Asymptotic Properties for Bayesian Neural Network in Besov Space
This provides theoretical foundations for Bayesian neural networks in nonparametric regression, addressing uncertainty quantification for practitioners, though it is incremental as it builds on existing priors and spaces.
The paper tackles the problem of ensuring Bayesian neural networks have theoretical guarantees like consistency and minimax convergence rates when the true regression function lies in Besov space, showing that spike-and-slab and shrinkage priors achieve these rates adaptively without knowing smoothness.
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.