The Laplace approximation provides a closed-form model selection objective for neural networks (NN). Online variants, which optimise NN parameters jointly with hyperparameters, like weight decay strength, have seen renewed interest in the Bayesian deep learning community. However, these methods violate Laplace's method's critical assumption that the approximation is performed around a mode of the loss, calling into question their soundness. This work re-derives online Laplace methods, showing them to target a variational bound on a mode-corrected variant of the Laplace evidence which does not make stationarity assumptions. Online Laplace and its mode-corrected counterpart share stationary points where 1. the NN parameters are a maximum a posteriori, satisfying the Laplace method's assumption, and 2. the hyperparameters maximise the Laplace evidence, motivating online methods. We demonstrate that these optima are roughly attained in practise by online algorithms using full-batch gradient descent on UCI regression datasets. The optimised hyperparameters prevent overfitting and outperform validation-based early stopping.

本文重新推导了在线 Laplace 方法，表明它们针对一种在模型选择问题中纠正了模态的 Laplace 证据的变分界。在线 Laplace 方法以及其纠正模态的版本在近乎平稳点上实现最优解，通过应用于UCI回归数据集，优化关键参数并防止过拟合，优于传统的早期停止方法。

在线拉普拉斯模型选择再探