← 研踪 PaperTrailarXiv:2601.22307 · v3
Exact Gaussian Moment Matching for Residual Networks: a Second-Order Method
首次提交 2026-01-30 04:47
版本更新 2026-11-03 15:26
雷达观测 2026-11-04 23:40
摘要
We study the problem of propagating the mean and covariance of a general multivariate Gaussian distribution through a deep (residual) neural network using layer-by-layer moment matching. We close a longstanding gap by deriving exact moment matching for the probit, GeLU, ReLU (as a limit of GeLU), Heaviside (as a limit of probit), and sine activation functions; for both feedforward and generalized residual layers. On large-variance random networks, median KL-divergence improvements over common baselines range from tens to thousands of times. On a variational Bayes neural network, we show that our method improves upon a state-of-the-art deterministic inference method by an order of magnitude in KL divergence from Monte Carlo ground truth. We also give a smooth-distance error bound showing that, under regularity assumptions, moment matching removes the leading low-variance errors and propagates higher-order local accuracy through the layers of a network.
作者备注：to appear at NeurIPS 2026, https://openreview.net/forum?id=1dsxDeHdpR
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v3 · 2026-11-03 15:26 · 最新观测版本
Exact Gaussian Moment Matching for Residual Networks: a Second-Order Method
We study the problem of propagating the mean and covariance of a general multivariate Gaussian distribution through a deep (residual) neural network using layer-by-layer moment matching. We close a longstanding gap by deriving exact moment matching for the probit, GeLU, ReLU (as a limit of GeLU), Heaviside (as a limit of probit), and sine activation functions; for both feedforward and generalized residual layers. On large-variance random networks, median KL-divergence improvements over common baselines range from tens to thousands of times. On a variational Bayes neural network, we show that our method improves upon a state-of-the-art deterministic inference method by an order of magnitude in KL divergence from Monte Carlo ground truth. We also give a smooth-distance error bound showing that, under regularity assumptions, moment matching removes the leading low-variance errors and propagates higher-order local accuracy through the layers of a network.
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