First-Order Decomposition Trees

Lifting attempts to speed up probabilistic inference by exploiting symmetries in the model. Exact lifted inference methods, like their propositional counterparts, work by recursively decomposing the model and the problem. In the propositional case, there exist formal structures, such as decomposition trees (dtrees), that represent such a decomposition and allow us to determine the complexity of inference a priori. However, there is currently no equivalent structure nor analogous complexity results for lifted inference. In this paper, we introduce FO-dtrees, which upgrade propositional dtrees to the first-order level. We show how these trees can characterize a lifted inference solution for a probabilistic logical model (in terms of a sequence of lifted operations), and make a theoretical analysis of the complexity of lifted inference in terms of the novel notion of lifted width for the tree

Symmetry-Aware Marginal Density Estimation

The Rao-Blackwell theorem is utilized to analyze and improve the scalability of inference in large probabilistic models that exhibit symmetries. A novel marginal density estimator is introduced and shown both analytically and empirically to outperform standard estimators by several orders of magnitude. The developed theory and algorithms apply to a broad class of probabilistic models including statistical relational models considered not susceptible to lifted probabilistic inference.Comment: To appear in proceedings of AAAI 201

Dimension Reduction via Colour Refinement

Colour refinement is a basic algorithmic routine for graph isomorphism testing, appearing as a subroutine in almost all practical isomorphism solvers. It partitions the vertices of a graph into "colour classes" in such a way that all vertices in the same colour class have the same number of neighbours in every colour class. Tinhofer (Disc. App. Math., 1991), Ramana, Scheinerman, and Ullman (Disc. Math., 1994) and Godsil (Lin. Alg. and its App., 1997) established a tight correspondence between colour refinement and fractional isomorphisms of graphs, which are solutions to the LP relaxation of a natural ILP formulation of graph isomorphism. We introduce a version of colour refinement for matrices and extend existing quasilinear algorithms for computing the colour classes. Then we generalise the correspondence between colour refinement and fractional automorphisms and develop a theory of fractional automorphisms and isomorphisms of matrices. We apply our results to reduce the dimensions of systems of linear equations and linear programs. Specifically, we show that any given LP L can efficiently be transformed into a (potentially) smaller LP L' whose number of variables and constraints is the number of colour classes of the colour refinement algorithm, applied to a matrix associated with the LP. The transformation is such that we can easily (by a linear mapping) map both feasible and optimal solutions back and forth between the two LPs. We demonstrate empirically that colour refinement can indeed greatly reduce the cost of solving linear programs

Plan Synthesis for Knowledge and Action Bases

We study plan synthesis for a variant of Knowledge and Action Bases (KABs), a rich, dynamic framework, where states are description logic (DL) knowledge bases (KBs) whose extensional part is manipulated by actions that possibly introduce new objects from an infinite domain. We show that plan existence over KABs is undecidable even under severe restrictions. We then focus on state-bounded KABs, a class for which plan existence is decidable, and provide sound and complete plan synthesis algorithms, which combine techniques based on standard planning, DL query answering, and finite-state abstraction. All results hold for any DL with decidable query answering. We finally show that for lightweight DLs, plan synthesis can be compiled into standard ADL planning

Graphical Models and Symmetries : Loopy Belief Propagation Approaches

Whenever a person or an automated system has to reason in uncertain domains, probability theory is necessary. Probabilistic graphical models allow us to build statistical models that capture complex dependencies between random variables. Inference in these models, however, can easily become intractable. Typical ways to address this scaling issue are inference by approximate message-passing, stochastic gradients, and MapReduce, among others. Exploiting the symmetries of graphical models, however, has not yet been considered for scaling statistical machine learning applications. One instance of graphical models that are inherently symmetric are statistical relational models. These have recently gained attraction within the machine learning and AI communities and combine probability theory with first-order logic, thereby allowing for an efficient representation of structured relational domains. The provided formalisms to compactly represent complex real-world domains enable us to effectively describe large problem instances. Inference within and training of graphical models, however, have not been able to keep pace with the increased representational power. This thesis tackles two major aspects of graphical models and shows that both inference and training can indeed benefit from exploiting symmetries. It first deals with efficient inference exploiting symmetries in graphical models for various query types. We introduce lifted loopy belief propagation (lifted LBP), the first lifted parallel inference approach for relational as well as propositional graphical models. Lifted LBP can effectively speed up marginal inference, but cannot straightforwardly be applied to other types of queries. Thus we also demonstrate efficient lifted algorithms for MAP inference and higher order marginals, as well as the efficient handling of multiple inference tasks. Then we turn to the training of graphical models and introduce the first lifted online training for relational models. Our training procedure and the MapReduce lifting for loopy belief propagation combine lifting with the traditional statistical approaches to scaling, thereby bridging the gap between statistical relational learning and traditional statistical machine learning