Energy-Based Diffusion Language Models (EDLMs)

• Energy-Based Diffusion Language Models (EDLMs) are a family of language models that blend discrete diffusion with energy-based techniques to improve training tractability and controlled generation.
• They leverage iterative denoising, parallel importance sampling, and energy correction to enable efficient inference with arbitrary-order decoding and enhanced sampling speeds.
• Empirical results demonstrate that EDLMs achieve competitive perplexity and planning performance, with models like Dream 7B showing 1.3× sampling speedup and improved constraint satisfaction.

Energy-Based Diffusion LLMs (EDLMs) span a family of LLMs that blend discrete diffusion generative paradigms with energy-based modeling. By viewing denoising transition operators as steps in an implicit energy landscape, EDLMs yield tractable training objectives, improved sequence modeling over standard diffusion approaches, and enable advanced capabilities such as parallel generation, arbitrary-order decoding, and enhanced controllability. These models build on both autoregressive and denoising formulations and admit both data-space and latent-space instantiations, achieving state-of-the-art performance on text generation and planning tasks across a range of corpus scales (Ye et al., 21 Aug 2025, Xu et al., 28 Oct 2024, Yu et al., 2022).

1. Foundations and Model Formulation

At the core of EDLMs is the combination of a discrete diffusion process and an energy-based model (EBM) over token sequences or latent representations. The forward diffusion corrupts text by masking tokens, inducing a sequence of random variables $X_0, X_1, \ldots, X_T$, where $X_0$ is the clean sequence and $X_T$ is fully masked. Each transition is defined by a masking kernel, for example,

$$q(X_t^n = v | X_{t-1}^n = u) = \alpha(t)\delta_{v,u} + (1-\alpha(t))1[v = \mathrm{MASK}]$$

for token positions $n = 1 \dots N$ using a linear schedule $\alpha(t) = 1-t$. The reverse (denoising) process is learned to reconstruct $X_{t-1}$ from $X_t$ by parallel prediction of true tokens at masked positions (Ye et al., 21 Aug 2025).

EDLMs explicitly introduce a residual energy function $E_\phi$, yielding a corrected denoiser: $$p_{\theta,\phi}(x_0 \mid x_t) = p_\theta(x_0 \mid x_t)\frac{\exp(-E_\phi(x_0, x_t, t))}{Z_\phi(x_t, t)}$$ Here, $p_\theta$ is usually a factorized diffusion model’s conditional predictor, and $E_\phi$ is a scalar sequence-level energy, typically modeled by a bidirectional transformer or using a pretrained autoregressive (AR) model (Xu et al., 28 Oct 2024).

In latent-space variants, the model defines a variational latent variable architecture with an encoder $q_\phi(z_0|x)$, a generative model $p_\beta(x|z_0)$, and an energy-based prior $p_\alpha(y, z)$ coupling discrete symbols and latent variables, regularized via information bottleneck and geometric clustering (Yu et al., 2022).

2. Training Objectives and Estimation Procedures

Training in EDLMs relies on tractable variational bounds. For token-space models, the objective is a weighted cross-entropy corresponding to a variational upper bound on $-\log p_\theta(X_0)$: $$L(\theta) = -\mathbb{E}_{X_0, t, X_t} \sum_{n=1}^N 1_{[X_t^n = \mathrm{MASK}]} w(t, X_t, n) \log p_\theta(X_0^n | X_t)$$ Weighting $w(t, X_t, n)$ can be context-adaptive (CART) to account for the informativeness of local context; in Dream 7B, it employs a geometric decay with respect to clean token proximity (Ye et al., 21 Aug 2025).

EDLMs introduce two key approaches for energy learning (Xu et al., 28 Oct 2024):

• AR-based Energy: The energy is the log-ratio $$E_\phi(x_0, x_t, t) = -\log p_\mathrm{AR}(x_0) + \log p_\theta(x_0|x_t)$$ utilizing a fixed pretrained AR model.
• Noise Contrastive Estimation (NCE): A learnable $E_\phi$ is updated via NCE, contrasting the energy on true denoised samples and negative samples produced by the denoiser, via a binary classification loss.

Latent-space models use a joint ELBO with terms for reconstruction, KL divergence to the energy-based prior, and “recovery likelihoods” from conditional EBMs at each diffusion step. Regularizations such as the information bottleneck and geometric clustering are added to enforce interpretable latent structures (Yu et al., 2022).

3. Inference Algorithms and Sampling

Inference proceeds by iterative denoising, with each step interpreted as a Gibbs move that relaxes the sequence toward lower energies. In Dream 7B, the number of denoising steps can be traded off against output quality: as few as 5–20 iterations outperform strong AR baselines in planning, while 50–100 steps provide finer energy minimization (Ye et al., 21 Aug 2025).

