Many real world data analysis problems exhibit invariant structure, and models that take advantage of this structure have shown impressive empirical performance, particularly in deep learning. While the literature contains a variety of methods to incorporate invariance into models, theoretical understanding is poor and there is no way to assess when one method should be preferred over another. In this work, we analyze the benefits and limitations of two widely used approaches in deep learning in the presence of invariance: data augmentation and feature averaging. We prove that training with data augmentation leads to better estimates of risk and gradients thereof, and we provide a PAC-Bayes generalization bound for models trained with data augmentation. We also show that compared to data augmentation, feature averaging reduces generalization error when used with convex losses, and tightens PAC-Bayes bounds. We provide empirical support of these theoretical results, including a demonstration of why generalization may not improve by training with data augmentation: the `learned invariance' fails outside of the training distribution.

本文分析比较了深度学习中两种广泛使用的处理不变性的方法：数据增强和特征平均，并针对其优缺点提出了相关理论结果和实验证明。其中，数据增强训练可以更好地估计风险和其梯度，并提供了PAC-Bayes泛化界；而特征平均可以在使用凸损失时降低泛化误差并收紧PAC-Bayes界。

神经网络中不变性的好处