arXiv：cs.AI· Luis Wiederhold, Nil Ayday, Debarghya Ghoshdastidar·· 20 小时前
将宽神经网络的有限样本高斯近似提升为函数收敛
Lifting Finite-Sample Gaussian Approximation of Wide Neural Networks to Functional Convergence
AI 导读
该论文将宽神经网络的有限样本高斯近似边界提升为函数型边界。作者通过逐样本度量误差使常数不随样本数增长，在多项式激活函数和任意紧输入集下，得到 sup 范数中 m^{-1/2} 的收敛速率及显式常数。
正文
Abstract: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 rates. In this paper, we therefore lift a finite-sample bound to a functional one. The lifting refines the sample set until it is dense in the domain, so the bound must first be made independent of the number of samples. We achieve this by measuring the error per sample, which makes the constant uniform. For a polynomial activation and any compact input set, this gives a rate of $m^{-1/2}$ in the supremum norm, with an explicit constant.
| Comments: | 19 pages, 1 table, 1 figure, preprint |
| Subjects: | Machine Learning (stat.ML); Artificial Intelligence (cs.AI); Machine Learning (cs.LG) |
| Cite as: | arXiv:2610.18166 [stat.ML] |
| (or arXiv:2610.18166v1 [stat.ML] for this version) | |
| https://doi.org/10.48550/arXiv.2610.18166 arXiv-issued DOI via DataCite (pending registration) |
Submission history
From: Nil Ayday [view email]
[v1]
Tue, 13 Oct 2026 11:10:44 UTC (34 KB)
来源：arXiv：cs.AI · arxiv.org