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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