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.

使用贝叶斯方法进行深度神经网络（BNNs）训练在广泛应用中受到了极大关注，并且已被有效地应用于各种情况。然而，大多数关于对 BNNs 的后验集中性质的研究仅在具有稀疏或重尾先验的 BNN 模型中证明结果。令人惊讶的是，目前还没有关于使用最常用的高斯先验进行 BNNs 的理论结果存在。这种理论缺乏是由于没有非稀疏且具有有界参数的深度神经网络（DNNs）的近似结果。在本文中，我们提出了一个新的近似理论，用于具有有界参数的非稀疏 DNNs。此外，基于该近似理论，我们表明具有非稀疏一般先验的 BNNs 可以以接近最小最优后验集中速率接近真实模型。