Discrete Diffusion Models are Minimax Optimal Under Sparsity

Discrete diffusion models have demonstrated strong performance across a range of datasets, including natural language data and graph-structured data. Among many variants, score-entropy discrete diffusion (SEDD) has achieved particularly strong empirical results. In SEDD, new samples are generated by iteratively evaluating a sequence of concrete score functions, which are learned by minimizing a score-entropy loss. While much of the prior theoretical literature on discrete diffusion has focused on the sampling efficiency of SEDD under the assumption of small score estimation error, recent work has begun to investigate the finite-sample properties of score estimation itself. In this work, we take a different route by investigating the fundamental statistical limits of concrete score estimation. We focus on uniform and masking discrete diffusions, two of the most widely adopted discrete diffusion models. We establish a minimax lower bound under the score-entropy loss, and propose an MLE-based thresholding estimator that matches this lower bound up to constant and polylogarithmic factors that depend on neighboring density ratios. We further show that, for any target distribution, this density ratio is naturally controlled under both uniform and masking discrete diffusion models, yielding nearly matching minimax lower and upper bounds for the aggregated score estimation error. Our results imply that, with appropriate initialization and discretization, SEDD can achieve nearly optimal minimax sample complexity, as measured by the KL divergence between the target and generated distributions.

Discrete diffusion models have emerged as a practically successful framework for generative modeling on discrete product spaces, yet their statistical generalization properties remain poorly understood. Discrete real-world data such as text or biological sequences often concentrate on a small fraction of the astronomically large ambient space because of semantic or physical constraints, but existing bounds fail to capture this distributional structure and instead scale with the size of the ambient space, giving rise to almost vacuous error bounds. We address this gap for uniform discrete diffusion, one of the two dominant discrete diffusion paradigms alongside masking diffusion, by deriving statistical guarantees governed by the effective support size $s_n(P_0)$, a sample-size-dependent measure of distributional complexity. Given $n$ independent and identically distributed (i.i.d.) samples from an unknown data distribution $P_0$ on $[K]^d$, we show that, with appropriate choices of network size and hyperparameters, the expected total variation (TV) loss scales as $O(\sqrt{s_n(P_0)/n})$, while the expected Kullback--Leibler (KL) divergence is bounded by $O(\frac{1}{n}s_n(P_0)\log(eK^d/s_n(P_0))\log n)$. Furthermore, we show that the TV rate is minimax optimal and that the KL rate is minimax optimal up to a factor of $\log n$. Together, these upper and lower bounds show that uniform discrete diffusion successfully avoids the curse of dimensionality for distributions with small effective support size: the TV error rate depends on the ambient state-space size only through $s_n(P_0)$, while the corresponding KL rate incurs only an additional logarithmic dependence on the ambient state-space size.

Discrete diffusion has become a leading framework for generative modeling in various applications including language, vision, and biology. Existing convergence theory, however, exhibits fundamental limitations. KL-based analyses diverge under singular priors such as the masked distribution, while bounds in total variation (TV) depend on the vocabulary size $S$ and become vacuous for modern language tasks, where vocabularies contain hundreds of thousands of tokens. We develop a unified adjoint-equation-based framework that establishes vocabulary-size-independent convergence guarantees in any integral probability metric (IPM). To the best of our knowledge, our bounds are the first to be entirely free of $S$ and applicable to both masked and uniform priors. Importantly, our results can extend existing step complexity guarantees to any IPM. Also, our theory relies only on a single standard rate-matrix regularity assumption and applies to general priors. Five novel techniques drive our improvements: 1. working in the space of observables via adjoint equations rather than directly with probability measures; 2. a regularity analysis that yields bounds on any IPM; 3. a coupling argument that removes $S$-dependence under uniform transitions; and 4. score-marginal cancellation and 5. exit-routing techniques that remove $S$-dependence under masked transitions. Our framework thus sharply departs from prior analyses and avoids the shortcomings of pathspace-KL and existing TV-based approaches. Beyond convergence bounds, our framework provides a versatile toolkit for further theoretical study of discrete diffusion models, including principled choices of loss functions and vocabulary-size-independent step complexity.