In this paper we consider the problem of learning undirected graphical models from data generated according to the Glauber dynamics. The Glauber dynamics is a Markov chain that sequentially updates individual nodes (variables) in a graphical model and it is frequently used to sample from the stationary distribution (to which it converges given sufficient time). Additionally, the Glauber dynamics is a natural dynamical model in a variety of settings. This work deviates from the standard formulation of graphical model learning in the literature, where one assumes access to i.i.d. samples from the distribution. Much of the research on graphical model learning has been directed towards finding algorithms with low computational cost. As the main result of this work, we establish that the problem of reconstructing binary pairwise graphical models is computationally tractable when we observe the Glauber dynamics. Specifically, we show that a binary pairwise graphical model on @math nodes with maximum degree @math can be learned in time @math , for a function @math , using nearly the information-theoretic minimum number of samples.
More broadly, a number of papers in the learning theory community have considered learning functions (or concepts) from examples generated by Markov chains, including @cite_41 @cite_15 @cite_20 @cite_22 . The present paper is similar in spirit to that of @cite_20 showing that it is relatively easy to learn DNF formulas from examples generated according to a random walk as compared to i.i.d. samples.
@cite_41 @cite_15 @cite_22 @cite_20
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Previous teaching models in the learning theory community have been batch models. That is, in these models the teacher has generated a single set of helpful examples to present to the learner. In this paper we present an interactive model in which the learner has the ability to ask queries as in the query learning model of Angluin. We show that this model is at least as powerful as previous teaching models. We also show that anything learnable with queries, even by a randomized learner, is teachable in our model. In all previous teaching models, all classes shown to be teachable are known to be efficiently learnable. An important concept class that is not known to be learnable is DNF formulas. We demonstrate the power of our approach by providing a deterministic teacher and learner for the class of DNF formulas. The learner makes only equivalence queries and all hypotheses are also DNF formulas. We consider a model of learning Boolean functions from examples generated by a uniform random walk on 0, 1 sup n . We give a polynomial time algorithm for learning decision trees and DNF formulas in this model. This is the first efficient algorithm for learning these classes in a natural passive learning model where the learner has no influence over the choice of examples used for learning. Using the recently developed framework of (2014), we show that under a natural assumption on the complexity of random K-SAT, learning DNF formulas is hard. Furthermore, the same assumption implies the hardness of various learning problems, including intersections of !(log(n)) halfspaces, agnostically learning conjunctions, as well as virtually all (distribution free) learning problems that were previously shown hard (under various complexity assumptions). Several Markov chain methods are available for sampling from a posterior distribution. Two important examples are the Gibbs sampler and the Metropolis algorithm. In addition, several strategies are available for constructing hybrid algorithms. This paper outlines some of the basic methods and strategies and discusses some related theoretical and practical issues. On the theoretical side, results from the theory of general state space Markov chains can be used to obtain convergence rates, laws of large numbers and central limit theorems for estimates obtained from Markov chain methods. These theoretical results can be used to guide the construction of more efficient algorithms. For the practical use of Markov chain methods, standard simulation methodology provides several variance reduction techniques and also gives guidance on the choice of sample size and allocation