Discrete State Diffusion Models: A Sample Complexity Perspective

TL;DR

This work addresses the theoretical understanding of discrete-state diffusion models by deriving the first sample complexity bound of $\widetilde{\mathcal{O}}(\epsilon^{-2})$ under practical training assumptions. It builds a principled framework using discrete CTMC forward/reverse dynamics on $[S]^d$, a score-entropy loss with clipping, and a structured error decomposition into approximation, statistical, optimization, and clipping components. A key technical novelty is leveraging the Polyak–Łojasiewicz condition to bound optimization dynamics and control the score-estimation error within a finite-sample regime, yielding a nearly dimension-free, order-optimal rate. The results establish theoretical tractability and practical relevance for discrete-state diffusion, with explicit sample-size requirements and architectural conditions (e.g., width $W \ge (S-1)d$) to achieve target KL accuracy, thereby enabling efficient learning for text, sequences, and combinatorial data generation.

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.

Theorems & Definitions (25)

• Theorem 1: Sample complexity and KL guarantee
• Corollary 1
• Lemma 1: Approximation Error
• Lemma 2: Statistical Error
• Lemma 3: Optimization Error
• Lemma 4
• proof
• Lemma 5
• Lemma 6: Score bound - follows from Lemma 4 of zhang2025convergencescorebaseddiscretediffusion
• proof