Energy Discrepancies: A Score-Independent Loss for Energy-Based Models

Tobias Schröder, Zijing Ou, Jen Ning Lim, Yingzhen Li, Sebastian Josef Vollmer, Andrew Duncan

Keywords: Energy-based models, statistical discrepancy, latent-variable model, density estimation

TL;DR: This paper proposes a new loss-function for energy-based modelling which can be evaluated without computing scores. We describe strong theoretical properties of the loss and demonstrate its efficacy for learning positive data distributions.

Abstract: Energy-based models are a simple yet powerful class of probabilistic models, but their widespread adoption has been limited by the computational burden of training them. We propose a novel loss function called Energy Discrepancy (ED) which does not rely on the computation of scores or expensive Markov chain Monte Carlo. We show that energy discrepancy approaches the explicit score matching and negative log-likelihood loss under different limits, effectively interpolating between both. Consequently, minimum energy discrepancy estimation overcomes the problem of nearsightedness encountered in score-based estimation methods, while also enjoying theoretical guarantees. Through numerical experiments, we demonstrate that ED learns low-dimensional data distributions faster and more accurately than explicit score matching or contrastive divergence. For high-dimensional image data, we describe how the manifold hypothesis puts limitations on our approach and demonstrate the effectiveness of energy discrepancy by training the energy-based model as a prior of a variational decoder model.

Submission Number: 8684