Theoretical Analysis of Engression and Reverse Markov Engression
This work addresses the lack of theoretical guarantees for a recently proposed conditional distribution learning method, providing rigorous bounds that are near-optimal for practitioners and theorists in nonparametric statistics.
The paper provides the first finite-sample statistical guarantees for Engression and its Reverse Markov extension, establishing near-optimal excess-risk bounds for conditional distribution learning under deep neural network parameterizations.
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.