Asymptotic Properties for Bayesian Neural Network in Besov Space

TL;DR

This paper demonstrates that Bayesian neural networks with spike-and-slab and shrinkage priors achieve nearly minimax convergence rates in Besov spaces, providing a practical method with strong theoretical guarantees.

Contribution

It establishes the asymptotic consistency and convergence rates of Bayesian neural networks with specific priors in Besov spaces, including adaptive properties for unknown smoothness.

Findings

Spike-and-slab prior achieves nearly minimax convergence rate.

Shrinkage prior also attains the same convergence rate.

Method provides practical Bayesian neural networks with theoretical guarantees.

Abstract

Neural networks have shown great predictive power when dealing with various unstructured data such as images and natural languages. The Bayesian neural network captures the uncertainty of prediction by putting a prior distribution for the parameter of the model and computing the posterior distribution. In this paper, we show that the Bayesian neural network using spike-and-slab prior has consistency with nearly minimax convergence rate when the true regression function is in the Besov space. Even when the smoothness of the regression function is unknown the same posterior convergence rate holds and thus the spike-and-slab prior is adaptive to the smoothness of the regression function. We also consider the shrinkage prior, which is more feasible than other priors, and show that it has the same convergence rate. In other words, we propose a practical Bayesian neural network with guaranteed asymptotic properties.