Statistical Efficiency of Score Matching: The View from Isoperimetry

TL;DR

This paper investigates the statistical efficiency of score matching for training energy-based models, revealing its dependence on the isoperimetric properties of the target distribution and comparing it to maximum likelihood.

Contribution

It establishes a theoretical connection between score matching efficiency and isoperimetric constants, providing conditions under which score matching is nearly optimal or substantially less efficient.

Findings

Score matching is efficient for distributions with small isoperimetric constants.

Large isoperimetric constants lead to lower efficiency of score matching compared to MLE.

The results extend to both finite sample and asymptotic regimes, with parallels in discrete settings.

Abstract

Deep generative models parametrized up to a normalizing constant (e.g. energy-based models) are difficult to train by maximizing the likelihood of the data because the likelihood and/or gradients thereof cannot be explicitly or efficiently written down. Score matching is a training method, whereby instead of fitting the likelihood $\log p(x)$ for the training data, we instead fit the score function $\nabla_x \log p(x)$ -- obviating the need to evaluate the partition function. Though this estimator is known to be consistent, its unclear whether (and when) its statistical efficiency is comparable to that of maximum likelihood -- which is known to be (asymptotically) optimal. We initiate this line of inquiry in this paper, and show a tight connection between statistical efficiency of score matching and the isoperimetric properties of the distribution being estimated -- i.e. the Poincar\'e, log-Sobolev and isoperimetric constant -- quantities which govern the mixing time of Markov processes like Langevin dynamics. Roughly, we show that the score matching estimator is statistically comparable to the maximum likelihood when the distribution has a small isoperimetric constant. Conversely, if the distribution has a large isoperimetric constant -- even for simple families of distributions like exponential families with rich enough sufficient statistics -- score matching will be substantially less efficient than maximum likelihood. We suitably formalize these results both in the finite sample regime, and in the asymptotic regime. Finally, we identify a direct parallel in the discrete setting, where we connect the statistical properties of pseudolikelihood estimation with approximate tensorization of entropy and the Glauber dynamics.