Discrete flow models offer a powerful framework for learning distributions over discrete state spaces and have demonstrated superior performance compared to the discrete diffusion models. However, their convergence properties and error analysis remain largely unexplored. In this work, we develop a unified framework grounded in stochastic calculus theory to systematically investigate the theoretical properties of discrete flow models. Specifically, by leveraging a Girsanov-type theorem for the path measures of two continuous-time Markov chains (CTMCs), we present a comprehensive error analysis that accounts for both transition rate estimation error and early stopping error. In fact, the estimation error of transition rates has received little attention in existing works. Unlike discrete diffusion models, discrete flow incurs no initialization error caused by truncating the time horizon in the noising process. Building on generator matching and uniformization, we establish non-asymptotic error bounds for distribution estimation without the boundedness condition on oracle transition rates. Furthermore, we derive a faster rate of total variation convergence for the estimated distribution with the boundedness condition, yielding a nearly optimal rate in terms of sample size. Our results provide the first error analysis for discrete flow models. We also investigate model performance under different settings based on simulation results.

翻译：离散流模型为学习离散状态空间上的分布提供了一个强大的框架，并且相较于离散扩散模型已展现出更优越的性能。然而，其收敛性质与误差分析在很大程度上仍未得到探索。在本工作中，我们基于随机分析理论建立了一个统一框架，以系统性地研究离散流模型的理论性质。具体而言，通过利用关于两个连续时间马尔可夫链路径测度的Girsanov型定理，我们提出了一个综合考虑转移速率估计误差与提前停止误差的全面误差分析。事实上，转移速率的估计误差在现有工作中极少受到关注。与离散扩散模型不同，离散流不会因在加噪过程中截断时间范围而产生初始化误差。基于生成器匹配与均匀化方法，我们在无需神谕转移速率有界性条件的情况下，建立了分布估计的非渐近误差界。此外，在转移速率有界的条件下，我们推导出了估计分布在全变差收敛意义上的更快收敛速率，从而在样本量意义上获得了近乎最优的收敛率。我们的结果为离散流模型提供了首个误差分析。基于仿真结果，我们还探究了模型在不同设置下的性能表现。