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Learning Latent Energy-Based Models via Interacting Particle Langevin Dynamics

Joanna Marks, Tim Y. J. Wang, O. Deniz Akyildiz

TL;DR

The paper addresses learning latent energy-based models (LEBMs) by formulating maximum marginal likelihood estimation (MMLE) and approximating the intractable posteriors with interacting particle Langevin dynamics. It introduces iplebm, a continuous-time SDE-based framework whose Euler–Maruyama discretisation yields a scalable algorithm that jointly updates model parameters and latent particles, with theoretical nonasymptotic convergence guarantees under strong convexity and smoothness. The authors provide exact-gradient and inexact-gradient analyses, detailing how the number of data points $M$, posterior particles $N$, step size $h$, and iteration count $k$ influence convergence, and they validate the method on synthetic data and image benchmarks, showing competitive performance and efficiency relative to existing EBMs. The work advances latent-space EBMs by marrying diffusion-inspired optimization with particle-based posterior approximation, offering practical gains and a foundation for future improvements with more advanced samplers and accelerated dynamics.

Abstract

We develop interacting particle algorithms for learning latent variable models with energy-based priors. To do so, we leverage recent developments in particle-based methods for solving maximum marginal likelihood estimation (MMLE) problems. Specifically, we provide a continuous-time framework for learning latent energy-based models, by defining stochastic differential equations (SDEs) that provably solve the MMLE problem. We obtain a practical algorithm as a discretisation of these SDEs and provide theoretical guarantees for the convergence of the proposed algorithm. Finally, we demonstrate the empirical effectiveness of our method on synthetic and image datasets.

Learning Latent Energy-Based Models via Interacting Particle Langevin Dynamics

TL;DR

The paper addresses learning latent energy-based models (LEBMs) by formulating maximum marginal likelihood estimation (MMLE) and approximating the intractable posteriors with interacting particle Langevin dynamics. It introduces iplebm, a continuous-time SDE-based framework whose Euler–Maruyama discretisation yields a scalable algorithm that jointly updates model parameters and latent particles, with theoretical nonasymptotic convergence guarantees under strong convexity and smoothness. The authors provide exact-gradient and inexact-gradient analyses, detailing how the number of data points , posterior particles , step size , and iteration count influence convergence, and they validate the method on synthetic data and image benchmarks, showing competitive performance and efficiency relative to existing EBMs. The work advances latent-space EBMs by marrying diffusion-inspired optimization with particle-based posterior approximation, offering practical gains and a foundation for future improvements with more advanced samplers and accelerated dynamics.

Abstract

We develop interacting particle algorithms for learning latent variable models with energy-based priors. To do so, we leverage recent developments in particle-based methods for solving maximum marginal likelihood estimation (MMLE) problems. Specifically, we provide a continuous-time framework for learning latent energy-based models, by defining stochastic differential equations (SDEs) that provably solve the MMLE problem. We obtain a practical algorithm as a discretisation of these SDEs and provide theoretical guarantees for the convergence of the proposed algorithm. Finally, we demonstrate the empirical effectiveness of our method on synthetic and image datasets.
Paper Structure (46 sections, 3 theorems, 55 equations, 5 figures, 2 tables, 3 algorithms)

This paper contains 46 sections, 3 theorems, 55 equations, 5 figures, 2 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Background
  3. Maximum marginal likelihood estimation
  4. Optimisation via Langevin dynamics
  5. The Algorithm
  6. An Interacting Particle Algorithm
  7. Practical Implementation Details
  8. Nonasymptotic Analysis
  9. Assumptions
  10. Convergence with exact gradient
  11. Convergence with inexact gradient
  12. Related Works
  13. Training EBMs.
  14. EBMs in the Latent Space.
  15. Interacting Particle Algorithms for Learning.
  16. ...and 31 more sections

Key Result

Theorem 1

Suppose ass:strong-convexity, ass:L-smooth hold. Let $\theta_k$ be the parameter marginal generated by iterates eq:ipla-algo-1--eq:ipla-algo-2 (equivalently, eq:alpha-update--eq:x-update), then given a step size $0 < h \leq 2 / (\mu + L)$, the following holds where $C_0$ is an explicit constant that only depends on the initial law of the system and and $\theta_\star$ denotes the unique maximiser

Figures (5)

  • Figure 1: Comparison of MMD (left) and runtime (right) for increasing numbers of posterior particles (iplebm) or MCMC steps (lebm) trained on the rotated Swiss roll dataset.
  • Figure 2: Samples generated with iplebm and lebm trained on the rotated Swiss roll dataset for increasing numbers of posterior particles (or MCMC steps) with energy landscapes.
  • Figure 3: Samples generated with iplebm trained on CIFAR-10, CelebA64, and SVHN.
  • Figure 4: Latent space interpolation of iplebm on the training (bottom) and validation set (top).
  • Figure 5: Visualisation of the rotated Swiss roll dataset

Theorems & Definitions (7)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • proof
  • Proposition B.1: Theorem 4 in dalalyan2019user adapted
  • proof