Learnable Uncertainty under Laplace Approximations

Abstract: Laplace approximations are classic, computationally lightweight means for constructing Bayesian neural networks (BNNs). As in other approximate BNNs, one cannot necessarily expect the induced predictive uncertainty to be calibrated. Here we develop a formalism to explicitly "train" the uncertainty in a decoupled way to the prediction itself. To this end, we introduce uncertainty units for Laplace-approximated networks: Hidden units associated with a particular weight structure that can be added to any pre-trained, point-estimated network. Due to their weights, these units are inactive -- they do not affect the predictions. But their presence changes the geometry (in particular the Hessian) of the loss landscape, thereby affecting the network's uncertainty estimates under a Laplace approximation. We show that such units can be trained via an uncertainty-aware objective, improving standard Laplace approximations' performance in various uncertainty quantification tasks.