MetaState: Persistent Working Memory Enhances Reasoning in Discrete Diffusion Language Models

Masked diffusion language models (dLLMs) generate text by iteratively denoising masked tokens with bidirectional attention. Extending reasoning across generation chunks normally requires keeping earlier generated text in context. We ask whether a dLLM can instead continue reasoning after that text is cleared, using only a fixed-size carried state. We implement this state as a small number of register tokens: dedicated fixed-position tokens whose continuous hidden states are trained to carry reasoning progress across generation chunks. We post-train dLLMs to decode a chunk of text, clear it while preserving the register values, and continue decoding from the prompt and carried state. In our main comparisons on LLaDA and Dream, registers outperform discrete-text carry on every benchmark, with gains of up to 8.5 points on math and 19.5 points on code. Registers are especially effective for bounded code generation, where correct programs usually span several chunks. Finally, registers can be further refined with reinforcement learning on long-horizon reasoning tasks.

Diffusion large language models (dLLMs) are a promising alternative to autoregressive generation. However, reasoning-oriented post-training for dLLMs remains challenging. Supervised fine-tuning (SFT) for dLLMs requires dense but often off-policy masked states, while reinforcement learning (RL) relies on sparse rewards or value modeling. This paper proposes trace-based on-policy distillation (TOPD), a teacher-supervised framework that transfers reasoning ability to a target dLLM without reward estimation. The key idea is to supervise a dLLM on its own denoising trajectory, focusing on the trace-aligned token decisions that form the final response. Specifically, TOPD samples on-policy diffusion trajectories from the target dLLM, obtains teacher token distributions from a teacher model on the corresponding partially denoised states, and updates the target dLLM with a token-level Reverse Kullback-Leibler (Reverse-KL) objective. This design preserves dense teacher supervision while aligning training with the model's own denoising states. On mathematical reasoning benchmarks, TOPD enables SDAR-4B-Chat to match the MATH500 accuracy of its RL-trained counterpart TraDo-4B-Instruct, with gains of +5.7 under static evaluation and +4.5 under dynamic evaluation. Compared with the RL-trained counterpart, TOPD achieves this with 4$\times$ fewer rollout rounds, corresponding to an estimated 96.0$\times$ to-accuracy model-compute speedup.

Despite the success of discrete diffusion language models (dLMs) for fast parallel decoding, their non-smooth, high-dimensional space hinders trajectory steering for reasoning and inference acceleration. To overcome this, we present Sigma, the first large-scale (3B/8B) continuous dLM built on steerable, low-dimensional ODE/SDE latent trajectories. Trained blockwise via likelihood optimization, Sigma jointly denoises Gaussian-corrupted token embeddings while learning an optimal embedding geometry. To accelerate training, Sigma leverages pre-trained weights from autoregressive (AR) models for warm-starting. During inference, we identify classifier-free guidance and score temperature as essential for high-fidelity reasoning and coding. Across comprehensive math reasoning and coding evaluations against state-of-the-art discrete counterparts (masked dLMs and AR baselines), Sigma achieves competitive performance with discrete models on standard benchmarks (e.g., GSM8K, Minerva, HumanEval, MBPP) after pre-training and on challenging reasoning tasks (e.g., MATH-500, AIME) after supervised fine-tuning. Beyond performance parity, we uncover key structural properties unique to continuous dLMs: (i) embedding-space steering effectively governs the quality-diversity trade-off, yielding strong pass@k performance and (ii) continuous trajectories enable graceful degradation for low NFEs and efficient distillation. These establish continuous dLMs as a promising paradigm for efficient language generation.