为条件分布学习的Engression方法提供理论保证，解析其误差传播机制。
Theoretical Analysis of Engression and Reverse Markov Engression
- 基于能量距离直接控制条件分布逼近误差
- 推导出接近最优的过拟合风险上界（近似最小最大率）
- 适用于关注生成模型理论分析的研究者
Engression 是一种新兴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一系列逆向转移，进一步提升了生成灵活性。尽管其在实验中表现优异，但目前缺乏严格的有限样本统计保证。本文在深度神经网络参数化假设下，通过直接控制学习分布与目标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，我们进一步提出了基于能量距离的链式法则，实现了对逆向步骤间误差传播的严格分析。所得结果给出了接近最优的过拟合风险上界，相对于一般 Hölder 类上的经典最小最大率，仅差对数因子。
原文摘要 · Abstract (English)
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.
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