发表机构
Zhejiang University; Westlake University; Fudan University（浙江大学; 西湖大学; 复旦大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过理论分解刻画了条件流匹配（CFM）可替代逐点负对数似然（NLL）的条件，指出其在离策略总体最优处的局限性，相关结论及分解为适配LLM方法至流匹配提供了理论基础。
AI 中文摘要
流匹配支持无需似然的训练，而对齐方法日益将条件流匹配（CFM）损失复用为端点负对数似然（NLL），并将其新旧差异用作对数似然比。本文刻画这些替换有效的情形：针对线性高斯路径，将端点NLL精确分解为熵、加权CFM目标、内部速度-得分残差及边界残差，仅当对应残差抵消时，仅CFM的估计值及差异才是精确的。在离策略总体最优处，普通CFM通常并非逐点NLL估计器，而权重w_sc(t)=(1-t)/t可消除内部残差；但该正向结果一般无法扩展至训练或在线策略对齐。即使端点分布相同或经代理优化，在线策略对数似然比仍可能存在偏差。跨维度、分布及几何结构的实验支持上述结论及使非精确比有用的机制。更广泛而言，该分解为将基于似然的大语言模型（LLM）方法适配至流匹配提供了理论基础，同时区分了精确替换与受控代理。
英文摘要
Flow matching enables likelihood-free training, yet alignment methods increasingly reuse conditional flow matching (CFM) losses as endpoint negative log-likelihoods (NLLs) and their old/new differences as log-likelihood ratios. We characterize when these substitutions are valid. For linear Gaussian paths, we exactly decompose endpoint NLL into entropy, a weighted CFM objective, an interior velocity--score residual, and a boundary residual. Thus CFM-only estimates and differences are exact only when the corresponding residuals cancel. At the off-policy population optimum, ordinary CFM is not generally a pointwise NLL estimator, whereas \(w_{\mathrm{sc}}(t)=(1-t)/t\) removes the interior residual; this positive result does not extend generally to training or on-policy alignment. On-policy log-ratios can remain biased even for identical endpoint laws or after surrogate optimization. Experiments across dimensions, distributions, and geometries support these conclusions and the mechanisms that make inexact ratios useful. **More broadly, the decomposition provides a theoretical basis for adapting likelihood-based LLM methods to flow matching, while distinguishing exact substitutions from controlled surrogates.**
