Fit Like You Sample: Sample-Efficient Generalized Score Matching from Fast Mixing Markov Chains
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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 an arbitrary Markov process with generator ℒ and an appropriately chosen generalized score matching loss that tries to fit 𝒪 p/p. If ℒ corresponds to a Markov process corresponding to a continuous version of simulated tempering, we show the corresponding generalized score matching loss is a Gaussian-convolution annealed score matching loss, akin to the one proposed in 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 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. This is the first result characterizing the benefits of annealing for score matching – a crucial component in more sophisticated score-based approaches like Song and Ermon 2019.
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