Optimal Non-Asymptotic Edgeworth Expansions for Multivariate Neural Network Outputs
Finite-width fully connected neural networks with Gaussian-initialized weights deviate from their infinite-width Gaussian limit, exhibiting non-vanishing higher-order cumulants. We approximate these deviations, for a neural network evaluated in a finite number of inputs, using multidimensional Edgeworth expansions of arbitrary order $4m-1$, with $m\in\mathbb{N}$. Assuming that the corresponding Gaussian limit has an invertible covariance matrix and that the activation function is polynomially bounded, we establish a bound of order $n^{-m}$ on the total variation distance between the law of the true network output and its Edgeworth approximation, with matching lower bounds. As an applicati...
| Authors | Lucia Celli |
|---|---|
| Primary category | stat.ML |
| Published | 22 May 2026 |
| Last updated at source | 22 May 2026 |
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