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

2025-10-02

AI Summary

This work addresses the lack of theoretical convergence guarantees for Persistent Contrastive Divergence (PCD) in energy-based models (EBMs) under maximum likelihood estimation (MLE) with unnormalized densities. We propose the first continuous-time PCD framework, modeling parameter updates and sampling jointly as a coupled multiscale stochastic differential equation (SDE) system. Through mean-field analysis, we derive a time-uniform convergence error bound, thereby establishing the first explicit, bounded approximation guarantee for MLE solutions under PCD. Technically, we introduce the S-ROCK numerical scheme for efficient and stable integration, integrating continuous-time modeling with multiscale dynamical analysis. Experiments demonstrate that our method significantly improves both convergence speed and generalization performance of EBMs while maintaining training stability.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Search and Optimization: Mixed Discrete/Continuous Search
• Reasoning under Uncertainty: Relational Probabilistic Models

Application Category

• Web Mining and Content Analysis: Models for Web evolution
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

We propose a continuous-time formulation of persistent contrastive divergence (PCD) for maximum likelihood estimation (MLE) of unnormalised densities. Our approach expresses PCD as a coupled, multiscale system of stochastic differential equations (SDEs), which perform optimisation of the parameter and sampling of the associated parametrised density, simultaneously. From this novel formulation, we are able to derive explicit bounds for the error between the PCD iterates and the MLE solution for the model parameter. This is made possible by deriving uniform-in-time (UiT) bounds for the difference in moments between the multiscale system and the averaged regime. An efficient implementation of the continuous-time scheme is introduced, leveraging a class of explicit, stable intregators, stochastic orthogonal Runge-Kutta Chebyshev (S-ROCK), for which we provide explicit error estimates in the long-time regime. This leads to a novel method for training energy-based models (EBMs) with explicit error guarantees.

Problem

• Proposes continuous-time PCD for maximum likelihood estimation of unnormalized densities
• Derives uniform-in-time convergence bounds between PCD iterates and MLE solution
• Develops efficient training method for energy-based models with explicit error guarantees

Innovation

• Continuous-time PCD formulation using coupled SDEs
• Uniform-in-time convergence bounds for MLE error
• Efficient implementation with S-ROCK integrators