{"ID":2894162,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2507.12686","arxiv_id":"2507.12686","title":"Finite-Dimensional Gaussian Approximation for Deep Neural Networks: Universality in Random Weights","abstract":"We study the Finite-Dimensional Distributions (FDDs) of deep neural networks with randomly initialized weights that have finite-order moments. Specifically, we establish Gaussian approximation bounds in the Wasserstein-$1$ norm between the FDDs and their Gaussian limit assuming a Lipschitz activation function and allowing the layer widths to grow to infinity at arbitrary relative rates. In the special case where all widths are proportional to a common scale parameter $n$ and there are $L-1$ hidden layers, we obtain convergence rates of order $n^{-({1}/{6})^{L-1} + ε}$, for any $ε\u003e 0$.","short_abstract":"We study the Finite-Dimensional Distributions (FDDs) of deep neural networks with randomly initialized weights that have finite-order moments. Specifically, we establish Gaussian approximation bounds in the Wasserstein-$1$ norm between the FDDs and their Gaussian limit assuming a Lipschitz activation function and allow...","url_abs":"https://arxiv.org/abs/2507.12686","url_pdf":"https://arxiv.org/pdf/2507.12686v2","authors":"[\"Krishnakumar Balasubramanian\",\"Nathan Ross\"]","published":"2025-07-16T23:41:09Z","proceeding":"stat.ML","tasks":"[\"stat.ML\",\"cs.LG\",\"math.PR\",\"math.ST\"]","methods":"[]","has_code":false}