Discrete flow models (DFMs) have been proposed to learn the data distribution on a finite state space, offering a flexible framework as an alternative to discrete diffusion models. A line of recent work has studied samplers for discrete diffusion models, such as tau-leaping and Euler solver. However, these samplers require a large number of iterations to control discretization error, since the transition rates are frozen in time and evaluated at the initial state within each time interval. Moreover, theoretical results for these samplers often require boundedness conditions of the transition rate or they focus on a specific type of source distributions. To address those limitations, we establish non-asymptotic discretization error bounds for those samplers without any restriction on transition rates and source distributions, under the framework of discrete flow models. Furthermore, by analyzing a one-step lower bound of the Euler sampler, we propose two corrected samplers: \textit{time-corrected sampler} and \textit{location-corrected sampler}, which can reduce the discretization error of tau-leaping and Euler solver with almost no additional computational cost. We rigorously show that the location-corrected sampler has a lower iteration complexity than existing parallel samplers. We validate the effectiveness of the proposed method by demonstrating improved generation quality and reduced inference time on both simulation and text-to-image generation tasks. Code can be found in https://github.com/WanZhengyan/Corrected-Samplers-for-Discrete-Flow-Models.

翻译：离散流模型（DFMs）被提出用于学习有限状态空间上的数据分布，为离散扩散模型提供了一种灵活的替代框架。近期一系列研究工作探讨了离散扩散模型的采样器，例如tau-leaping和Euler求解器。然而，由于这些采样器在每个时间区间内将转移速率冻结在初始状态进行评估，因此需要大量迭代次数来控制离散化误差。此外，这些采样器的理论结果通常要求转移速率满足有界性条件，或仅关注特定类型的源分布。为克服这些局限性，我们在离散流模型框架下，建立了无需任何转移速率限制和源分布约束的采样器非渐近离散化误差界。进一步地，通过分析Euler采样器的一步下界，我们提出了两种修正采样器：\textit{时间修正采样器}和\textit{位置修正采样器}，它们能以几乎零额外计算成本降低tau-leaping和Euler求解器的离散化误差。我们严格证明了位置修正采样器相比现有并行采样器具有更低的迭代复杂度。通过在仿真实验和文本到图像生成任务中展示提升的生成质量与减少的推理时间，我们验证了所提方法的有效性。代码可在 https://github.com/WanZhengyan/Corrected-Samplers-for-Discrete-Flow-Models 获取。