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

TL;DR

This paper shows that trainable and periodic activations are essential for Bayesian Neural Networks to accurately replicate Gaussian Process priors, improving performance and theoretical understanding.

Contribution

It introduces trainable and periodic activations for BNNs, enabling better GP prior mapping and model selection, with a solid theoretical basis.

Findings

Outperforms existing methods in empirical tests

Matches heuristic approaches with stronger theory

Uses Wasserstein distance for efficient optimization

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.