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Automorphism groups of graphical models and lifted variational inference
"... Using the theory of group action, we first introduce the concept of the automorphism group of an exponential family or a graphical model, thus formalizing the general notion of symmetry of a probabilistic model. This automorphism group provides a precise mathematical framework for lifted inference i ..."
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Using the theory of group action, we first introduce the concept of the automorphism group of an exponential family or a graphical model, thus formalizing the general notion of symmetry of a probabilistic model. This automorphism group provides a precise mathematical framework for lifted inference in the general exponential family. Its group action partitions the set of random variables and feature functions into equivalent classes (called orbits) having identical marginals and expectations. Then the inference problem is effectively reduced to that of computing marginals or expectations for each class, thus avoiding the need to deal with each individual variable or feature. We demonstrate the usefulness of this general framework in lifting two classes of variational approximation for MAP inference: local LP relaxation and local LP relaxation with cycle constraints; the latter yields the first lifted inference that operate on a bound tighter than local constraints. Initial experimental results demonstrate that lifted MAP inference with cycle constraints achieved the state of the art performance, obtaining much better objective function values than local approximation while remaining relatively efficient. 1
Lifted probabilistic inference
 In
, 2012
"... Abstract. Many AI problems arising in a wide variety of fields such as machine learning, semantic web, network communication, computer vision, and robotics can elegantly be encoded and solved using probabilistic graphical models. Often, however, we are facing inference problems with symmetries and r ..."
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Abstract. Many AI problems arising in a wide variety of fields such as machine learning, semantic web, network communication, computer vision, and robotics can elegantly be encoded and solved using probabilistic graphical models. Often, however, we are facing inference problems with symmetries and redundancies only implicitly captured in the graph structure and, hence, not exploitable by efficient inference approaches. A prominent example are probabilistic logical models that tackle a long standing goal of AI, namely unifying firstorder logic — capturing regularities and symmetries — and probability — capturing uncertainty. Although they often encode large, complex models using few rules only and, hence, symmetries and redundancies abound, inference in them was originally still at the propositional representation level and did not exploit symmetries. This paper is intended to give a (not necessarily complete) overview and invitation to the emerging field of lifted probabilistic inference, inference techniques that exploit these symmetries in graphical models in order to speed up inference, ultimately orders of magnitude. 1
On the Completeness of FirstOrder Knowledge Compilation for Lifted Probabilistic Inference
"... Probabilistic logics are receiving a lot of attention today because of their expressive power for knowledge representation and learning. However, this expressivity is detrimental to the tractability of inference, when done at the propositional level. To solve this problem, various lifted inference a ..."
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Cited by 19 (7 self)
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Probabilistic logics are receiving a lot of attention today because of their expressive power for knowledge representation and learning. However, this expressivity is detrimental to the tractability of inference, when done at the propositional level. To solve this problem, various lifted inference algorithms have been proposed that reason at the firstorder level, about groups of objects as a whole. Despite the existence of various lifted inference approaches, there are currently no completeness results about these algorithms. The key contribution of this paper is that we introduce a formal definition of lifted inference that allows us to reason about the completeness of lifted inference algorithms relative to a particular class of probabilistic models. We then show how to obtain a completeness result using a firstorder knowledge compilation approach for theories of formulae containing up to two logical variables. 1 Introduction and related work Probabilistic logic models build on firstorder logic to capture relational structure and on graphical
Markov chains on orbits of permutation groups
 In UAI2012
, 2012
"... We present a novel approach to detecting and utilizing symmetries in probabilistic graphical models with two main contributions. First, we present a scalable approach to computing generating sets of permutation groups representing the symmetries of graphical models. Second, we introduce orbital Mark ..."
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Cited by 18 (4 self)
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We present a novel approach to detecting and utilizing symmetries in probabilistic graphical models with two main contributions. First, we present a scalable approach to computing generating sets of permutation groups representing the symmetries of graphical models. Second, we introduce orbital Markov chains, a novel family of Markov chains leveraging model symmetries to reduce mixing times. We establish an insightful connection between model symmetries and rapid mixing of orbital Markov chains. Thus, we present the first lifted MCMC algorithm for probabilistic graphical models. Both analytical and empirical results demonstrate the effectiveness and efficiency of the approach. 1
RockIt: Exploiting Parallelism and Symmetry for MAP Inference in Statistical Relational Models
 In Proceedings of the 27th Conference on Artificial Intelligence (AAAI
"... ROCKIT is a maximum aposteriori (MAP) query engine for statistical relational models. MAP inference in graphical models is an optimization problem which can be compiled to integer linear programs (ILPs). We describe several advances in translating MAP queries to ILP instances and present the novel ..."
