# d2: Improving Reasoning in Diffusion Language Models via Trajectory Likelihood Estimation

arXiv: https://arxiv.org/abs/2509.21474 · HTML: https://mlsanity.com/paper/2509.21474/

## Key facts

- Cited by 27 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion reasoning and test-time scaling, one of 17 directions in LLM reasoning.
- Builds on 38 of its 56 references that are in the map.
- First posted on arXiv 25 Sep 2025.

## Placed in

- LLM reasoning / [Diffusion reasoning and test-time scaling](https://mlsanity.com/lr/diffusion-reasoning-and-test-time-scaling/)
- Diffusion models / [Discrete and text diffusion](https://mlsanity.com/diffusion/discrete-and-text-diffusion/)

## Builds on (most cited first, 30 of 38)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Training language models to follow instructions with human feedback](https://mlsanity.com/paper/2203.02155/) (2022)
- [Proximal Policy Optimization Algorithms](https://mlsanity.com/paper/1707.06347/) (2017)
- [LoRA: Low-Rank Adaptation of Large Language Models](https://mlsanity.com/paper/2106.09685/) (2021)
- [Chain-of-Thought Prompting Elicits Reasoning in Large Language Models](https://mlsanity.com/paper/2201.11903/) (2022)
- [Training Verifiers to Solve Math Word Problems](https://mlsanity.com/paper/2110.14168/) (2021)
- [DeepSeekMath: Pushing the Limits of Mathematical Reasoning in Open Language Models](https://mlsanity.com/paper/2402.03300/) (2024)
- [Evaluating Large Language Models Trained on Code](https://mlsanity.com/paper/2107.03374/) (2021)
- [Direct Preference Optimization: Your Language Model is Secretly a Reward Model](https://mlsanity.com/paper/2305.18290/) (2023)
- [Let's Verify Step by Step](https://mlsanity.com/paper/2305.20050/) (2023)
- [Training a Helpful and Harmless Assistant with Reinforcement Learning from Human Feedback](https://mlsanity.com/paper/2204.05862/) (2022)
- [Program Synthesis with Large Language Models](https://mlsanity.com/paper/2108.07732/) (2021)
- [Trust Region Policy Optimization](https://mlsanity.com/paper/1502.05477/) (2015)
- [HuggingFace's Transformers: State-of-the-art Natural Language Processing](https://mlsanity.com/paper/1910.03771/) (2019)
- [Structured Denoising Diffusion Models in Discrete State-Spaces](https://mlsanity.com/paper/2107.03006/) (2021)
- [Training Diffusion Models with Reinforcement Learning](https://mlsanity.com/paper/2305.13301/) (2023)
- [Large Language Diffusion Models](https://mlsanity.com/paper/2502.09992/) (2025)
- [Simple and Effective Masked Diffusion Language Models](https://mlsanity.com/paper/2406.07524/) (2024)
- [Simplified and Generalized Masked Diffusion for Discrete Data](https://mlsanity.com/paper/2406.04329/) (2024)
- [Dream 7B: Diffusion Large Language Models](https://mlsanity.com/paper/2508.15487/) (2025)
- [Fast-dLLM: Training-free Acceleration of Diffusion LLM by Enabling KV Cache and Parallel Decoding](https://mlsanity.com/paper/2505.22618/) (2025)
- [Your Absorbing Discrete Diffusion Secretly Models the Conditional Distributions of Clean Data](https://mlsanity.com/paper/2406.03736/) (2024)
- [LLaDA 1.5: Variance-Reduced Preference Optimization for Large Language Diffusion Models](https://mlsanity.com/paper/2505.19223/) (2025)
- [Mercury: Ultra-Fast Language Models Based on Diffusion](https://mlsanity.com/paper/2506.17298/) (2025)
- [A General Framework for Inference-time Scaling and Steering of Diffusion Models](https://mlsanity.com/paper/2501.06848/) (2025)
- [Autoregressive Diffusion Models](https://mlsanity.com/paper/2110.02037/) (2021)
- [Remasking Discrete Diffusion Models with Inference-Time Scaling](https://mlsanity.com/paper/2503.00307/) (2025)
- [ReMax: A Simple, Effective, and Efficient Reinforcement Learning Method for Aligning Large Language Models](https://mlsanity.com/paper/2310.10505/) (2023)
- [DiffuCoder: Understanding and Improving Masked Diffusion Models for Code Generation](https://mlsanity.com/paper/2506.20639/) (2025)
- [d1: Scaling Reasoning in Diffusion Large Language Models via Reinforcement Learning](https://mlsanity.com/paper/2504.12216/) (2025)

