Title: A Theoretical Examination of Engression and Reverse Markov Engression

Abstract:

Engression has emerged as a highly effective methodology for learning conditional distributions, recently introduced to the field. Its capability is further enhanced by the Reverse Markov extension, a multi-step approach that boosts generative flexibility by breaking down complex conditional sampling tasks into a series of sequential reverse transitions. Although these techniques have demonstrated robust empirical results, they have yet to be supported by rigorous finite-sample statistical guarantees.

In this study, we address this gap by establishing nonasymptotic convergence bounds for Engression within the context of deep neural network parameterizations. Our approach involves directly managing the Energy Distance between the target conditional distribution and the one learned by the model. Furthermore, for the Reverse Markov framework, we introduce an Energy-Distance-based chain rule. This tool facilitates a rigorous examination of how errors propagate across the various reverse steps. Consequently, our analysis derives excess-risk bounds that achieve near-optimality, differing from the classical minimax rate only by logarithmic factors, across a broad H\"older class.

Source: arXiv Generated at: 2026-06-02 00:00:00 UTC