Fit Like You Sample: Sample-Efficient Generalized Score Matching from Fast Mixing Diffusions
Yilong Qin, Andrej Risteski
TL;DR
This work addresses the inefficiency of score matching for learning complex multimodal distributions by connecting the statistical performance of generalized score matching to the mixing properties of underlying Markov diffusions. By framing GSM losses through an operator that preconditions the score, the authors show how diffusion preconditioning, lifting, and continuous annealing translate into more favorable sample complexity, closely tied to the Poincaré constant of the diffusion. They develop a general theoretical framework that bounds GSM performance in terms of the diffusion’s $C_P$, and prove a polynomial-time, annealed GSM estimator for finite mixtures of Gaussians with shared covariance using Continuously Tempered Langevin Dynamics (CTLD). The results formally demonstrate the statistical benefits of annealing for GSM and provide a path toward scalable, sample-efficient density estimation for multimodal data, with implications for diffusion-based generative modeling and energy-based learning.
Abstract
Score matching is an approach to learning probability distributions parametrized up to a constant of proportionality (e.g. Energy-Based Models). The idea is to fit the score of the distribution, rather than the likelihood, thus avoiding the need to evaluate the constant of proportionality. While there's a clear algorithmic benefit, the statistical "cost'' can be steep: recent work by Koehler et al. 2022 showed that for distributions that have poor isoperimetric properties (a large Poincaré or log-Sobolev constant), score matching is substantially statistically less efficient than maximum likelihood. However, many natural realistic distributions, e.g. multimodal distributions as simple as a mixture of two Gaussians in one dimension -- have a poor Poincaré constant. In this paper, we show a close connection between the mixing time of a broad class of Markov processes with generator $\mathcal{L}$ and an appropriately chosen generalized score matching loss that tries to fit $\frac{\mathcal{O} p}{p}$. This allows us to adapt techniques to speed up Markov chains to construct better score-matching losses. In particular, ``preconditioning'' the diffusion can be translated to an appropriate ``preconditioning'' of the score loss. Lifting the chain by adding a temperature like in simulated tempering can be shown to result in a Gaussian-convolution annealed score matching loss, similar to Song and Ermon, 2019. Moreover, we show that if the distribution being learned is a finite mixture of Gaussians in $d$ dimensions with a shared covariance, the sample complexity of annealed score matching is polynomial in the ambient dimension, the diameter of the means, and the smallest and largest eigenvalues of the covariance -- obviating the Poincaré constant-based lower bounds of the basic score matching loss shown in Koehler et al. 2022.