后验贝叶斯神经网络在无限宽时可近似为更灵活的t过程。
Student-t processes as infinite-width limits of posterior Bayesian neural networks
- 用逆伽马先验建模最后一层与似然方差，使后验BNN收敛到t过程
- 通过Wasserstein距离控制近似误差，实现对收敛速率的量化分析
- 适合关注不确定性建模与理论保证的研究者
贝叶斯神经网络（BNN）的渐近性质已被广泛研究，尤其关注其在无限宽度极限下对高斯过程的逼近。本文拓展了这些结果，表明若BNN参数服从高斯先验，且最后一层隐藏层和高斯似然函数的方差均服从逆伽马先验，则后验BNN在无限宽度极限下可被近似为t过程。该近似提供了更强的不确定性建模能力。我们的证明利用Wasserstein度量，建立了对t过程近似的收敛速率的控制。这一结果为贝叶斯深度学习中的不确定性量化提供了新的理论支持。
原文摘要 · Abstract (English)
The asymptotic properties of Bayesian Neural Networks (BNNs) have been extensively studied, particularly regarding their approximations by Gaussian processes in the infinite-width limit. We extend these results by showing that posterior BNNs can be approximated by Student-t processes, which offer greater flexibility in modeling uncertainty. Specifically, we show that, if the parameters of a BNN follow a Gaussian prior distribution, and the variance of both the last hidden layer and the Gaussian likelihood function follows an Inverse-Gamma prior distribution, then the resulting posterior BNN converges to a Student-t process in the infinite-width limit. Our proof leverages the Wasserstein metric to establish control over the convergence rate of the Student-t process approximation.
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