Neural Generative Distributional Regression

2026-05-03

AI Summary

This work proposes a neural generative modeling approach to approximate arbitrary continuous conditional distributions $Y|X$ by leveraging known noise distributions (e.g., uniform or Gaussian). The method learns a transformation function $g$ such that $Y = g(X, U)$ minimizes the empirical energy distance to the true conditional distribution. It introduces energy distance—a theoretically grounded and computationally efficient metric—into neural conditional density estimation for the first time. This framework adaptively captures low-dimensional structures in the data and achieves minimax optimal convergence rates under nonparametric assumptions. Empirical evaluations on both synthetic and real-world datasets demonstrate its effectiveness in conditional moment estimation, prediction interval construction, and conditional density estimation, highlighting its practical utility alongside strong theoretical guarantees.

Abstract

Any continuous conditional distribution of $Y$ given $X$ can be generated from a transform of a known noise distribution $U$ such as the uniform or normal distribution via $Y = g(X, U)$. This paper provides an estimator of such a generative transformation $g$ by minimizing the empirical energy distance between distributions of $Y$ and $g(X, U)$, and implements it via neural networks. The estimated distribution can then be readily applied to downstream tasks such as conditional moment estimation, predictive interval construction, and conditional density estimation. By leveraging the representation power of neural networks, the estimator can adaptively exploit low-dimensional structures in a purely algorithmic manner. Theoretically, we establish an oracle inequality attaining the adaptive optimal nonparametric rates. Numerical simulations and real data analysis further demonstrate the practical effectiveness of the proposed method.

Problem

• conditional distribution
• generative modeling
• distributional regression
• neural networks
• nonparametric estimation

Innovation

• Neural Generative Regression
• Energy Distance
• Conditional Distribution Estimation
• Adaptive Nonparametric Rate
• Low-dimensional Structure