Learning Hard-Constrained Models with One Sample

We consider the problem of estimating the parameters of a Markov Random Field with hard-constraints using a single sample. As our main running examples, we use the k-SAT and the proper coloring models, as well as general H-coloring models; for all of these we obtain both positive and negative results. In contrast to the soft-constrained case, we show in particular that single-sample estimation is not always possible, and that the existence of an estimator is related to the existence of non-satisfiable instances. Our algorithms are based on the pseudo-likelihood estimator. We show variance bounds for this estimator using coupling techniques inspired, in the case of k-SAT, by Moitra's sampling algorithm (JACM, 2019); our positive results for colorings build on this new coupling approach. For q-colorings on graphs with maximum degree d, we give a linear-time estimator exists when q > d + 1, whereas the problem is non-identifiable when q <= d + 1. For general H-colorings, we show that standard conditions that guarantee sampling, such as Dobrushin's condition, are insufficient for one-sample learning; on the positive side, we provide a general condition that is sufficient to guarantee linear-time learning and obtain applications for proper colorings and permissive models. For the k-SAT model on formulas with maximum degree d, we provide a linear-time estimator when k greater than or similar to 6.45 log d, whereas the problem becomes non-identifiable when k less than or similar to log d.