Absorb and Converge: Provable Convergence Guarantee for Absorbing Discrete Diffusion Models

2025-06-02

AI Summary

The generative performance of discrete-state-space diffusion models critically depends on the choice of the transition rate matrix; however, existing theoretical analyses are limited to uniform rate matrices, leaving convergence properties and error bounds under absorbing rate matrices long unaddressed. Method: We establish the first finite-time convergence theory for absorbing discrete diffusion models: (i) we propose a proxy initial distribution to circumvent KL divergence blowup caused by absorption into stationary states; (ii) we develop a Jensen-type forward-process analysis framework, introducing novel tools including absorption fraction function bounds and non-explosive initial-value upper bounds; (iii) we design τ-jump and uniformization samplers to accelerate convergence. Results: We derive the first explicit finite-time KL divergence upper bound under general absorbing rate matrices and prove global convergence—without early stopping—under mild assumptions.

Technology Category

• Computer Vision: Diffusion Models for Vision
• Machine Learning: Deep Generative Models & Autoencoders
• Reasoning under Uncertainty: Relational Probabilistic Models

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems

Abstract

Discrete state space diffusion models have shown significant advantages in applications involving discrete data, such as text and image generation. It has also been observed that their performance is highly sensitive to the choice of rate matrices, particularly between uniform and absorbing rate matrices. While empirical results suggest that absorbing rate matrices often yield better generation quality compared to uniform rate matrices, existing theoretical works have largely focused on the uniform rate matrices case. Notably, convergence guarantees and error analyses for absorbing diffusion models are still missing. In this work, we provide the first finite-time error bounds and convergence rate analysis for discrete diffusion models using absorbing rate matrices. We begin by deriving an upper bound on the KL divergence of the forward process, introducing a surrogate initialization distribution to address the challenge posed by the absorbing stationary distribution, which is a singleton and causes the KL divergence to be ill-defined. We then establish the first convergence guarantees for both the $ au$-leaping and uniformization samplers under absorbing rate matrices, demonstrating improved rates over their counterparts using uniform rate matrices. Furthermore, under suitable assumptions, we provide convergence guarantees without early stopping. Our analysis introduces several new technical tools to address challenges unique to absorbing rate matrices. These include a Jensen-type argument for bounding forward process convergence, novel techniques for bounding absorbing score functions, and a non-divergent upper bound on the score near initialization that removes the need of early-stopping.

Problem

• Analyzing convergence of absorbing discrete diffusion models
• Providing error bounds for absorbing rate matrices
• Comparing performance between uniform and absorbing matrices

Innovation

• Upper bound on KL divergence for forward process
• Convergence guarantees for τ-leaping and uniformization samplers
• Non-divergent upper bound on score near initialization