Abstract. We propose a simple approach to combining first-order logic and probabilistic graphical models in a single representation. A Markov logic network (MLN) is a first-order knowledge base with a weight attached to each formula (or clause). Together with a set of constants representing objects in the domain, it specifies a ground Markov network containing one feature for each possible grounding of a first-order formula in the KB, with the corresponding weight. Inference in MLNs is performed by MCMC over the minimal subset of the ground network required for answering the query. Weights are efficiently learned from relational databases by iteratively optimizing a pseudo-likelihood measure. Optionally, additional clauses are learned using inductive logic programming techniques. Experiments with a real-world database and knowledge base in a university domain illustrate the promise of this approach.

Many machine learning applications require a combination of probability and first-order logic. Markov logic networks (MLNs) accomplish this by attaching weights to first-order clauses, and viewing these as templates for features of Markov networks. Model parameters (i.e., clause weights) can be learned by maximizing the likelihood of a relational database, but this can be quite costly and lead to suboptimal results for any given prediction task. In this paper we propose a discriminative approach to training MLNs, one which optimizes the conditional likelihood of the query predicates given the evidence ones, rather than the joint likelihood of all predicates. We extend Collins’s (2002) voted perceptron algorithm for HMMs to MLNs by replacing the Viterbi algorithm with a weighted satisfiability solver. Experiments on entity resolution and link prediction tasks show the advantages of this approach compared to generative MLN training, as well as compared to purely probabilistic and purely logical approaches.

Entity resolution is the problem of determining which records in a database refer to the same entities, and is a crucial and expensive step in the data mining process. Interest in it has grown rapidly in recent years, and many approaches have been proposed. However, they tend to address only isolated aspects of the problem, and are often ad hoc. This paper proposes a well-founded, integrated solution to the entity resolution problem based on Markov logic. Markov logic combines first-order logic and probabilistic graphical models by attaching weights to first-order formulas, and viewing them as templates for features of Markov networks. We show how a number of previous approaches can be formulated and seamlessly combined in Markov logic, and how the resulting learning and inference problems can be solved efficiently. Experiments on two citation databases show the utility of this approach, and evaluate the contribution of the different components. 1

Abstract. Markov logic networks (MLNs) combine Markov networks and first-order logic, and are a powerful and increasingly popular representation for statistical relational learning. The state-of-the-art method for discriminative learning of MLN weights is the voted perceptron algorithm, which is essentially gradient descent with an MPE approximation to the expected sufficient statistics (true clause counts). Unfortunately, these can vary widely between clauses, causing the learning problem to be highly ill-conditioned, and making gradient descent very slow. In this paper, we explore several alternatives, from per-weight learning rates to second-order methods. In particular, we focus on two approaches that avoid computing the partition function: diagonal Newton and scaled conjugate gradient. In experiments on standard SRL datasets, we obtain order-of-magnitude speedups, or more accurate models given comparable learning times. 1

Machine learning approaches to coreference resolution are typically supervised, and require expensive labeled data. Some unsupervised approaches have been proposed (e.g., Haghighi and Klein (2007)), but they are less accurate. In this paper, we present the first unsupervised approach that is competitive with supervised ones. This is made possible by performing joint inference across mentions, in contrast to the pairwise classification typically used in supervised methods, and by using Markov logic as a representation language, which enables us to easily express relations like apposition and predicate nominals. On MUC and ACE datasets, our model outperforms Haghigi and Klein’s one using only a fraction of the training data, and often matches or exceeds the accuracy of state-of-the-art supervised models. 1

Propositionalization of a first-order theory followed by satisfiability testing has proved to be a remarkably efficient approach to inference in relational domains such as planning (Kautz & Selman 1996) and verification (Jackson 2000). More recently, weighted satisfiability solvers have been used successfully for MPE inference in statistical relational learners (Singla & Domingos 2005). However, fully instantiating a finite first-order theory requires memory on the order of the number of constants raised to the arity of the clauses, which significantly limits the size of domains it can be applied to. In this paper we propose LazySAT, a variation of the Walk-SAT solver that avoids this blowup by taking advantage of the extreme sparseness that is typical of relational domains (i.e., only a small fraction of ground atoms are true, and most clauses are trivially satisfied). Experiments on entity resolution and planning problems show that LazySAT reduces memory usage by orders of magnitude compared to Walk-SAT, while taking comparable time to run and producing the same solutions.

Many real-world problems are characterized by complex relational structure, which can be succinctly represented in firstorder logic. However, many relational inference algorithms proceed by first fully instantiating the first-order theory and then working at the propositional level. The applicability of such approaches is severely limited by the exponential time and memory cost of propositionalization. Singla and Domingos (2006) addressed this by developing a “lazy ” version of the WalkSAT algorithm, which grounds atoms and clauses only as needed. In this paper we generalize their ideas to a much broader class of algorithms, including other types of SAT solvers and probabilistic inference methods like MCMC. Lazy inference is potentially applicable whenever variables and functions have default values (i.e., a value that is much more frequent than the others). In relational domains, the default is false for atoms and true for clauses. We illustrate our framework by applying it to MC-SAT, a state-of-the-art MCMC algorithm. Experiments on a number of real-world domains show that lazy inference reduces both space and time by several orders of magnitude, making probabilistic relational inference applicable in previously infeasible domains.

Markov logic networks (MLNs) combine first-order logic and Markov networks, allowing us to handle the complexity and uncertainty of real-world problems in a single consistent framework. However, in MLNs all variables and features are discrete, while most real-world applications also contain continuous ones. In this paper we introduce hybrid MLNs, in which continuous properties (e.g., the distance between two objects) and functions over them can appear as features. Hybrid MLNs have all distributions in the exponential family as special cases (e.g., multivariate Gaussians), and allow much more compact modeling of non-i.i.d. data than propositional representations like hybrid Bayesian networks. We also introduce inference algorithms for hybrid MLNs, by extending the MaxWalkSAT and MC-SAT algorithms to continuous domains. Experiments in a mobile robot mapping domain—involving joint classification, clustering and regression—illustrate the power of hybrid MLNs as a modeling language, and the accuracy and efficiency of the inference algorithms.

Intelligent agents must be able to handle the complexity and uncertainty of the real world. Logical AI has focused mainly on the former, and statistical AI on the latter. Markov logic combines the two by attaching weights to first-order formulas and viewing them as templates for features of Markov networks. Inference algorithms for Markov logic draw on ideas from satisfiability, Markov chain Monte Carlo and knowledge-based model construction. Learning algorithms are based on the voted perceptron, pseudo-likelihood and inductive logic programming. Markov logic has been successfully applied to problems in entity resolution, link prediction, information extraction and others, and is the basis of the open-source Alchemy system.

In this paper we consider the problem of finding minimal models of a set of propositional clauses. This problem has many applications: for instance, finding minimal-length plans, minimal diagnosis, and solving many other minimization and maximization problems. We have adapted the Davis-Putnam algorithm (that finds generic models of a set of clauses) in order to find minimal models. Experiments have been performed to determine which heuristics can be useful, and to decide in which cases the problem is hard and in which cases it is easy. A comparison with a local search algorithm is also shown.