This characteristic enables simulation-free training, bypassing the necessity to simulate the underlying diffusion processes by directly sampling the diffused data.

We present a systematic theoretical framework that interprets masked diffusion models (MDMs) as solutions to energy minimization problems in discrete optimal transport. Specifically, we prove that three distinct energy formulations--kinetic, conditional kinetic, and geodesic energy--are mathematically equivalent under the structure of MDMs, and that MDMs minimize all three when the mask schedule satisfies a closed-form optimality condition. This unification not only clarifies the theoretical foundations of MDMs, but also motivates practical improvements in sampling. By parameterizing interpolation schedules via Beta distributions, we reduce the schedule design space to a tractable 2D search, enabling efficient post-training tuning without model modification. Experiments on synthetic and real-world benchmarks demonstrate that our energy-inspired schedules outperform hand-crafted baselines, particularly in low-step sampling settings.

Masked Diffusion Models (MDMs) have emerged as one of the most promising paradigms for generative modeling over discrete domains. It is known that MDMs effectively train to decode tokens in a random order, and that this ordering has significant performance implications in practice. This observation raises a fundamental question: can we design a training framework that optimizes for a favorable decoding order? We answer this in the affirmative, showing that the continuous-time variational objective of MDMs, when equipped with multivariate noise schedules, can identify and optimize for a decoding order during training. We establish a direct correspondence between decoding order and the multivariate noise schedule and show that this setting breaks invariance of the MDM objective to the noise schedule. Furthermore, we prove that the MDM objective decomposes precisely into a weighted auto-regressive losses over these orders, which establishes them as auto-regressive models with learnable orders.

Masked diffusion models (MDMs) are a promising alternative to autoregressive models (ARMs), but they suffer from inherently much higher training variance. High variance leads to noisier gradient estimates and unstable optimization, so even equally strong pretrained MDMs and ARMs that are competitive at initialization often diverge after task-specific training, with MDMs falling far behind. There has been no theoretical explanation or systematic solution. We derive the first decomposition of MDM training variance into three sources: (A) masking pattern noise, (B) masking rate noise, and (C) data noise, while ARMs are only affected by (C). This explains the fundamental training gap. Building on this foundation, we design six variance-reduction methods, including two core methods: (1) P-POTS, a Pareto-optimal t sampler that minimizes training variance by sampling harder t values more often with appropriately smaller update steps, and (2) MIRROR, which uses negatively correlated samples to reduce (A). Experiments show that compared to standard MDM training, our methods improve accuracy by 7-8% on complex reasoning tasks, while simultaneously reducing run-to-run variability to near ARM levels, substantially narrowing the gap with strong ARM baselines; in most settings, even the best baseline runs remain below the worst run of our method.

Diffusion language models have recently emerged as a competitive alternative to autoregressive language models. Beyond next-token generation, they are more efficient and flexible by enabling parallel and any-order token generation. However, despite empirical successes, their computational power and fundamental limitations remain poorly understood. In this paper, we formally study whether non-autoregressive generation in Masked Diffusion Models (MDM) enables solving problems beyond the reach of Auto-Regressive Models (ARM). Our results show that MDM with sufficiently large context length is computationally universal with decoding steps matching the optimal parallel time complexity in PRAM. However, when controlling for other factors, MDM's flexibility to generate in any-order does not expand what ARM can already solve. To address this, we propose a new form of generation called any-process generation, which extends MDM with capabilities to remask, insert and delete tokens, allowing self-correction, length-variable editing, and adaptive parallelism. Theoretically and empirically, we demonstrate these capabilities enable scalability to significantly harder reasoning problems that are otherwise intractable for ARM and vanilla MDM. Additionally, they prove essential for generation tasks where objects naturally evolve through non-sequential processes, crucial for extending current LLMs beyond natural language to domains such as coding and science.

Manhattan Distance Mapping (MDM) is a post-training deep neural network (DNN) weight mapping technique for memristive bit-sliced compute-in-memory (CIM) crossbars that reduces parasitic resistance (PR) nonidealities. PR limits crossbar efficiency by mapping DNN matrices into small crossbar tiles, reducing CIM-based speedup. Each crossbar executes one tile, requiring digital synchronization before the next layer. At this granularity, designers either deploy many small crossbars in parallel or reuse a few sequentially-both increasing analog-to-digital conversions, latency, I/O pressure, and chip area. MDM alleviates PR effects by optimizing active-memristor placement. Exploiting bit-level structured sparsity, it feeds activations from the denser low-order side and reorders rows according to the Manhattan distance, relocating active cells toward regions less affected by PR and thus lowering the nonideality factor (NF). Applied to DNN models on ImageNet-1k, MDM reduces NF by up to 46% and improves accuracy under analog distortion by an average of 3.6% in ResNets. Overall, it provides a lightweight, spatially informed method for scaling CIM DNN accelerators.

Masked diffusion models (MDMs) for text offer a compelling alternative to traditional autoregressive language models. Parallel generation makes them efficient, but their computational capabilities and the limitations inherent to their parallelism remain largely unexplored. To this end, we characterize what types of reasoning problems MDMs can provably solve and how efficiently. We do this by connecting MDMs to the well-understood reasoning frameworks of chain of thought (CoT) and padded looped transformers (PLTs) in the finite-precision log-width setting: We show that MDMs and polynomially-padded PLTs are, in fact, equivalent in this setting, and that MDMs can solve all problems that CoT-augmented transformers can. Moreover, we showcase classes of problems (including regular languages) for which MDMs are inherently more efficient than CoT transformers, where parallel generation allows for substantially faster reasoning.

While Masked Diffusion Models (MDMs), such as LLaDA, present a promising paradigm for language modeling, there has been relatively little effort in aligning these models with human preferences via reinforcement learning. The challenge primarily arises from the high variance in Evidence Lower Bound (ELBO)-based likelihood estimates required for preference optimization. To address this issue, we propose Variance-Reduced Preference Optimization (VRPO), a framework that formally analyzes the variance of ELBO estimators and derives bounds on both the bias and variance of preference optimization gradients. Building on this theoretical foundation, we introduce unbiased variance reduction strategies, including optimal Monte Carlo budget allocation and antithetic sampling, that significantly improve the performance of MDM alignment. We demonstrate the effectiveness of VRPO by applying it to LLaDA, and the resulting model, LLaDA 1.5, outperforms its SFT-only predecessor consistently and significantly across mathematical (GSM8K +4.7), code (HumanEval +3.0, MBPP +1.8), and alignment benchmarks (IFEval +4.0, Arena-Hard +4.3). Furthermore, LLaDA 1.5 demonstrates a highly competitive mathematical performance compared to strong language MDMs and ARMs. Project page: https://ml-gsai.github.io/LLaDA-1.5-Demo/.