(Left) Our proposed masked diffusion language model (MDLM) is trained using a weighted average of masked cross entropy losses. (Top Right) In comparison to masked language models (MLM), MDLM's objective correspond to a principled variational lower bound, and supports generation via ancestral sampling.(Bottom Right) Perplexity (PPL) on One Billion Words benchmark.

A descriptive text for the GIF

The sample generation process begins with a sequence of all masked tokens. MDLM then replaces these masked tokens with actual tokens in a random order.

A Simple and Effective tutorial on our paper by Sasha Rush.


While diffusion models excel at generating high-quality images, prior work reports a significant performance gap between diffusion and autoregressive (AR) methods on language modeling. In this work, we show that simple masked discrete diffusion is more performant than previously thought. We apply an effective training recipe that improves the performance of masked diffusion models and derive a simplified, Rao-Blackwellized objective that results in additional improvements. Our objective has a simple form—it is a mixture of classical masked language modeling losses— and can be used to train encoder-only language models that admit efficient samplers, including ones that can generate arbitrary lengths of text semi-autoregressively like a traditional language model. On language modeling benchmarks, a range of masked diffusion models trained with modern engineering practices achieves a new state-of-the-art among diffusion models, and approaches AR perplexity.

Diffusion Models

Diffusion models excel at producing realistic, high-quality images and have received significant attention as potential tools for generating discrete data such as text, biological sequences, and graphs. Unlike autoregressive (AR) approaches, diffusion-based methods are not constrained to generate data sequentially, and therefore have the potential to improve long-term planning, controllable generation, and sampling speed. However, discrete diffusion methods exhibit a performance gap relative to AR models, especially in language modeling. The standard measure of language modeling performance is log-likelihood: when controlling for parameter count, prior work reports a sizable log-likelihood gap between AR and diffusion models.

Discrete Diffusion

Applications of diffusion modeling to discrete data can be categorized into two broad areas. The first involves embedding discrete structures in continuous space and then performing the Gaussian diffusion defined above on these continuous representations. More related to our method are works that define a diffusion process directly on discrete structures. D3PM introduces a framework with a Markov forward process \( q(z_t|z_{t−1}) = \text{Cat}(z_t; Q_t z_{t−1}) \), defined by the multiplication of matrices \( Q_t \in \mathbb{R}^{n \times n} \) over \( T \) discrete time steps. The matrix \( Q_t \) is designed such that \( Q_T \cdot Q_{T-1} \cdots Q_1 \mathbf{x} \) converges to a stationary distribution.

Simple Masked Diffusion Models

While previous work on discrete diffusion supports general forward processes (e.g., general \(Q_t\) in D3PM), absorbing state (i.e., masking) diffusion consistently achieves the best performance. In this work, instead of supporting general noise processes, we focus on masking and derive tight Rao-Blackwellized objectives that outperform general approaches and do not require CTMC theory (for ex. SEDD). We denote our overall approach as Masked Diffusion Language Models (MDLM).

Interpolating Masked Diffusion

We forcus on forward processes q that interpolate between clean data \(\mathbf{x} \in \mathcal{V}\), where \(\mathcal{V}\) is a set of all one-hot vectors, and a target distribution \(\text{Cat}(.; \boldsymbol{\mathit{\pi}})\). This approach is a direct extention of Gaussian diffusion where the intermediate sample interpolates between the clean data and white noise. The \(q\) defines a sequence of increasingly noisy latent variables \(\mathbf{z}_t \in \mathcal{V}\), where the time step \(t\) runs from \(t = 0\) (least noisy) to \(t = 1\) (most noisy). The marginal of \(\mathbf{z}_t\) conditioned on \(\mathbf{x}\) at time \(t\) is given by \(q(\mathbf{z}_t|x) = \text{Cat}(\mathbf{z}_t;\alpha_t \mathbf{x}+(1 - \alpha_t) \boldsymbol{\mathit{\pi}})\), where \(\alpha_t \in [0, 1]\) is a strictly decreasing function in t, with \(\alpha_{t=0} \approx 0\) and \(\alpha_{t=1} \approx 1\). In absorbing state diffusion, we set \(\boldsymbol{\mathit{\pi}} = \mathbf{m}\) where \(\mathbf{m} \in \mathcal{V}\) and \(\mathbf{m}_K=1\), with \(K^\text{th}\)category representing the special masked token, [MASK].

Reverse process

The specific parameterization for \(p_\theta(\mathbf{z}_s | \mathbf{z}_t)\) with \(0 \leq s < t \leq 1\) that we use is: \begin{align}\label{eqn:approx_posterior} p_\theta(\mathbf{z}_s | \mathbf{z}_t) = q(\mathbf{z}_s | \mathbf{z}_t, \mathbf{x} = \mathbf {x}_\theta (\mathbf{z}_t, t)) = \begin{cases} \text{Cat} (\mathbf{z}_s; \mathbf{z}_t), & \mathbf{z}_t \neq \mathbf{m}, \\ \text{Cat} \left( \mathbf{z}_s; \frac{ (1 - \alpha_s)\mathbf{m} + (\alpha_s - \alpha_t) \mathbf{x}_\theta (\mathbf{z}_t, t)}{1 - \alpha_t}\right). & \mathbf{z}_t= \mathbf{m}, \end{cases} \end{align} Furthermore, we induce 2 key properties of the absorbing state diffusion process into our denoising model, \( \mathbf{x}_\theta (\mathbf{z}_t, t): \mathcal{V} \times [0, 1] \rightarrow \Delta^{K}\), where \(\Delta^{K}\) denotes the \(K\)-simplex:

