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MDNS: Masked Diffusion Neural Sampler via Stochastic Optimal Control

NeurIPS 2025 · Image generation › Fast diffusion sampling · approach: Diffusion theory and learned samplers · 0 citations (OpenAlex)

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Proposes MDNS, a discrete neural sampler trained via stochastic optimal control of CTMCs with masked diffusion as reference, using gradient-free objectives including a weighted denoising cross-entropy (WDCE) loss, evaluated on Ising and Potts models.

Problem

Learning neural samplers that draw samples from unnormalized discrete distributions (e.g., Ising/Potts models) with large state spaces and multimodality, where CTMC trajectories make direct gradient-based training non-differentiable.

Approach

Cast discrete sampling as SOC over CTMCs with a memoryless masked-diffusion reference path measure, then train the controlled generator with gradient-free path-measure objectives (RERF, LV, CE) and a scalable weighted denoising cross-entropy loss that importance-weights final samples and applies masked denoising supervision.

Key results

• On 16x16 Ising at beta_critical, MDNS achieves magnetization error 3.7e-3 and ESS 0.933 vs LEAPS 7.4e-3 and 0.384.
• On 16x16 Potts at beta_low=1.2, MDNS reaches correlation error 8.8e-5 and ESS 0.933 while LEAPS gets 2.5e-1 correlation error and ESS 0.012.
• On 4x4 Ising, the LV objective ranks best (ESS 0.9713, TV 0.0748), with the ranking LV > WDCE > RERF > CE on most metrics.

Measured effects reported in the paper

Surprising findings

• Metropolis-Hastings cannot mix on 16x16 Potts even after more than 20 hours of simulation, while it works on all Ising experiments.
• The Gumbel-softmax relaxation approach fails to converge to the correct target even in low dimensions.

Limitations

• Evaluated only on Ising and Potts lattice models; performance on graph or combinatorial optimization distributions is unknown.
• Warm-up annealing schedule is heuristic and not systematically studied.
• LV/RERF are not valid loss functions; reduction in objective does not guarantee improved performance.

Experimental setup

Datasets: Ising model samples (self-generated via target energy) Potts model samples (self-generated via target energy)

Benchmarks: 2D Ising model 4x4 and 16x16 2D Potts model 16x16 (q=3)

Metrics: ESS total variation KL divergence chi-squared divergence magnetization error 2-point correlation error absolute error of log Z

Baselines: LEAPS Metropolis-Hastings Gumbel-softmax relaxation objective

Builds on: LEAPS Path Integral Sampler denoising cross-entropy masked diffusion random order autoregressive sampler

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• Scalable Discrete Diffusion Samplers: Combinatorial Optimization and Statistical Physics (ICLR 2025 · similarity 0.90)
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