VGB for Masked Diffusion Model: Efficient Test-time Scaling for Reward Satisfaction and Sample Editing

Masked diffusion models (MDMs) have recently emerged as a promising paradigm for sequence generation. Scaling MDMs is conventionally achieved by increasing the parameter count or the number of denoising steps. We introduce Recursive Masked Diffusion Models (R-MDMs), which add recursive depth as a third scaling axis by repeatedly applying the same denoising transformer within each diffusion step. Recursion enables iterative refinement of the output through parameter reuse, increasing effective model depth without increasing parameter count. Across structured generation tasks, including Sudoku and Countdown, we show that R-MDMs achieve substantially improved parameter efficiency: a model with $L$ recursive iterations often matches the performance of non-recursive baselines with roughly $L\times$ more parameters. Moreover, recursive refinement can partially substitute for additional denoising steps, allowing recursive models to reach the same generation quality with fewer forward passes at inference time. These results suggest that recursive depth is a practically useful scaling mechanism for MDMs, improving both parameter efficiency and the allocation of test-time compute.

In this paper, we propose using random walks on graphs as a verifiable sandbox to study different parallel sampling strategies in masked diffusion models (MDMs). We train an MDM on random walk samples from a fixed graph. The graph or the transition kernel is never shown to the model explicitly and plays the role of latent structure in the sequences, albeit one that is controllable and can be used for quantitative evaluation. Thus, this framework enjoys a Sudoku-like validity check: verifying that an output is a valid walk and estimating the Markov kernel from the walks to measure distribution fidelity. Using simple graphs, we theoretically prove that parallel unmasking via widely used scores like lowest entropy is not uniformly better than a random parallel sampler; the performance critically depends on the structure of the underlying graph. We develop a new bisection sampler for random walks, which takes logarithmic steps in the sequence length and is provably exact under perfect training. Experiments on various graph walk tasks show that different parallel samplers are better for different graphs even in practice. Our initial experiments on a pretrained OpenWebText MDM show that the bisection-style samplers improve speed-quality tradeoffs even for language generation. Together, these results position graph random walks as a mechanistic benchmark for diagnosing and designing parallel samplers for masked diffusion models.

Reinforcement learning has proven effective for improving reasoning in large language models, but extending it to Masked Diffusion Language Models (MDLMs) remains challenging due to the intractability of the log-likelihood estimation. Existing approaches approximate this log-likelihood by modeling only the token predictions, ignoring the order in which positions are unmasked during generation. We observe that MDLM generation involves two decisions at each step: what tokens to place at each masked position and which positions to remask. We formalize this as a two-stage action MDP, showing that the policy gradient naturally decomposes into a token term and a masking term. Combining optimization of both terms leads to state-of-the-art outcomes on mathematical reasoning and coding benchmarks, with scores of 87.1% on GSM8K and 53.4% on MBPP.