity, or optimizer mismatch. The authors propose a “training-under-challenge” framework that enables reproducible and verifiable diagnosis and repair by constructing alternative solutions within the same certification class via pre-declared executable certificate programs and re-evaluating the objective. Central to this approach are the notions of challenge-closed optimality and challenge capacity modulus, integrated with a squared-loss block descent operator, a channel-gated ResNet-18 architecture, and output-direction coverage verification. In distillation tasks with known optima, eight internal challenges cover all 240 output directions, achieving residual bounds within 1.74–3.02× of the true optimality gap, effectively disentangling underutilization of the decoder from genuine representational insufficiency.

Abstract

A flat training curve does not reveal whether a neural network has reached a global optimum, is locally trapped, is representation-limited, or is mismatched to its trainer. We introduce Training Under Challenge, an executable-certificate framework in which predeclared, architecture-valid procedures construct complete alternatives in the same certified class and reevaluate the same objective. Any lower-valued candidate is a replayable witness that lower-bounds the checkpoint's empirical global-optimality gap. Passing a finite suite is only suite-relative; global-gap conclusions require a separately justified coverage mechanism. We define a resource-indexed challenge-power modulus that characterizes the largest gap compatible with passage. For squared loss, current block-decrease operators make coverage checkable and yield uniform and realized-residual bounds. We prove the converse frontier: without coverage, a first-order ReLU trainer can reach infinitely many exact conditional head optima while converging to a non-global point. On a channel-gated ResNet-18 distillation problem with known optimum, eight internal challenges cover all 240 audited output directions, and realized-residual bounds lie within factors of 1.74--3.02 of the true gap. Paired predictive certificates separate decoder under-use from representation insufficiency, while quantized-denoising studies demonstrate diagnosis, repair, and current-state recertification.

Problem

• neural network optimization
• global optimality
• training diagnostics
• representation limitation
• empirical gap

Innovation

• executable certificates
• challenge-closed optimality
• global-optimality gap
• coverage mechanism
• realized-residual bounds