Lifting Finite-Sample Gaussian Approximation of Wide Neural Networks to Functional Convergence
Randomly initialised neural networks of width $m$ converge to Gaussian processes as $m \to \infty$, with two types of quantitative bounds characterising this convergence. Finite-sample bounds compare the network with its Gaussian limit at a fixed finite sample set, on an arbitrary domain, in the quadratic Wasserstein distance $\mathcal W_2$, achieving a convergence rate of $m^{-1/2}$ with explicit constants. However, these constants grow with the number of samples, so the bound becomes uninformative as the sample set is refined. Functional bounds remove that dependence by comparing the two as random functions, at the price of spherical input, metrics weaker than $\mathcal W_2$, or slower ...
| Authors | Luis Wiederhold, Nil Ayday, Debarghya Ghoshdastidar |
|---|---|
| Primary category | stat.ML |
| Published | 13 October 2026 |
| Last updated at source | 13 October 2026 |
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