Equilibrium Training of Energy-Based Models with Parallel Trajectory Tempering

Efficient sampling from Boltzmann distributions over discrete variables is a fundamental operation in a wide range of applications. While fast non-MCMC samplers have recently emerged as promising alternatives to conventional MCMC methods, their practical use for probabilistic learning remains hindered by the difficulty of estimating the effective temperature of the generated samples. In this work, we begin by introducing Langevin simulated bifurcation (LSB), a Boltzmann sampler that enables fast and parallel sampling with accuracy comparable to sequential MCMC methods. To address the challenge of unknown effective temperature, we propose conditional expectation matching (CEM), an efficient estimation method applicable to energy-based models (EBMs) with exploitable conditional independence structures. Building on these components, we further develop a learning framework, termed sampler adaptive learning (SAL), which adaptively adjusts the model temperature to match that of the distribution induced by fast non-MCMC sampling. We demonstrate the effectiveness of LSB, CEM, and SAL on semi-restricted Boltzmann machines (SRBMs), a class of EBMs that are difficult to train using conventional approaches. LSB achieves orders-of-magnitude acceleration over Gibbs sampling while maintaining comparable or higher accuracy, and CEM enables accurate temperature estimation of the resulting distribution with negligible computational overhead. As a consequence, SAL enables efficient training of SRBMs and outperforms conventional Boltzmann machine learning methods on synthetic spin-glass datasets. In addition, the trained models achieve strong performance across multiple tasks. These results establish LSB as a fast and accurate Boltzmann sampler and provide key insights that enable practical applications of fast non-MCMC sampling methods via efficient temperature estimation with CEM.

Energy-based models (EBMs) are flexible generative architectures inspired by statistical physics, but their learning and generative properties remain poorly understood. Here, we analyze a solvable EBM in the high-dimensional limit: the spherical Boltzmann machine (SBM). Combining tools from random matrix theory and dynamical mean-field theory, we: solve exact equations describing the training dynamics of the SBM; compute the Bayesian evidence, which acts as a partition function in parameter space and encodes global properties of the trained model; and uncover cascades of phase transitions that occur both during training and as a function of hyperparameters, related to successive alignment and condensation of the top modes of the coupling matrix to the data. We connect these transitions to sampling-time generative phenomena in a teacher-student scenario, including: sampling temperature tuning, double descent as a function of regularization strength, tempered posterior effects, and out-of-equilibrium effects during training that induce biases in the trained model. We provide numerical evidence demonstrating that all these phenomena appear in standard generative architectures, beyond the SBM.

Energy-based models (EBMs) are a flexible class of deep generative models and are well-suited to capture complex dependencies in multimodal data. However, learning multimodal EBM by maximum likelihood requires Markov Chain Monte Carlo (MCMC) sampling in the joint data space, where noise-initialized Langevin dynamics often mixes poorly and fails to discover coherent inter-modal relationships. Multimodal VAEs have made progress in capturing such inter-modal dependencies by introducing a shared latent generator and a joint inference model. However, both the shared latent generator and joint inference model are parameterized as unimodal Gaussian (or Laplace), which severely limits their ability to approximate the complex structure induced by multimodal data. In this work, we study the learning problem of the multimodal EBM, shared latent generator, and joint inference model. We present a learning framework that effectively interweaves their MLE updates with corresponding MCMC refinements in both the data and latent spaces. Specifically, the generator is learned to produce coherent multimodal samples that serve as strong initial states for EBM sampling, while the inference model is learned to provide informative latent initializations for generator posterior sampling. Together, these two models serve as complementary models that enable effective EBM sampling and learning, yielding realistic and coherent multimodal EBM samples. Extensive experiments demonstrate superior performance for multimodal synthesis quality and coherence compared to various baselines. We conduct various analyses and ablation studies to validate the effectiveness and scalability of the proposed multimodal framework.