Variational Noise-Contrastive Estimation

TL;DR

This paper introduces variational noise-contrastive estimation (VNCE), a novel method for parameter estimation and posterior inference in unnormalised latent variable models, extending noise-contrastive estimation with variational techniques.

Contribution

VNCE combines NCE with variational lower bounds, enabling effective parameter estimation and inference in unnormalised models, a significant advancement over existing methods.

Findings

VNCE successfully estimates parameters in toy models.

VNCE effectively infers latent variables in complex models.

Application to real data demonstrates practical utility.

Abstract

Unnormalised latent variable models are a broad and flexible class of statistical models. However, learning their parameters from data is intractable, and few estimation techniques are currently available for such models. To increase the number of techniques in our arsenal, we propose variational noise-contrastive estimation (VNCE), building on NCE which is a method that only applies to unnormalised models. The core idea is to use a variational lower bound to the NCE objective function, which can be optimised in the same fashion as the evidence lower bound (ELBO) in standard variational inference (VI). We prove that VNCE can be used for both parameter estimation of unnormalised models and posterior inference of latent variables. The developed theory shows that VNCE has the same level of generality as standard VI, meaning that advances made there can be directly imported to the unnormalised setting. We validate VNCE on toy models and apply it to a realistic problem of estimating an undirected graphical model from incomplete data.