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.

翻译：Engression是一种近期提出且有效的条件分布学习框架。其多步反向马尔可夫扩展通过将复杂条件采样分解为序列反向转移，进一步提升了生成灵活性。尽管这些方法在实证上表现优异，但尚缺乏严格的有限样本统计保证。本文中，在深度神经网络参数化下，我们通过直接控制学习到的条件分布与目标条件分布之间的能量距离，建立了Engression的非渐近收敛界。针对反向马尔可夫框架，我们进一步提出基于能量距离的链式法则，从而能够严格分析反向步骤间的误差传播。我们的分析得出了相应的超额风险界，该界相对于一般Hölder类上的经典极小化最优速率而言，在对数因子范围内达到近乎最优。