EDLMs accelerate sampling by parallel importance sampling. At each stage, the denoiser proposes multiple candidates $x_0^i \sim p_\theta(\cdot | x_t)$, energies $E_\phi$ are scored in parallel, and one candidate is resampled using self-normalized weights

$w_i = \frac{\exp(-E_i)}{\sum_{j}\exp(-E_j)}$

This correction is only applied in the early part of the diffusion trajectory (window fraction $\rho$), with later steps defaulting to $p_\theta$ proposals. This reduces the required number of steps (e.g., halving from 1024 to 512 on OpenWebText) and leads to 1.3× sampling speedup in practice (Xu et al., 28 Oct 2024).

In latent-space frameworks, reverse-time generation is conducted by running Langevin dynamics on localized conditional energies before decoding through the AR generator (Yu et al., 2022).

4. Model Initialization, Architectural Innovations, and Regularization

Scaling EDLMs to large LLMs requires leveraging pretrained AR models, as in Dream 7B, where existing AR weights (e.g., Qwen2.5-7B) are loaded, causal masks are replaced with full attention, and a positional shift operation realigns hidden states to predict current masked tokens. This initialization preserves left-to-right consistency while enabling parallel denoising (Ye et al., 21 Aug 2025).

Token-level noise rescheduling (CART) addresses heterogeneity in context, dynamically adjusting loss weighting based on geometric proximity to clean tokens, yielding faster convergence and improved final accuracies. In latent-space EDLMs, information bottleneck and geometric clustering regularization encourage interpretability and discrete-structure in $\{z_0, y\}$, aligning latent clusters with semantic classes (Yu et al., 2022).

5. Empirical Performance and Qualitative Insights

EDLMs consistently outperform prior discrete diffusion models and approach or match AR perplexity on language modeling benchmarks. For instance, on OpenWebText, the AR baseline achieves PPL = 17.56, whereas EDLM-coAR achieves PPL = 17.58 and EDLM-NCE achieves PPL ≤ 21.52, surpassing prior diffusion approaches (e.g., MDLM PPL = 23.83) (Xu et al., 28 Oct 2024).

Dream 7B demonstrates strong reasoning and planning: it closes the gap to Qwen2.5-7B on general NLP (MMLU, BBH, ARC), exceeds LLaDA-8B by +6 points on math (GSM8K, MATH), and achieves 2–3× better constraint satisfaction in non-autoregressive planning tasks (Countdown, Sudoku, Trip Planning) with only 0.6T pretraining tokens versus 2.3T (Ye et al., 21 Aug 2025).

Crucially, arbitrary infilling and flexible decoding emerge without architectural modifications: masking any span at inference followed by denoising enables gap-filling, prefix extension, and blockwise or arbitrary-order completion.

Latent-space EDLMs achieve improved interpretable clustering and attribute control. On the DailyDialog dataset, mutual information between latent $y$ and dialog action/emotion increases from ~2.4 to 3.94, and reconstruction BLEU rises from 10.0 to 28.8 with LDEBM. Sentiment-controlled generation accuracy increases to ~99% (Yu et al., 2022).

6. Limitations and Open Challenges

Remaining challenges include the tractability of the partition function $Z_\phi$ in general EBMs. EDLM-coAR side-steps this via a carry-over trick, but general approaches for self-normalizing EBMs at scale are still lacking (Xu et al., 28 Oct 2024). Computational overhead is mild as $k$ (candidates per step) and $\rho$ (fraction of steps corrected) are kept small, but further scaling NCE-trained energy models is a promising avenue. For latent space models, interpretability and mode discovery are limited by clustering and information bottleneck capacities.

This suggests that future advances may include continuous-time inference, integration with ODE/SDE solver-based sampling, and new score-matching or flow-matching objectives. Improved self-normalization and robust scaling strategies will be required to deploy sequence-level EBMs in even larger LMs.

7. Outlook and Impact

Energy-Based Diffusion LLMs establish a distinct axis from purely autoregressive and classic diffusion approaches by unifying sequence-level energy shaping with parallel denoising. This results in models that rival AR LLMs in perplexity and controllability, but offer enhanced bidirectional context, programmable inference-order flexibility, and improved performance in tasks dominated by planning and global structure. These properties position EDLMs as prime candidates for the next generation of interpretable, efficient, and controllable language generation systems (Ye et al., 21 Aug 2025, Xu et al., 28 Oct 2024, Yu et al., 2022).