Full-Covariance Smoothing of Bayesian Neural Networks for Online Adaptation

A neural network's layers can be treated as time steps of a state-space model, turning Bayesian training into a smoothing problem: a forward pass propagates Gaussian moments through the network, and a backward Rauch--Tung--Striebel pass updates the weight posteriors in closed form. Such methods learn from each observation in a single pass, in an uncertainty-aware manner, and without gradient-based iterations or replay, which makes them well suited for online adaptation and data-efficient learning. Existing smoothing-based methods, however, are restricted to diagonal covariances across activations, discarding correlations between neurons. We overcome this limitation via a cross-covariance identity that enables full-covariance propagation through a network's nonlinear activations. We derive a one-step-per-layer smoother that approximates as Gaussian only each layer's affine output, and that applies both to deterministic systems with noisy observations and to stochastic systems described by output statistics. We demonstrate this method in non-stationary classification, online dynamics learning, and policy adaptation of a vision-language-action model, and find that it is generally more accurate than other smoothing-based methods.