Fit Like You Sample: Sample-Efficient Generalized Score Matching from Fast Mixing Diffusions

• URL: http://arxiv.org/abs/2306.09332v3
• Date: Wed, 13 Dec 2023 16:32:49 GMT
• Title: Fit Like You Sample: Sample-Efficient Generalized Score Matching from Fast Mixing Diffusions
• Authors: Yilong Qin, Andrej Risteski
• Abstract summary: We show a close connection between the mixing time of a broad class of Markov processes with generator $mathcalL$. We 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.
• Score: 29.488555741982015
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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 Poincar\'e 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 Poincar\'e 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 Poincar\'e constant-based lower bounds of the basic score matching loss shown in Koehler et al. 2022.

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