Revisiting the Equivalence of Bayesian Neural Networks and Gaussian Processes: On the Importance of Learning Activations

Marcin Sendera, Amin Sorkhei, Tomasz Kuśmierczyk

Keywords: function space priors, Bayesian Neural Networks, Gaussian Processes, optimal transport

TL;DR: In this paper, we consider embedding of functional priors in shallow BNNs and demonstrate that learning activations increases their fidelity and faithfulness.

Abstract:

Gaussian Processes (GPs) provide a convenient framework for specifying function-space priors, making them a natural choice for modeling uncertainty. In contrast, Bayesian Neural Networks (BNNs) offer greater scalability and extendability but lack the advantageous properties of GPs. This motivates the development of BNNs capable of replicating GP-like behavior. However, existing solutions are either limited to specific GP kernels or rely on heuristics.

We demonstrate that trainable activations are crucial for effective mapping of GP priors to wide BNNs. Specifically, we leverage the closed-form 2-Wasserstein distance for efficient gradient-based optimization of reparameterized priors and activations. Beyond learned activations, we also introduce trainable periodic activations that ensure global stationarity by design, and functional priors conditioned on GP hyperparameters to allow efficient model selection.

Empirically, our method consistently outperforms existing approaches or matches performance of the heuristic methods, while offering stronger theoretical foundations.

Code Link: https://github.com/gmum/bnn-functional-priors

Readers: auai.org/UAI/2025/Conference, auai.org/UAI/2025/Conference/Area_Chairs, auai.org/UAI/2025/Conference/Reviewers, auai.org/UAI/2025/Conference/Submission105/Authors, auai.org/UAI/2025/Conference/Submission105/Reproducibility_Reviewers

Submission Number: 105