## Built on by (most cited first, 27 of 27)

- [WeDLM: Reconciling Diffusion Language Models with Standard Causal Attention for Fast Inference](https://mlsanity.com/paper/2512.22737/) (2025)
- [From Next-Token to Next-Block: A Principled Adaptation Path for Diffusion LLMs](https://mlsanity.com/paper/2512.06776/) (2025)
- [d-TreeRPO: Towards More Reliable Policy Optimization for Diffusion Language Models](https://mlsanity.com/paper/2512.09675/) (2025)
- [DUEL: Exact Likelihood for Masked Diffusion via Deterministic Unmasking](https://mlsanity.com/paper/2603.01367/) (2026)
- [LaDi-RL: Latent Diffusion Reasoning Prevents Entropy Collapse in Reinforcement Learning](https://mlsanity.com/paper/2602.01705/) (2026)
- [LaViDa-R1: Advancing Reasoning for Unified Multimodal Diffusion Language Models](https://mlsanity.com/paper/2602.14147/) (2026)
- [GDSD: Reinforcement Learning as Guided Denoiser Self-Distillation for Diffusion Language Models](https://mlsanity.com/paper/2605.29398/) (2026)
- [A Tale of Two Temperatures: Simple, Efficient, and Diverse Sampling from Diffusion Language Models](https://mlsanity.com/paper/2604.09921/) (2026)
- [Diffusion-State Policy Optimization for Masked Diffusion Language Models](https://mlsanity.com/paper/2602.06462/) (2026)
- [Reinforcement Learning for Diffusion LLMs with Entropy-Guided Step Selection and Stepwise Advantages](https://mlsanity.com/paper/2603.12554/) (2026)
- [Reasoning in Diffusion Large Language Models is Concentrated in Dynamic Confusion Zones](https://mlsanity.com/paper/2511.15208/) (2025)
- [Dimension-free error estimate for diffusion model and optimal scheduling](https://mlsanity.com/paper/2512.01820/) (2025)
- [dTRPO: Trajectory Reduction in Policy Optimization of Diffusion Large Language Models](https://mlsanity.com/paper/2603.18806/) (2026)
- [ReflectDrive-2: Reinforcement-Learning-Aligned Self-Editing for Discrete Diffusion Driving](https://mlsanity.com/paper/2605.04647/) (2026)
- [Learnability-Informed Fine-Tuning of Diffusion Language Models](https://mlsanity.com/paper/2605.22939/) (2026)
- [Discrete Tilt Matching](https://mlsanity.com/paper/2604.18739/) (2026)
- [Unlocking Lossless Speedups in LLMs via Discrete Diffusion](https://mlsanity.com/paper/2609.04010/) (2026)
- [VDLM: Variable Diffusion LMs via Robust Latent-to-Text Rendering](https://mlsanity.com/paper/2602.15870/) (2026)
- [Break the Block: Dynamic-size Reasoning Blocks for Diffusion Large Language Models via Monotonic Entropy Descent with Reinforcement Learning](https://mlsanity.com/paper/2605.02263/) (2026)
- [The Confidence Shortcut: A Reasoning Failure Mode of Masked Diffusion Models](https://mlsanity.com/paper/2605.29123/) (2026)
- [Hesitation-Aware On-Policy Distillation for Diffusion Language Models](https://mlsanity.com/paper/2609.33301/) (2026)
- [AR-MAP: Are Autoregressive Large Language Models Implicit Teachers for Diffusion Large Language Models?](https://mlsanity.com/paper/2602.02178/) (2026)
- [Block-R1: Rethinking the Role of Block Size in Multi-domain Reinforcement Learning for Diffusion Large Language Models](https://mlsanity.com/paper/2605.11726/) (2026)
- [SLIM-RL: Risk-Budgeted Random-Masking RL for Diffusion LLMs Without Trajectory Slicing](https://mlsanity.com/paper/2607.00208/) (2026)
- [CanvasAnneal: Curriculum Reinforcement Learning for Diffusion Language Models](https://mlsanity.com/paper/2609.13060/) (2026)
- [Low-Confidence Remasking Traps Flexibility: Realizing Arbitrary-Order Potential for Diverse Rollouts in Diffusion LLMs](https://mlsanity.com/paper/2609.34509/) (2026)
- [Conditional Generation of Creative Chess Puzzles with Diffusion Models](https://mlsanity.com/paper/2609.38577/) (2026)

## Abstract (arXiv)

While diffusion language models (DLMs) have achieved competitive performance in text generation, improving their reasoning ability with reinforcement learning remains an active research area. Here, we introduce d2, a reasoning framework tailored for masked DLMs. Central to our framework is a new policy gradient algorithm that relies on accurate estimates of the sampling trajectory likelihoods. Because computing these likelihoods naively is computationally expensive for masked DLMs, we develop a family of estimators tailored to distinct model classes. For DLMs that support a sampling algorithm called any-order decoding, we propose d2-AnyOrder, which achieves exact trajectory likelihood with a single model pass. Through an empirical study of widely used DLMs, we show that any-order decoding is not universally supported in practice. For standard masked diffusion models, we propose d2-StepMerge, which approximates the trajectory likelihood, trading off compute for approximation accuracy in an analytically tractable manner. Empirically, d2 significantly outperforms widely-used RL baselines when applied to popular DLMs, and sets a new state-of-the-art performance for DLMs on logical reasoning tasks (Countdown and Sudoku) and math reasoning benchmarks (GSM8K and MATH500). We provide the code along with a blog post on the project page: https://guanghanwang.com/d2