# Constrained Discrete Diffusion

arXiv: https://arxiv.org/abs/2503.09790 · HTML: https://mlsanity.com/paper/2503.09790/

## Key facts

- Cited by 17 papers in the map (data through 14 Oct 2026).
- Placed in Discrete and text diffusion, one of 14 directions in Diffusion models.
- Builds on 50 of its 66 references that are in the map.
- First posted on arXiv 12 Mar 2025.

## Placed in

- Diffusion models / [Discrete and text diffusion](https://mlsanity.com/diffusion/discrete-and-text-diffusion/)

## Builds on (most cited first, 30 of 50)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [Training language models to follow instructions with human feedback](https://mlsanity.com/paper/2203.02155/) (2022)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Judging LLM-as-a-Judge with MT-Bench and Chatbot Arena](https://mlsanity.com/paper/2306.05685/) (2023)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Classifier-Free Diffusion Guidance](https://mlsanity.com/paper/2207.12598/) (2022)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Categorical Reparameterization with Gumbel-Softmax](https://mlsanity.com/paper/1611.01144/) (2016)
- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Structured Denoising Diffusion Models in Discrete State-Spaces](https://mlsanity.com/paper/2107.03006/) (2021)
- [RealToxicityPrompts: Evaluating Neural Toxic Degeneration in Language Models](https://mlsanity.com/paper/2009.11462/) (2020)
- [Simple and Effective Masked Diffusion Language Models](https://mlsanity.com/paper/2406.07524/) (2024)
- [Plug and Play Language Models: A Simple Approach to Controlled Text Generation](https://mlsanity.com/paper/1912.02164/) (2019)
- [Discrete Diffusion Modeling by Estimating the Ratios of the Data Distribution](https://mlsanity.com/paper/2310.16834/) (2023)
- [Simplified and Generalized Masked Diffusion for Discrete Data](https://mlsanity.com/paper/2406.04329/) (2024)
- [Sora: A Review on Background, Technology, Limitations, and Opportunities of Large Vision Models](https://mlsanity.com/paper/2402.17177/) (2024)
- [ToxiGen: A Large-Scale Machine-Generated Dataset for Adversarial and Implicit Hate Speech Detection](https://mlsanity.com/paper/2203.09509/) (2022)
- [PICARD: Parsing Incrementally for Constrained Auto-Regressive Decoding from Language Models](https://mlsanity.com/paper/2109.05093/) (2021)
- [FUDGE: Controlled Text Generation With Future Discriminators](https://mlsanity.com/paper/2104.05218/) (2021)
- [Text2Motion: From Natural Language Instructions to Feasible Plans](https://mlsanity.com/paper/2303.12153/) (2023)
- [Masked Diffusion Models are Secretly Time-Agnostic Masked Models and Exploit Inaccurate Categorical Sampling](https://mlsanity.com/paper/2409.02908/) (2024)
- [Grammar-Constrained Decoding for Structured NLP Tasks without Finetuning](https://mlsanity.com/paper/2305.13971/) (2023)
- [LINC: A Neurosymbolic Approach for Logical Reasoning by Combining Language Models with First-Order Logic Provers](https://mlsanity.com/paper/2310.15164/) (2023)
- [Constrained Language Models Yield Few-Shot Semantic Parsers](https://mlsanity.com/paper/2104.08768/) (2021)
- [Fine-Tuning Large Vision-Language Models as Decision-Making Agents via Reinforcement Learning](https://mlsanity.com/paper/2405.10292/) (2024)
- [Synchromesh: Reliable code generation from pre-trained language models](https://mlsanity.com/paper/2201.11227/) (2022)
- [Simple Guidance Mechanisms for Discrete Diffusion Models](https://mlsanity.com/paper/2412.10193/) (2024)
- [NeuroLogic A*esque Decoding: Constrained Text Generation with Lookahead Heuristics](https://mlsanity.com/paper/2112.08726/) (2021)
- [NeuroLogic Decoding: (Un)supervised Neural Text Generation with Predicate Logic Constraints](https://mlsanity.com/paper/2010.12884/) (2020)

## Built on by (most cited first, 17 of 17)

- [Calibrating Generative Models to Distributional Constraints](https://mlsanity.com/paper/2510.10020/) (2025)
- [Simple Self-Conditioning Adaptation for Masked Diffusion Models](https://mlsanity.com/paper/2604.26985/) (2026)
- [Primal-Dual Guided Decoding for Constrained Discrete Diffusion](https://mlsanity.com/paper/2605.09749/) (2026)
- [Plug-and-Play Guidance for Discrete Diffusion Models via Gradient-Informed Logit Correction](https://mlsanity.com/paper/2606.06303/) (2026)
- [Search-Augmented Masked Diffusion Models for Constrained Generation](https://mlsanity.com/paper/2602.02727/) (2026)
- [Large Neighborhood Search meets Iterative Neural Constraint Heuristics](https://mlsanity.com/paper/2603.20801/) (2026)
- [Constrained Code Generation with Discrete Diffusion](https://mlsanity.com/paper/2605.16829/) (2026)
- [Accelerating Masked Diffusion Large Language Models: A Survey of Efficient Inference Techniques](https://mlsanity.com/paper/2607.12829/) (2026)
- [Focus on the Core: Empowering Diffusion Large Language Models by Self-Contrast](https://mlsanity.com/paper/2605.01373/) (2026)
- [Masked Diffusion Modeling for Anomaly Detection](https://mlsanity.com/paper/2605.30046/) (2026)
- [Beyond the Autoregressive Horizon: A Comprehensive Survey of Diffusion Models, World Modelling, and State Space Models for Code](https://mlsanity.com/paper/2606.23690/) (2026)
- [Ready to Speak: Aligning LLMs for TTS-Friendly Text Generation](https://mlsanity.com/paper/2609.01246/) (2026)
- [Let It Go or Learn to Self-Correct: Continuous Diffusion for Constrained Discrete Tasks](https://mlsanity.com/paper/2609.09009/) (2026)
- [Extracting Arguments, Not Just Classifying Them: Instruction-Tuned LLMs for Generative Component Detection](https://mlsanity.com/paper/2609.24855/) (2026)
- [Grab a Coffee: Future-Aware Guidance for Discrete Diffusion with Compiled Objectives](https://mlsanity.com/paper/2609.35924/) (2026)
- [pCoMole: Pareto-Constrained Molecule Editing with Discrete Flows](https://mlsanity.com/paper/2610.01663/) (2026)
- [Only What You've Signed up For: Secure Workflows with Constrained Diffusion Agents](https://mlsanity.com/paper/2610.15177/) (2026)

## Abstract (arXiv)

Discrete diffusion models are a class of generative models that construct sequences by progressively denoising samples from a categorical noise distribution. Beyond their rapidly growing ability to generate coherent natural language, these models present a new and important opportunity to enforce sequence-level constraints, a capability that current autoregressive models cannot natively provide. This paper capitalizes on this opportunity by introducing Constrained Discrete Diffusion (CDD), a novel integration of differentiable constraint optimization within the diffusion process to ensure adherence to constraints, logic rules, or safety requirements for generated sequences. Unlike conventional text generators that often rely on post-hoc filtering or model retraining for controllable generation, CDD directly imposes constraints into the discrete diffusion sampling process, resulting in a training-free and effective approach. Experiments in toxicity-controlled text generation, property-constrained molecule design, and instruction-constrained text completion demonstrate that CDD achieves zero constraint violations in a diverse array of tasks while preserving fluency, novelty, and coherence while outperforming autoregressive and existing discrete diffusion approaches.