Probabilistic graphical models are powerful tools for analyzing constrained, continuous domains. However, finding most-probable explanations (MPEs) in these models can be computationally expensive. In this paper, we improve the scalability of MPE inference in a class of graphical models with piecewise-linear and piecewise-quadratic dependencies and linear constraints over continuous domains. We derive algorithms based on a consensus-optimization framework and demonstrate their superior performance over state of the art. We show empirically that in a large-scale voter-preference modeling problem our algorithms scale linearly in the number of dependencies and constraints.

A fundamental challenge in developing high-impact machine learning technologies is balancing the need to model rich, structured domains with the ability to scale to big data. Many important problem areas are both richly structured and large scale, from social and biological networks, to knowledge graphs and the Web, to images, video, and natural language. In this paper, we introduce two new formalisms for modeling structured data, and show that they can both capture rich structure and scale to big data. The first, hinge-loss Markov random fields (HL-MRFs), is a new kind of probabilistic graphical model that generalizes different approaches to convex inference. We unite three approaches from the randomized algorithms, probabilistic graphical models, and fuzzy logic communities, showing that all three lead to the same inference objective. We then define HL-MRFs by generalizing this unified objective. The second new formalism, probabilistic soft logic (PSL), is a probabilistic programming language that makes HL-MRFs easy to define using a syntax based on first-order logic. We introduce an algorithm for inferring most-probable variable assignments (MAP inference) that is much more scalable than general-purpose convex optimization methods, because it uses message passing to take advantage of sparse dependency structures. We then show how to learn the parameters of HL-MRFs. The learned HL-MRFs are as accurate as analogous discrete models, but much more scalable. Together, these algorithms enable HL-MRFs and PSL to model rich, structured data at scales not previously possible.

Probabilistic graphical models are powerful tools for analyzing constrained, continuous domains. However, finding most-probable explanations (MPEs) in these models can be computationally expensive. In this paper, we improve the scalability of MPE inference in a class of graphical models with piecewise-linear and piecewise-quadratic dependencies and linear constraints over continuous domains. We derive algorithms based on a consensus-optimization framework and demonstrate their superior performance over state of the art. We show empirically that in a large-scale voter-preference modeling problem our algorithms scale linearly in the number of dependencies and constraints.

Continuous Markov random fields are a general formalism to model joint probability distributions over events with continuous outcomes. We prove that marginal computation for constrained continuous MRFs is #P-hard in general and present a polynomial-time approximation scheme under mild assumptions on the structure of the random field. Moreover, we introduce a sampling algorithm to compute marginal distributions and develop novel techniques to increase its efficiency. Continuous MRFs are a general purpose probabilistic modeling tool and we demonstrate how they can be applied to statistical relational learning. On the problem of collective classification, we evaluate our algorithm and show that the standard deviation of marginals serves as a useful measure of confidence.