Error Analysis of Neural-Network-Based Engression

2026-07-30

AI Summary

This work addresses the lack of theoretical error analysis in existing neural network–based engression methods, which hinders a clear understanding of their generalization and convergence behavior. For the first time, it establishes a systematic theoretical error decomposition for the engression framework, splitting the excess risk into three components: approximation error, stochastic error, and Monte Carlo error. Under a composite smoothness assumption on the target conditional generator, the study derives explicit convergence rates. By integrating deep neural networks, energy scores, function approximation theory, and nonparametric estimation, this research elucidates how each error source influences overall performance, thereby providing a rigorous theoretical foundation and practical guidance for engression methods.

Abstract

Engression (Shen and Meinshausen, 2024) learns a conditional distribution by fitting a generative model $Y = f(X,\varepsilon)$ under the energy score, a strictly proper scoring rule. We provide a theoretical error analysis of engression implemented with deep neural networks. We decompose the excess risk into three components: the approximation error, the stochastic error, and the Monte Carlo error. Based on this decomposition, we establish convergence rates under the assumption that the target conditional generator admits a compositional smoothness structure.

Problem

• error analysis
• neural networks
• conditional distribution
• engression
• convergence rates

Innovation

• engression
• error decomposition
• convergence rates
• compositional smoothness
• energy score