When Can Conditional Flow Matching Replace Pointwise Negative Log-Likelihood?

arXiv:2608.28010v1 Announce Type: new Abstract: 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_{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.

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Source: arXiv cs.LG | 2026-08-31