Theoretical Analysis of Engression and Reverse Markov Engression

2026-05-31

AI Summary

This work addresses the lack of finite-sample statistical guarantees for Engression and its Reverse Markov extension by establishing the first non-asymptotic convergence bounds. The authors quantify the error in conditional distribution learning using energy distance and introduce an energy-distance-based chain rule to develop a framework for analyzing error propagation in multi-step backward processes. Under parameterization by deep neural networks and assuming the target functions belong to a Hölder class, the derived excess risk bound nearly achieves the classical minimax rate, differing only by a logarithmic factor. This result provides rigorous theoretical validation of the method's effectiveness in distributional modeling.

Abstract

Engression is a recently proposed and effective framework for conditional distribution learning. Its multi-step Reverse Markov extension further improves generative flexibility by decomposing complex conditional sampling into sequential reverse transitions. Despite their strong empirical performance, rigorous finite-sample statistical guarantees for these methods remain unavailable. In this paper, under deep neural network parameterizations, we establish nonasymptotic convergence bounds for Engression by directly controlling the Energy Distance between the learned and target conditional distributions. For the Reverse Markov framework, we further develop an Energy-Distance-based chain rule that enables a rigorous analysis of error propagation across reverse steps. Our analysis yields corresponding excess-risk bounds that are near-optimal up to logarithmic factors relative to the classical minimax rate over a general Hölder class.

Problem

• Engression
• Reverse Markov
• conditional distribution learning
• finite-sample guarantees
• Energy Distance

Innovation

• Engression
• Reverse Markov
• Energy Distance
• nonasymptotic convergence
• excess-risk bound