Quantitative convergence of trained single layer neural networks to Gaussian processes

Eloy Mosig; Andrea Agazzi; Dario Trevisan

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arXiv · 2509.24544

Quantitative convergence of trained single layer neural networks to Gaussian processes

Eloy Mosig·Andrea Agazzi·Dario Trevisan

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Abstract

In this paper, we study the quantitative convergence of shallow neural networks trained via gradient descent to their associated Gaussian processes in the infinite-width limit. While previous work has established qualitative convergence under broad settings, precise, finite-width estimates remain limited, particularly during training. We provide explicit upper bounds on the quadratic Wasserstein distance between the network output and its Gaussian approximation at any training time $t \ge 0$, demonstrating polynomial decay with network width. Our results quantify how architectural parameters, such as width and input dimension, influence convergence, and how training dynamics affect the approximation error.

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Eloy Mosig, Andrea Agazzi, Dario Trevisan. 2025-09-29. Quantitative convergence of trained single layer neural networks to Gaussian processes. https://arxiv.org/abs/2509.24544

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