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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 for the training data, we instead fit the score function, obviating the need to evaluate the partition function. Though this estimator is known to be consistent, it's 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. 

We prove a new generalization bound that shows for any class of linear predictors in Gaussian space, the Rademacher complexity of the class and the training error under any continuous loss ℓ can control the test error under all Moreau envelopes of the loss ℓ . We use our finite-sample bound to directly recover the “optimistic rate” of Zhou et al. (2021) for linear regression with the square loss, which is known to be tight for minimal ℓ2-norm interpolation, but we also handle more general settings where the label is generated by a potentially misspecified multi-index model. The same argument can analyze noisy interpolation of max-margin classifiers through the squared hinge loss, and establishes consistency results in spiked-covariance settings. More generally, when the loss is only assumed to be Lipschitz, our bound effectively improves Talagrand’s well-known contraction lemma by a factor of two, and we prove uniform convergence of interpolators (Koehler et al. 2021) for all smooth, non-negative losses. Finally, we show that application of our generalization bound using localized Gaussian width will generally be sharp for empirical risk minimizers, establishing a non-asymptotic Moreau envelope theory 

Graphical models are powerful tools for modeling high-dimensional data, but learning graphical models in the presence of latent variables is well-known to be difficult. In this work we give new results for learning Restricted Boltzmann Machines, probably the most well-studied class of latent variable models. Our results are based on new connections to learning two-layer neural networks under ℓ∞ bounded input; for both problems, we give nearly optimal results under the conjectured hardness of sparse parity with noise. Using the connection between RBMs and feedforward networks, we also initiate the theoretical study of supervised RBMs [Hinton, 2012], a version of neural-network learning that couples distributional assumptions induced from the underlying graphical model with the architecture of the unknown function class. We then give an algorithm for learning a natural class of supervised RBMs with better runtime than what is possible for its related class of networks without distributional assumptions.