Exact closed-form Gaussian moments of residual layers

Abstract

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 random networks, we find orders-of-magnitude improvements in the KL divergence error metric, up to a millionfold, over popular alternatives. On real data, we find competitive statistical calibration for inference under epistemic uncertainty in the input. On a variational Bayes network, we show that our method attains hundredfold improvements in KL divergence from Monte Carlo ground truth over a state-of-the-art deterministic inference method. We also give an a priori error bound and a preliminary analysis of stochastic feedforward neurons, which have recently attracted general interest.