Convergence of Score-Based Discrete Diffusion Models: A Discrete-Time Analysis

2024-10-03

AI Summary

This work establishes, for the first time, a rigorous convergence theory for score-based discrete diffusion models on the discrete state space $[S]^d$, formulated within the continuous-time Markov chain (CTMC) framework. To address the lack of theoretical guarantees for discrete diffusion, the paper introduces a novel definition of discrete fractional scores that captures essential smoothness and estimability properties. It further develops a Girsanov-type analysis tailored to CTMCs, yielding explicit bounds on the KL divergence and total variation distance between the generated and true data distributions. The derived KL bound scales nearly linearly in dimension $d$, matching the current best rate in discrete diffusion theory; moreover, it uniformly handles both early-stopping and non-early-stopping regimes. These results confirm the theoretical feasibility and sampling fidelity of high-dimensional discrete diffusion models, aligning with recent theoretical conclusions for continuous diffusion models.

Technology Category

• Computer Vision: Diffusion Models for Vision
• Machine Learning: Information Theory
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• Security and Privacy: Large-scale security measurements

Abstract

Diffusion models have achieved great success in generating high-dimensional samples across various applications. While the theoretical guarantees for continuous-state diffusion models have been extensively studied, the convergence analysis of the discrete-state counterparts remains under-explored. In this paper, we study the theoretical aspects of score-based discrete diffusion models under the Continuous Time Markov Chain (CTMC) framework. We introduce a discrete-time sampling algorithm in the general state space $[S]^d$ that utilizes score estimators at predefined time points. We derive convergence bounds for the Kullback-Leibler (KL) divergence and total variation (TV) distance between the generated sample distribution and the data distribution, considering both scenarios with and without early stopping under specific assumptions. Notably, our KL divergence bounds are nearly linear in dimension $d$, aligning with state-of-the-art results for diffusion models. Our convergence analysis employs a Girsanov-based method and establishes key properties of the discrete score function, which are essential for characterizing the discrete-time sampling process.

Problem

• Analyzing convergence of discrete diffusion models
• Deriving KL and TV bounds for sample distributions
• Establishing discrete-time sampling algorithm properties

Innovation

• Discrete-time sampling algorithm in general state space
• Convergence bounds for KL divergence and TV distance
• Girsanov-based method for discrete-time analysis