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

Lifted graphical models: a survey

Lifted graphical models provide a language for expressing dependencies between different types of entities, their attributes, and their diverse relations, as well as techniques for probabilistic reasoning in such multi-relational domains. In this survey, we review a general form for a lifted graphical model, a par-factor graph, and show how a number of existing statistical relational representations map to this formalism. We discuss inference algorithms, including lifted inference algorithms, that efficiently compute the answers to probabilistic queries over such models. We also review work in learning lifted graphical models from data. There is a growing need for statistical relational models (whether they go by that name or another), as we are inundated with data which is a mix of structured and unstructured, with entities and relations extracted in a noisy manner from text, and with the need to reason effectively with this data. We hope that this synthesis of ideas from many different research groups will provide an accessible starting point for new researchers in this expanding field

Lifted Bayesian filtering in multi-entity systems

This thesis focuses on Bayesian filtering for systems that consist of multiple, interacting entites (e.g. agents or objects), which can naturally be described by Multiset Rewriting Systems (MRSs). The main insight is that the state space that is underling an MRS exhibits a certain symmetry, which can be exploited to increase inference efficiency. We provide an efficient, lifted filtering algorithm, which is able to achieve a factorial reduction in space and time complexity, compared to conventional, ground filtering.Diese Arbeit betrachtet Bayes'sche Filter in Systemen, die aus mehreren, interagierenden Entitäten (z.B. Agenten oder Objekten) bestehen. Die Systemdynamik solcher Systeme kann auf natürliche Art durch Multiset Rewriting Systems (MRS) spezifiziert werden. Die wesentliche Erkenntnis ist, dass der Zustandraum Symmetrien aufweist, die ausgenutzt werden können, um die Effizienz der Inferenz zu erhöhen. Wir führen einen effizienten, gelifteten Filter-Algorithmus ein, dessen Zeit- und Platzkomplexität gegenüber dem grundierten Algorithmus um einen faktoriellen Faktor reduziert ist

Graphical models beyond standard settings: lifted decimation, labeling, and counting

With increasing complexity and growing problem sizes in AI and Machine Learning, inference and learning are still major issues in Probabilistic Graphical Models (PGMs). On the other hand, many problems are specified in such a way that symmetries arise from the underlying model structure. Exploiting these symmetries during inference, which is referred to as "lifted inference", has lead to significant efficiency gains. This thesis provides several enhanced versions of known algorithms that show to be liftable too and thereby applies lifting in "non-standard" settings. By doing so, the understanding of the applicability of lifted inference and lifting in general is extended. Among various other experiments, it is shown how lifted inference in combination with an innovative Web-based data harvesting pipeline is used to label author-paper-pairs with geographic information in online bibliographies. This results is a large-scale transnational bibliography containing affiliation information over time for roughly one million authors. Analyzing this dataset reveals the importance of understanding count data. Although counting is done literally everywhere, mainstream PGMs have widely been neglecting count data. In the case where the ranges of the random variables are defined over the natural numbers, crude approximations to the true distribution are often made by discretization or a Gaussian assumption. To handle count data, Poisson Dependency Networks (PDNs) are introduced which presents a new class of non-standard PGMs naturally handling count data

On lifted pagerank, kalman filter and towards lifted linear program solving

Lifted message passing algorithms exploit repeated structure within a given graphical model to answer queries efficiently. Given evidence, they construct a lifted network of supernodes and superpotentials corresponding to sets of nodes and potentials that are indistinguishable given the evidence. Recently, efficient algorithms were presented for updating the structure of an existing lifted network with incremental changes to the evidence. In the inference stage, however, current algorithms need to construct a separate lifted network for each evidence case and run a modified message passing algorithm on each lifted network separately. Consequently, symmetries across the inference tasks are not exploited. In this paper, we present a novel lifted message passing technique that exploits symmetries across multiple evidence cases. The benefits of this multi-evidence lifted inference are shown for several important AI tasks such as solving linear programs, computing personaliz ed PageRanks and Kalman filters via multi-evidence lifted Gaussian belief propagation

Query Answering in Probabilistic Data and Knowledge Bases

Probabilistic data and knowledge bases are becoming increasingly important in academia and industry. They are continuously extended with new data, powered by modern information extraction tools that associate probabilities with knowledge base facts. The state of the art to store and process such data is founded on probabilistic database systems, which are widely and successfully employed. Beyond all the success stories, however, such systems still lack the fundamental machinery to convey some of the valuable knowledge hidden in them to the end user, which limits their potential applications in practice. In particular, in their classical form, such systems are typically based on strong, unrealistic limitations, such as the closed-world assumption, the closed-domain assumption, the tuple-independence assumption, and the lack of commonsense knowledge. These limitations do not only lead to unwanted consequences, but also put such systems on weak footing in important tasks, querying answering being a very central one. In this thesis, we enhance probabilistic data and knowledge bases with more realistic data models, thereby allowing for better means for querying them. Building on the long endeavor of unifying logic and probability, we develop different rigorous semantics for probabilistic data and knowledge bases, analyze their computational properties and identify sources of (in)tractability and design practical scalable query answering algorithms whenever possible. To achieve this, the current work brings together some recent paradigms from logics, probabilistic inference, and database theory

On the Combination of Game-Theoretic Learning and Multi Model Adaptive Filters

This paper casts coordination of a team of robots within the framework of game theoretic learning algorithms. In particular a novel variant of fictitious play is proposed, by considering multi-model adaptive filters as a method to estimate other players’ strategies. The proposed algorithm can be used as a coordination mechanism between players when they should take decisions under uncertainty. Each player chooses an action after taking into account the actions of the other players and also the uncertainty. Uncertainty can occur either in terms of noisy observations or various types of other players. In addition, in contrast to other game-theoretic and heuristic algorithms for distributed optimisation, it is not necessary to find the optimal parameters a priori. Various parameter values can be used initially as inputs to different models. Therefore, the resulting decisions will be aggregate results of all the parameter values. Simulations are used to test the performance of the proposed methodology against other game-theoretic learning algorithms.</p