In this work, we improve upon the stepwise analysis of noisy iterative learning algorithms initiated by Pensia, Jog, and Loh (2018) and recently extended by Bu, Zou, and Veeravalli (2019). Our main contributions are significantly improved mutual information bounds for Stochastic Gradient Langevin Dynamics via data-dependent estimates. Our approach is based on the variational characterization of mutual information and the use of data-dependent priors that forecast the mini-batch gradient based on a subset of the training samples. Our approach is broadly applicable within the information-theoretic framework of Russo and Zou (2015) and Xu and Raginsky (2017). Our bound can be tied to a measure of flatness of the empirical risk surface. As compared with other bounds that depend on the squared norms of gradients, empirical investigations show that the terms in our bounds are orders of magnitude smaller.

本文改进了 Pensia，Jog 和 Loh (2018) 开始的有噪声迭代学习算法的逐步分析，并在 Bu，Zou 和 Veeravalli (2019) 的基础上最近扩展。我们主要的贡献是通过数据相关估计显著提高了随机梯度 Langevin 动力学的互信息界限。我们的方法基于互信息的变分特性和使用基于训练样本子集的数据相关先验来预测小批量梯度。我们的方法在 Russo 和 Zou (2015)、Xu 和 Raginsky (2017) 的信息论框架内广泛适用。与其他依赖于梯度平方范数的边界相比，我们的边界项的数量级要小得多，同时可以与经验风险面的平坦度指标相关联。

通过数据依赖估计的信息理论广义绑定对SGLD的应用