Stochastic Attention via Langevin Dynamics on the Modern Hopfield Energy

Paper: arXiv 2603.06875

Authors: Abdulrahman Alswaidan, Jeffrey D. Varner

Abstract

Attention heads retrieve: given a query, they return a weighted average of stored values. We showed that this computation is one step of gradient descent on the modern Hopfield energy, and that Langevin sampling from the corresponding Boltzmann distribution yielded stochastic attention, a training-free sampler controlled by a single temperature parameter. Lowering the temperature gave exact retrieval; raising it gave open-ended generation. Because the energy gradient equals the attention map, no score network, training loop, or learned model was required, making the approach particularly suited to the low-data regime where learned generative models are starved of training signal. We derived an entropy inflection condition that identified the retrieval-to-generation transition temperature for any memory geometry and validated the sampler on five domains spanning two orders of magnitude in dimension. A single Boolean mask on the attention softmax, identical to the causal mask used in transformers but applied along the memory axis rather than the sequence axis, turned the sampler into a zero-shot class-conditional generator on Olivetti faces with no retraining and no learned classifier. On MNIST digit images, stochastic attention produced samples that were markedly more novel and more diverse than the best learned baseline while matching a Metropolis-corrected gold standard. On protein sequences from a small Pfam family, the generation regime preserved amino acid composition far more faithfully than a variational autoencoder at matched novelty, indicating that the training-free score function retained family-level fidelity that learned models lost. A denoising diffusion baseline failed across all memory sizes tested, producing samples indistinguishable from isotropic noise. The approach required no architectural changes to the underlying attention mechanism.

Complexity vs Empirical Score

• Math Complexity: 6.5/10
• Empirical Rigor: 6.0/10
• Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper presents an elegant synthesis of modern Hopfield networks and Langevin dynamics to create a training-free generative sampler via stochastic attention. It supports its theoretical derivations with analytical proofs (such as the entropy inflection condition) and demonstrates strong empirical validation across multiple domains including protein sequences and image datasets.

Research Flowchart

  flowchart TD
    A["Research Goal: Generative Sampling via Modern Hopfield Energy"] --> B["Key Insight: Attention Map Equals Energy Gradient"]
    B --> C["Method: Langevin Sampling on Boltzmann Distribution<br/><i>(Controlled by Temperature T)</i>"]
    
    C --> D["Inputs: Stored Memory Matrix & Query Vectors<br/><i>(No Score Network / No Training Loop)</i>"]
    
    D --> E{"Temperature Control & Axis Masking"}
    
    E -- "Low T" --> F["Exact Retrieval Regime"]
    E -- "High T + Boolean Memory Mask" --> G["Open-Ended & Zero-Shot Conditional Generation"]
    
    F & G --> H["Outcomes: Superior Novelty/Fidelity vs VAE/Diffusion<br/><i>(MNIST, Olivetti Faces, Pfam Proteins)</i>"]

Cited References (44)

1. Attention is All You Need (2017)
2. Neural Networks and Physical Systems with Emergent Collective Computational Abilities (1982) — link
3. Hopfield Networks is All You Need (2021)
4. Dense Associative Memory for Pattern Recognition (2016)
5. Exponential Convergence of Langevin Distributions and Their Discrete Approximations (1996) — link
6. Storing Infinite Numbers of Patterns in a Spin-Glass Model of Neural Networks (1985) — link
7. Bayesian Learning via Stochastic Gradient Langevin Dynamics (2011)
8. Nonasymptotic Convergence Analysis for the Unadjusted Langevin Algorithm (2017) — link

0 of 44 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.