Predictability Analysis of Regression Problems via Conditional Entropy Estimations

In the field of machine learning, regression problems are pivotal due to their ability to predict continuous outcomes. Traditional error metrics like mean squared error, mean absolute error, and coefficient of determination measure model accuracy. The model accuracy is the consequence of the selected model and the features, which blurs the analysis of contribution. Predictability, in the other hand, focus on the predictable level of a target variable given a set of features. This study introduces conditional entropy estimators to assess predictability in regression problems, bridging this gap. We enhance and develop reliable conditional entropy estimators, particularly the KNIFE-P estimator and LMC-P estimator, which offer under- and over-estimation, providing a practical framework for predictability analysis. Extensive experiments on synthesized and real-world datasets demonstrate the robustness and utility of these estimators. Additionally, we extend the analysis to the coefficient of determination \(R^2 \), enhancing the interpretability of predictability. The results highlight the effectiveness of KNIFE-P and LMC-P in capturing the achievable performance and limitations of feature sets, providing valuable tools in the development of regression models. These indicators offer a robust framework for assessing the predictability for regression problems.

A Step-by-step Introduction to the Implementation of Automatic Differentiation

Automatic differentiation is a key component in deep learning. This topic is well studied and excellent surveys such as Baydin et al. (2018) have been available to clearly describe the basic concepts. Further, sophisticated implementations of automatic differentiation are now an important part of popular deep learning frameworks. However, it is difficult, if not impossible, to directly teach students the implementation of existing systems due to the complexity. On the other hand, if the teaching stops at the basic concept, students fail to sense the realization of an implementation. For example, we often mention the computational graph in teaching automatic differentiation, but students wonder how to implement and use it. In this document, we partially fill the gap by giving a step by step introduction of implementing a simple automatic differentiation system. We streamline the mathematical concepts and the implementation. Further, we give the motivation behind each implementation detail, so the whole setting becomes very natural.