PAC-learning bounded tree-width Graphical Models

We show that the class of strongly connected graphical models with treewidth at most k can be properly efficiently PAC-learnt with respect to the Kullback-Leibler Divergence. Previous approaches to this problem, such as those of Chow ([1]), and Ho gen ([7]) have shown that this class is PAC-learnable by reducing it to a combinatorial optimization problem. However, for k > 1, this problem is NP-complete ([15]), and so unless P=NP, these approaches will take exponential amounts of time. Our approach differs significantly from these, in that it first attempts to find approximate conditional independencies by solving (polynomially many) submodular optimization problems, and then using a dynamic programming formulation to combine the approximate conditional independence information to derive a graphical model with underlying graph of the tree-width specified. This gives us an efficient (polynomial time in the number of random variables) PAC-learning algorithm which requires only polynomial number of samples of the true distribution, and only polynomial running time.

A submodular-supermodular procedure with applications to discriminative structure learning

In this paper, we present an algorithm for minimizing the difference between two submodular functions using a variational framework which is based on (an extension of) the concave-convex procedure [17]. Because several commonly used metrics in machine learning, like mutual information and conditional mutual information, are submodular, the problem of minimizing the difference of two submodular problems arises naturally in many machine learning applications. Two such applications are learning discriminatively structured graphical models and feature selection under computational complexity constraints. A commonly used metric for measuring discriminative capacity is the EAR measure which is the difference between two conditional mutual information terms. Feature selection taking complexity considerations into account also fall into this framework because both the information that a set of features provide and the cost of computing and using the features can be modeled as submodular functions. This problem is NP-hard, and we give a polynomial time heuristic for it. We also present results on synthetic data to show that classifiers based on discriminative graphical models using this algorithm can significantly outperform classifiers based on generative graphical models.