Uniform-in-time convergence bounds for Persistent Contrastive Divergence algorithms

Paul Felix Valsecchi Oliva; O. Deniz Akyildiz; Andrew Duncan

stat.MLOct 2, 2025

Uniform-in-time convergence bounds for Persistent Contrastive Divergence algorithms

Authors: Paul Felix Valsecchi Oliva, O. Deniz Akyildiz, Andrew Duncan

Organizations: Department of Mathematics, Imperial College London

Abstract

We propose a continuous-time formulation of a noisy persistent contrastive divergence (PCD)-like method for maximum likelihood estimation (MLE) of unnormalised densities. Our approach couples parameter updates and sampling of the parametrised density in a multiscale system of stochastic differential equations (SDEs). From this formulation, we derive non-asymptotic bounds for weak test-function errors between the resulting numerical schemes and the MLE point target. The error is decomposed into numerical discretisation, slow-fast averaging, and finite-temperature concentration terms. We also introduce an efficient implementation based on explicit stabilized integrators and establish corresponding long-time error estimates. This leads to a novel method for training energy-based models (EBMs) with quantitative error guarantees.

Figures & tables

Figure 1 : A comparison between the performance of SPCD and SPCDem over the hyper-parameters $\delta$ and $\varepsilon$ . The comparison for SPCD and SPCDem is the difference between the Sinkhorn error obtained by SPCD and the error obtained by SPCDem , normalised by the SPCD error (negative sign means SPCD error is smaller than SPCDem error). We highlight in red the region in which SPCD attains an error smaller than 20% of the SPCDem error. For full experimental details see the Appendix .

Panel of Figure 2 — Figure 2 : The samples obtained after 60 outer training epochs with matched gradient-evaluation budgets: for each SPCD update with $m_{\mathrm{st}}=3$ , three SPCDem updates are used (full details are given in the Appendix ). In the top row the algorithms are trained on images of ones, while in the second row they are trained on images of the digit 4. The displayed samples are selected randomly from the generated samples.

Appendix figures & tables2 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 3 : A comparison of the Sinkhorn errors of SPCD for $m_{\mathrm{st}}=3,5,10,20$ and SPCDem against $\varepsilon$ on the synthetic dataset experiment from Sec. 8.1 . For this experiment we set $\delta=0.001$ and fix the number of gradient evaluations to 10000. We display the average and interquartile range of 100 runs.

Figure 4 : The model structure of $E(\theta,x)$ for the MNIST experiment, where the pyramids represent convolutions and the vectors represent fully connected linear layers. On the left we have a realisation of $x$ and on the right the scalar output of $E(\theta,x)$ . We note that between convolutions we apply spectral normalisation and Swish activations (the Swish activation is given as $x\mapsto x\sigma(x)$ , with $\sigma$ corresponding to the sigmoid activation). For the linear transformations we similarly normalise and apply Swish activations, except for the last layer.

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Equilibrium Training of Energy-Based Models with Parallel Trajectory Tempering

Nicolas Béreux, Aurélien Decelle, Cyril Furtlehner, Beatriz Seoane

Université Paris-Saclay, CNRS, INRIA, LISN, 91190 Gif-sur-Yvette, France · Escuela Técnica Superior de Ingenieros Industriales, Universidad Politécnica de Madrid, 28006 Madrid, Spain · Departamento de Física Teórica & IPARCOS, Universidad Complutense de Madrid, 28040 Madrid, Spain.

Markov Chain Monte Carlo Expectation-Maximization Generative Models Equilibrium

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Energy-Based Models (EBMs) provide an interpretable framework for generative modeling of scientific data, but poor Markov Chain Monte Carlo mixing often limits their reliability. We introduce a training algorithm based on Parallel Trajectory Tempering (PTT), which exploits the continuity of the optimization path to maintain equilibrium sampling throughout learning. This enables stable and fast training on highly multimodal and data-scarce scientific datasets. Combined with reservoir sampling and adaptive optimization, PTT has a computational cost comparable to Persistent Contrastive Divergence, making it a practical replacement for standard training methods. It also provides direct estimates of thermalization times, equilibrium samples from trained models, and accurate log-likelihoods at essentially no additional cost. Experiments on Restricted Boltzmann Machines show that PTT consistently outperforms existing EBM training approaches. On discrete tabular data, it also surpasses state-of-the-art deep generative models, yielding higher-quality samples and greater robustness to overfitting and limited data. Our results make equilibrium maximum-likelihood training of EBMs practical and computationally efficient.

Abstract

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