Discrete State Diffusion Models: A Sample Complexity Perspective

2025-10-12

AI Summary

Discrete-state diffusion models lack rigorous theoretical foundations, particularly regarding sample complexity analysis. Method: We establish the first formal theoretical framework for text and combinatorial structure generation, decomposing the score estimation error into four components—statistical, approximation, optimization, and clipping—and integrating discrete diffusion process modeling, score matching, error decomposition, and sample complexity analysis grounded in statistical learning and optimization theory. Contribution/Results: We derive the first sample complexity upper bound $widetilde{mathcal{O}}(varepsilon^{-2})$ for discrete diffusion models. Our analysis identifies key factors governing training efficiency and rigorously establishes their learnability and computational efficiency under finite samples. This work bridges a critical gap between the empirical success and theoretical understanding of discrete diffusion models, providing foundational guarantees for their practical deployment in structured generation tasks.

Technology Category

• Computer Vision: Diffusion Models for Vision
• Machine Learning: Deep Generative Models & Autoencoders
• Search and Optimization: Learning to 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
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Diffusion models have demonstrated remarkable performance in generating high-dimensional samples across domains such as vision, language, and the sciences. Although continuous-state diffusion models have been extensively studied both empirically and theoretically, discrete-state diffusion models, essential for applications involving text, sequences, and combinatorial structures, remain significantly less understood from a theoretical standpoint. In particular, all existing analyses of discrete-state models assume score estimation error bounds without studying sample complexity results. In this work, we present a principled theoretical framework for discrete-state diffusion, providing the first sample complexity bound of $widetilde{mathcal{O}}(ε^{-2})$. Our structured decomposition of the score estimation error into statistical, approximation, optimization, and clipping components offers critical insights into how discrete-state models can be trained efficiently. This analysis addresses a fundamental gap in the literature and establishes the theoretical tractability and practical relevance of discrete-state diffusion models.

Problem

• Analyzing sample complexity of discrete-state diffusion models theoretically
• Establishing first sample complexity bound for discrete-state diffusion models
• Providing theoretical framework for discrete-state diffusion model training

Innovation

• First sample complexity bound for discrete diffusion
• Structured decomposition of score estimation error
• Theoretical framework for discrete-state diffusion models