Quantitative CLTs in deep neural networks

• 成果类型: Article
• 署名作者: Favaro, S.; Hanin, B.; Marinucci, D.; Nourdin, I.; Peccati, G.
• 署名单位: University of Turin; Princeton University; University of Rome Tor Vergata; University of Luxembourg
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-025-01360-1
• 发表日期: 2025
• 页码: 933-977
• 关键词: stein kernels; CONVERGENCE; limit

摘要

We study the distribution of a fully connected neural network with random Gaussian weights and biases in which the hidden layer widths are proportional to a large constant n. Under mild assumptions on the non-linearity, we obtain quantitative bounds on normal approximations valid at large but finite n and any fixed network depth. Our theorems show both for the finite-dimensional distributions and the entire process, that the distance between a random fully connected network (and its derivatives) to the corresponding infinite width Gaussian process scales liken(-gamma )for gamma > 0, with the exponent depending on the metric used to measure discrepancy. Our bounds are strictly stronger in terms of their dependence on network width than any previously available in the literature; in the one-dimensional case, we also prove that they are optimal, i.e., we establish matching lower bounds.