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.
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Training Discrete Deep Generative Models via Gapped Straight-Through Estimator
While deep generative models have succeeded in image processing, natural language processing, and reinforcement learning, training that involves discrete random variables remains challenging due to the high variance of its gradient estimation process. Monte Carlo is a common solution used in most variance reduction approaches. However, this involves time-consuming resampling and multiple function evaluations. We propose a Gapped Straight-Through (GST) estimator to reduce the variance without incurring resampling overhead. This estimator is inspired by the essential properties of Straight-Through Gumbel-Softmax. We determine these properties and show via an ablation study that they are essential. Experiments demonstrate that the proposed GST estimator enjoys better performance compared to strong baselines on two discrete deep generative modeling tasks, MNIST-VAE and ListOps.
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- Award ID(s):
- 1919452
- NSF-PAR ID:
- 10380235
- Editor(s):
- Chaudhuri, Kamalika and
- Date Published:
- Journal Name:
- Proceedings of Machine Learning Research
- Volume:
- 162
- ISSN:
- 2640-3498
- Page Range / eLocation ID:
- 6059-6073
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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