  1. Zero Masking Probabilities: First, notice that by definition, \( \langle \mathbf{x}, \mathbf{m} \rangle = 0 \). For this reason, we design the denoising network such that \( \langle \mathbf{x}_\theta (\mathbf{z}_t,t),\mathbf{m} \rangle = 0 \), i.e., we substitute the logit index corresponding to the [MASK] token with \(-\infty\).
  2. Carry-Over Unmasking: Second, if \( \mathbf{z}_t \) is unmasked, then we desire \( \mathbf{x}_\theta (\mathbf{z}_t, t) = \mathbf{z}_t \), i.e., unmasked latents are "carried over". We accomplish this by substituting the output of our network to simply copy unmasked inputs.
As discussed in the paper, each of these properties plays a crucial role simplifying the Diffusion objetive. We implement these as substitutions to the output of \(\mathbf{x}_\theta (\mathbf{z}_t, t)\), hence we call our parameterization SUBS.


For a sequence \(\mathbf{x}^{1: L}\) of length \(L\), SUBS parameterization simplifies the Negative Evidence Lower Bound (NELBO) to the following: \begin{align} \mathcal{L}_{\text{NELBO}}^\infty = \mathbb{E}_{q}\int_{t=0}^{t=1} \frac{\alpha_{t}'}{1 - \alpha_t} \sum_{\ell = 1}^{L} \log \langle \mathbf {x}_\theta^\ell(\mathbf{z}_t), \mathbf{x}^\ell \rangle \text{d} t \end{align} where \(\alpha_{t}'\) denotes the time derivative of \(\alpha_{t}\). As shown in the table below, MDLM outperforms the previous diffusion models and nearly matches the performance of AR models in text generation on the LM1B dataset.

Test perplexities (PPL; ↓) on LM1B. †. Best diffusion value is bolded.
Parameters PPL (↓)
Autoregressive Transformer-X Base 0.46B 23.5
OmniNetT 100M 21.5
Diffusion BERT-Mouth 110M ≤142.89
D3PM (absorb) 70M ≤77.50
Diffusion-LM 80M ≤118.62
DiffusionBert 110M ≤63.78
SEDD 110M ≤32.79
Autoregressive (Retrained) Transformer (33B tokens) 110M 22.32
Transformer (327B tokens) 20.86
Diffusion (Ours) MDLM (33B tokens) 110M ≤27.04
MDLM (327B tokens) 23.00

We also explore models' ability to generalize by taking models trained on OWT and evaluating how well they model unseen datasets. We compare the zero-shot perplexities of MDLM with SEDD and an AR Transformer language model on the validation splits of Penn Tree Bank (PTB), Wikitext, LM1B, Lambada, AG News, and Scientific Papers (Pubmed and Arxiv subsets). MDLM consistently outperforms SEDD. In some cases, e.g., for Lambada and Scientific Papers, MDLM attains better perplexity than AR. We hypothesize that these datasets are farther from OWT, and that diffusion models may be more robust to out-of-domain evaluation due to the unmasking-based objective.

Zero-shot validation perplexities ( ↓) of models trained for 524B tokens on OpenWebText. All perplexities for diffusion models are upper bounds.
PTB Wikitext LM1B Lambada AG News Pubmed Arxiv
AR (Retrained) 82.05 25.75 51.25 51.28 52.09 49.01 41.73
SEDD (Retrained) 100.09 34.28 68.20 49.86 62.09 44.53 38.48
MDLM (Ours) 95.26 32.83 67.01 47.52 61.15 41.89 37.37

Training Considerations for Masked Diffusion

One of the key contributions of our work is a well-engineered implementation of masked diffusion models. Our experiments demonstrate that these improvements greatly boost performance even for methods previously thought to perform poorly, e.g., D3PM . Below we briefly summarize these implementation details.

  1. We find that tokenization is critical to performance. Small vocabularies, such as the 8k vocabulary in D3PM, result in longer-range dependencies that decrease the performance of both diffusion and AR models.
  2. By focusing on masked diffusion, we are able to provide a numerically stable implementation of the objective function. Namely, since previous formulations of discrete diffusion were constructed to accommodate a wide range of limiting distributions, the objective was implemented by materializing the full transition matrices \(Q_t\) and posterior probabilities. In contrast, we evaluate \(\text{D}_{\text{KL}}[q(\mathbf{z}_s | \mathbf{z}_t,\mathbf{x}) \| p_\theta (\mathbf{z}_s | \mathbf{z}_t)]\) by examining only the masked token indices rather than comparing the full true and approximate posterior distributions.
  3. Furthermore, we modernize the architecture for the denoising network relative to D3PM. In lieu of the T5 architecture used in D3PM, we use the diffusion transformer (DiT), which integrates time step conditioning into a standard encoder-only transformer and uses rotary positional embeddings.
  4. In addition, we implement a low-discrepancy sampler that reduces the variance of the ELBO, similar to Kingma et al. and draws correlated samples \(t \in [0, 1]\) rather than performing i.i.d. sampling along the batch dimension.


      title={Simple and Effective Masked Diffusion Language Models}, 
      author={Subham Sekhar Sahoo and Marianne Arriola and Yair Schiff and Aaron Gokaslan and Edgar Marroquin and Justin T Chiu and Alexander Rush and Volodymyr Kuleshov},