Few-sample Variational Inference of Bayesian Neural Networks with Arbitrary Nonlinearities

Bayesian Neural Networks (BNNs) extend traditional neural networks to provide uncertainties associated with their outputs. On the forward pass through a BNN, predictions (and their uncertainties) are made either by Monte Carlo sampling network weights from the learned posterior or by analytically propagating statistical moments through the network. Though flexible, Monte Carlo sampling is computationally expensive and can be infeasible or impractical under resource constraints or for large networks. While moment propagation can ameliorate the computational costs of BNN inference, it can be difficult or impossible for networks with arbitrary nonlinearities, thereby restricting the possible set of network layers permitted with such a scheme. In this work, we demonstrate a simple yet effective approach for propagating statistical moments through arbitrary nonlinearities with only 3 deterministic samples, enabling few-sample variational inference of BNNs without restricting the set of network layers used. Furthermore, we leverage this approach to demonstrate a novel nonlinear activation function that we use to inject physics-informed prior information into output nodes of a BNN.

A Framework for Variational Inference of Lightweight Bayesian Neural Networks with Heteroscedastic Uncertainties

Obtaining heteroscedastic predictive uncertainties from a Bayesian Neural Network (BNN) is vital to many applications. Often, heteroscedastic aleatoric uncertainties are learned as outputs of the BNN in addition to the predictive means, however doing so may necessitate adding more learnable parameters to the network. In this work, we demonstrate that both the heteroscedastic aleatoric and epistemic variance can be embedded into the variances of learned BNN parameters, improving predictive performance for lightweight networks. By complementing this approach with a moment propagation approach to inference, we introduce a relatively simple framework for sampling-free variational inference suitable for lightweight BNNs.