Information theoretic quantities play a central role in machine learning. The recent surge in the complexity of data and models has increased the demand for accurate estimation of these quantities. However, as the dimension grows the estimation presents significant challenges, with existing methods struggling already in relatively low dimensions. To address this issue, in this work, we introduce $\texttt{REMEDI}$ for efficient and accurate estimation of differential entropy, a fundamental information theoretic quantity. The approach combines the minimization of the cross-entropy for simple, adaptive base models and the estimation of their deviation, in terms of the relative entropy, from the data density. Our approach demonstrates improvement across a broad spectrum of estimation tasks, encompassing entropy estimation on both synthetic and natural data. Further, we extend important theoretical consistency results to a more generalized setting required by our approach. We illustrate how the framework can be naturally extended to information theoretic supervised learning models, with a specific focus on the Information Bottleneck approach. It is demonstrated that the method delivers better accuracy compared to the existing methods in Information Bottleneck. In addition, we explore a natural connection between $\texttt{REMEDI}$ and generative modeling using rejection sampling and Langevin dynamics.

翻译：信息论量在机器学习中扮演核心角色。近年来数据和模型复杂性的急剧增长，提升了对这些量进行精确估计的需求。然而，随着维度的增加，估计面临显著挑战，现有方法甚至在相对低维的情况下已难以应对。为解决该问题，本文引入$\texttt{REMEDI}$以实现微分熵（一种基础信息论量）的高效精确估计。该方法结合了简单自适应基模型的交叉熵最小化，以及通过相对熵对模型与数据密度偏差的估计。我们的方法在广泛的估计任务中展现出改进，包括对合成数据与自然数据的熵估计。此外，我们将重要的理论一致性结果推广至方法所需的更泛化场景。我们展示了该框架如何自然扩展至信息论监督学习模型，特别聚焦于信息瓶颈方法。实验证明，该方法在信息瓶颈问题中相较于现有方法具有更优的准确性。最后，我们探讨了$\texttt{REMEDI}$与基于拒绝采样和朗之万动力学的生成建模之间的自然联系。