Bayesian approaches for training deep neural networks (BNNs) have received significant interest and have been effectively utilized in a wide range of applications. There have been several studies on the properties of posterior concentrations of BNNs. However, most of these studies only demonstrate results in BNN models with sparse or heavy-tailed priors. Surprisingly, no theoretical results currently exist for BNNs using Gaussian priors, which are the most commonly used one. The lack of theory arises from the absence of approximation results of Deep Neural Networks (DNNs) that are non-sparse and have bounded parameters. In this paper, we present a new approximation theory for non-sparse DNNs with bounded parameters. Additionally, based on the approximation theory, we show that BNNs with non-sparse general priors can achieve near-minimax optimal posterior concentration rates to the true model.

翻译：贝叶斯方法在深度神经网络训练（BNNs）中引起了广泛关注，并已成功应用于众多领域。关于BNNs后验集中性的研究已有若干成果，然而这些研究大多仅证明了使用稀疏或重尾先验的BNN模型的性质。令人意外的是，目前尚无针对最常用的高斯先验BNN的理论结果。该理论缺失源于缺乏针对非稀疏且参数有界的深度神经网络（DNNs）的逼近理论。本文提出了一种新的非稀疏参数有界DNN逼近理论，并基于该理论证明了使用非稀疏一般先验的BNN能够实现对真实模型的近极小化最优后验集中速率。