Error Analysis of Discrete Flow with Generator Matching

2025-09-26

AI Summary

This work addresses the weak theoretical foundations of discrete flow models by systematically analyzing their convergence properties and error sources in discrete-state distribution learning. We propose a unified analytical framework based on the KL divergence between path measures of continuous-time Markov chains (CTMCs), establishing the first non-asymptotic upper bound on estimation error—precisely characterizing both transition-rate estimation bias and early-stopping-induced error. Innovatively, we construct a Girsanov-type theorem that circumvents the reliance on time discretization inherent in conventional discrete diffusion models, thereby enabling rigorous error control of the original flow model. Our analysis further integrates generator matching and uniformization techniques to complete the theoretical derivation. This work delivers the first systematic, non-asymptotic error analysis for discrete flow models, filling a critical theoretical gap in this emerging area.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Relational Probabilistic Models
• Computer Vision: Diffusion Models for Vision

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation
• Web Mining and Content Analysis: Models for Web evolution

Abstract

Discrete flow models offer a powerful framework for learning distributions over discrete state spaces and have demonstrated superior performance compared to the discrete diffusion model. 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. Specifically, we derive the KL divergence of two path measures regarding two continuous-time Markov chains (CTMCs) with different transition rates by developing a novel Girsanov-type theorem, and provide a comprehensive analysis that encompasses the error arising from transition rate estimation and early stopping, where the first type of error has rarely been analyzed by existing works. Unlike discrete diffusion models, discrete flow incurs no truncation 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. Our results provide the first error analysis for discrete flow models.

Problem

• Analyzing convergence properties of discrete flow models
• Investigating error from transition rate estimation and early stopping
• Establishing non-asymptotic error bounds for distribution estimation

Innovation

• Developed Girsanov theorem for CTMC divergence analysis
• Analyzed transition rate estimation and early stopping errors
• Established non-asymptotic bounds via generator matching