Nicholas Ruozzi

Message-Passing Algorithms: Reparameterizations and Splittings

The max-product algorithm, a local message-passing scheme that attempts to compute the most probable assignment (MAP) of a given probability distribution, has been successfully employed as a method of approximate inference for applications arising in coding theory, computer vision, and machine learning. However, the max-product algorithm is not guaranteed to converge, and if it does, it is not guaranteed to recover the MAP assignment. Alternative convergent message-passing schemes have been proposed to overcome these difficulties. This paper provides a systematic study of such message-passing algorithms that extends the known results by exhibiting new sufficient conditions for convergence to local and/or global optima, providing a combinatorial characterization of these optima based on graph covers, and describing a new convergent and correct message-passing algorithm whose derivation unifies many of the known convergent message-passing algorithms. While convergent and correct message-passing algorithms represent a step forward in the analysis of max-product style message-passing algorithms, the conditions needed to guarantee convergence to a global optimum can be too restrictive in both theory and practice. This limitation of convergent and correct message-passing schemes is characterized by graph covers and illustrated by example.

Graph covers and quadratic minimization

We formulate a new approach to understanding the behavior of the min-sum algorithm by exploiting the properties of graph covers. First, we present a new, natural characterization of scaled diagonally dominant matrices in terms of graph covers; this result motivates our approach because scaled diagonal dominance is a known sufficient condition for the convergence of min-sum in the case of quadratic minimization. We use our understanding of graph covers to characterize the periodic behavior of the min-sum algorithm on a single cycle. Lastly, we explain how to extend the single cycle results to understand the 2-periodic behavior of min-sum for general pairwise MRFs. Some of our techniques apply more broadly, and we believe that by capturing the notion of indistinguishability, graph covers represent a valuable tool for understanding the abilities and limitations of general message-passing algorithms.

Applications of metric coinduction

Metric coinduction is a form of coinduction that can be used to establish properties of objects constructed as a limit of finite approximations. One can prove a coinduction step showing that some property is preserved by one step of the approximation process, then automatically infer by the coinduction principle that the property holds of the limit object. This can often be used to avoid complicated analytic arguments involving limits and convergence, replacing them with simpler algebraic arguments. This paper examines the application of this principle in a variety of areas, including infinite streams, Markov chains, Markov decision processes, and non-well-founded sets. These results point to the usefulness of coinduction as a general proof technique.

s-t paths using the min-sum algorithm

Solving the distributed shortest path problem has important applications in the theory of distributed systems, most notably routing. In this paper, we provide and prove the convergence of a min-sum algorithm to compute the shortest path between two nodes in a graph with positive edge weights. Unlike the standard distributed shortest path algorithms, the rate of convergence depends on the weight of the minimal path and not necessarily the number of nodes in the network.

Applications of Metric Coinduction

Metric coinduction is a form of coinduction that can be used to establish properties of objects constructed as a limit of finite approximations. One can prove a coinduction step showing that some property is preserved by one step of the approximation process, then automatically infer by the coinduction principle that the property holds of the limit object. This can often be used to avoid complicated analytic arguments involving limits and convergence, replacing them with simpler algebraic arguments. This paper examines the application of this principle in a variety of areas, including infinite streams, Markov chains, Markov decision processes, and non-well-founded sets. These results point to the usefulness of coinduction as a general proof technique.

Unconstrained minimization of quadratic functions via min-sum

Gaussian belief propagation is an iterative algorithm for computing the mean of a multivariate Gaussian distribution. Equivalently, the min-sum algorithm can be used to compute the minimum of a multivariate positive definite quadratic function. Although simple sufficient conditions that guarantee the convergence and correctness of these algorithms are known, the algorithms may fail to converge to the correct solution even when restricted to only positive definite quadratic functions. In this work, we propose a novel change to the typical factorization used in GaBP that allows us to construct a variant of GaBP that can solve the minimization problem for arbitrary positive semidefinite matrices while still preserving the distributed message passing nature of GaBP. We prove that the new factorization avoids the major pitfalls of the standard factorization, and we demonstrate empirically that the algorithm can be used to solve problems for which the standard GaBP algorithm would have failed. As quadratic minimization is equivalent to solving a system of linear equations, this work can be applied to solve large positive semidefinite linear systems in many application areas.

Convergent message-passing algorithms in the presence of erasures

We examine how to design convergent and correct message-passing schemes, similar to the min-sum algorithm, for maximum a posteriori (MAP) estimation in the case that the messages passed between two nodes of the network may never be delivered. The proposed solution creates a new, but equivalent, graphical model over which the convergence of a specific message-passing algorithm is guaranteed. We then show that the messages passed on this new model can be reduced to message passing over the original model if we allow some additional state at each node of the network.

Efficient Inference for Untied MLNs

We address the problem of scaling up localsearch or sampling-based inference in Markov logic networks (MLNs) that have large shared substructures but no (or few) tied weights. Such untied MLNs are ubiquitous in practical applications. However, they have very few symmetries, and as a result lifted inference algorithms–the dominant approach for scaling up inference–perform poorly on them. The key idea in our approach is to reduce the hard, time-consuming sub-task in sampling algorithms, computing the sum of weights of features that satisfy a full assignment, to the problem of computing a set of partition functions of graphical models, each defined over the logical variables in a first-order formula. The importance of this reduction is that when the treewidth of all the graphical models is small, it yields an order of magnitude speedup. When the treewidth is large, we propose an over-symmetric approximation and experimentally demonstrate that it is both fast and accurate.