A unifying view of contrastive learning, importance sampling, and bridge sampling for energy-based models

TL;DR

This paper unifies various estimation methods for energy-based models, clarifying their relationships and proposing new estimators to enhance efficiency, supported by MATLAB code for reproducibility.

Contribution

It provides a unified framework connecting NCE, RLR, MIS, and bridge sampling, revealing their equivalences and enabling new estimator development.

Findings

Methods are shown to be equivalent under certain conditions.

The unified framework clarifies the strengths of NCE.

New estimators can improve statistical and computational efficiency.

Abstract

In the last decades, energy-based models (EBMs) have become an important class of probabilistic models in which a component of the likelihood is intractable and therefore cannot be evaluated explicitly. Consequently, parameter estimation in EBMs is challenging for conventional inference methods. In this work, we provide a unified framework that connects noise contrastive estimation (NCE), reverse logistic regression (RLR), multiple importance sampling (MIS), and bridge sampling within the context of EBMs. We further show that these methods are equivalent under specific conditions. This unified perspective clarifies relationships among existing methods and enables the development of new estimators, with the potential to improve statistical and computational efficiency. Furthermore, this study helps elucidate the success of NCE in terms of its flexibility and robustness, while also identifying scenarios in which its performance can be further improved. Hence, rather than being a purely descriptive review, this work offers a unifying perspective and additional methodological contributions. The MATLAB code used in the numerical experiments is also made freely available to support the reproducibility of the results.