# Vocabulary-size-independent Convergence of Discrete Diffusion Models: adjoint equations induce the right space

arXiv: https://arxiv.org/abs/2605.17232 · HTML: https://mlsanity.com/paper/2605.17232/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 6 of its 61 references that are in the map.
- First posted on arXiv 17 May 2026.

## Placed in

- Diffusion models / [Diffusion foundations and sampling](https://mlsanity.com/diffusion/diffusion-foundations-and-sampling/)

## Builds on (most cited first, 6 of 6)

- [Muse: Text-To-Image Generation via Masked Generative Transformers](https://mlsanity.com/paper/2301.00704/) (2023)
- [Optimal Inference Schedules for Masked Diffusion Models](https://mlsanity.com/paper/2511.04647/) (2025)
- [Efficient Sampling with Discrete Diffusion Models: Sharp and Adaptive Guarantees](https://mlsanity.com/paper/2602.15008/) (2026)
- [Non-Asymptotic Convergence of Discrete Diffusion Models: Masked and Random Walk dynamics](https://mlsanity.com/paper/2512.00580/) (2025)
- [Sharp Convergence Rates for Masked Diffusion Models](https://mlsanity.com/paper/2602.22505/) (2026)
- [Neural Continuous-Time Markov Chain: Discrete Diffusion via Decoupled Jump Timing and Direction](https://mlsanity.com/paper/2604.15694/) (2026)

## Built on by (most cited first, 3 of 3)

- [From Scores to Gibbs Correctors: Accelerating Uniform-Rate Discrete Diffusion Models](https://mlsanity.com/paper/2605.27352/) (2026)
- [Parallelism, critical windows, and separations among diffusion language models](https://mlsanity.com/paper/2609.20539/) (2026)
- [Sample complexity bounds for categorical Markov random fields via Discrete Diffusions](https://mlsanity.com/paper/2610.02128/) (2026)

## Abstract (arXiv)

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.