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Cited by 16 (5 self)
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ROCKIT is a maximum aposteriori (MAP) query engine for statistical relational models. MAP inference in graphical models is an optimization problem which can be compiled to integer linear programs (ILPs). We describe several advances in translating MAP queries to ILP instances and present the novel metaalgorithm cutting plane aggregation (CPA). CPA exploits local contextspecific symmetries and bundles up sets of linear constraints. The resulting counting constraints lead to more compact ILPs and make the symmetry of the ground model more explicit to stateoftheart ILP solvers. Moreover, ROCKIT parallelizes most parts of the MAP inference pipeline taking advantage of ubiquitous sharedmemory multicore architectures. We report on extensive experiments with Markov logic network (MLN) benchmarks showing that ROCKIT outperforms the stateoftheart systems ALCHEMY, MARKOV THEBEAST, and TUFFY both in terms of efficiency and quality of results.
On Lifting the Gibbs Sampling Algorithm
"... Firstorder probabilistic models combine the power of firstorder logic, the de facto tool for handling relational structure, with probabilistic graphical models, the de facto tool for handling uncertainty. Lifted probabilistic inference algorithms for them have been the subject of much recent resea ..."
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Cited by 15 (10 self)
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Firstorder probabilistic models combine the power of firstorder logic, the de facto tool for handling relational structure, with probabilistic graphical models, the de facto tool for handling uncertainty. Lifted probabilistic inference algorithms for them have been the subject of much recent research. The main idea in these algorithms is to improve the accuracy and scalability of existing graphical models’ inference algorithms by exploiting symmetry in the firstorder representation. In this paper, we consider blocked Gibbs sampling, an advanced MCMC scheme, and lift it to the firstorder level. We propose to achieve this by partitioning the firstorder atoms in the model into a set of disjoint clusters such that exact lifted inference is polynomial in each cluster given an assignment to all other atoms not in the cluster. We propose an approach for constructing the clusters and show how it can be used to trade accuracy with computational complexity in a principled manner. Our experimental evaluation shows that lifted Gibbs sampling is superior to the propositional algorithm in terms of accuracy, scalability and convergence. 1
Programming with Personalized PageRank: A Locally Groundable FirstOrder Probabilistic Logic
"... Many informationmanagement tasks (including classification, retrieval, information extraction, and information integration) can be formalized as inference in an appropriate probabilistic firstorder logic. However, most probabilistic firstorder logics are not efficient enough for realisticallysize ..."
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Cited by 15 (6 self)
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Many informationmanagement tasks (including classification, retrieval, information extraction, and information integration) can be formalized as inference in an appropriate probabilistic firstorder logic. However, most probabilistic firstorder logics are not efficient enough for realisticallysized instances of these tasks. One key problem is that queries are typically answered by “grounding ” the query— i.e., mapping it to a propositional representation, and then performing propositional inference—and with a large database of facts, groundings can be very large, making inference and learning computationally expensive. Here we present a firstorder probabilistic language which is wellsuited to approximate “local ” grounding: in particular, every query Q can be approximately grounded with a small graph. The language is an extension of stochastic logic programs where inference is performed by a variant of personalized PageRank. Experimentally, we show that the approach performs well on an entity resolution task, a classification task, and a joint inference task; that the cost of inference is independent of database size; and that speedup in learning is possible by multithreading.
Probabilistic Databases with MarkoViews
"... Most of the work on query evaluation in probabilistic databases has focused on the simple tupleindependent data model, where all tuples are independent random events. Several efficient query evaluation techniques exists in this setting, such as safe plans, algorithms based on OBDDs, treedecomposit ..."
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Cited by 12 (5 self)
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Most of the work on query evaluation in probabilistic databases has focused on the simple tupleindependent data model, where all tuples are independent random events. Several efficient query evaluation techniques exists in this setting, such as safe plans, algorithms based on OBDDs, treedecomposition and a variety of approximation algorithms. However, complex data analytics tasks often require complex correlations between tuples, and here query evaluation is significantly more expensive, or more restrictive. In this paper, we propose MVDB as a framework both for representing complex correlations and for efficient query evaluation. An MVDB specifies correlations by views, called MarkoViews, on the probabilistic relations and declaring the weights of the view’s outputs. An MVDB is a (very large) Markov Logic Network. We make two sets of contributions. First, we show that query evaluation on an MVDB is equivalent to evaluating a Union of Conjunctive Query(UCQ) over a tupleindependent database. The translation is exact (thus allowing the techniques developed for tuple independent databases to be carried over to MVDB), yet it is novel and quite nonobvious (some resulting probabilities may be negative!). This translation in itself though may not lead to much gain since the translated query gets complicated as we try to capture more correlations. Our second contribution is to propose a new query evaluation strategy that exploits offline compilation to speed up online query evaluation. Here we utilize and extend our prior work on compilation of UCQ. We validate experimentally our techniques on a large probabilistic database with MarkoViews inferred from the DBLP data. 1.
Advances in Lifted Importance Sampling
"... We consider lifted importance sampling (LIS), a previously proposed approximate inference algorithm for statistical relational learning (SRL) models. LIS achieves substantial variance reduction over conventional importance sampling by using various lifting rules that take advantage of the symmetry i ..."
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Cited by 10 (6 self)
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We consider lifted importance sampling (LIS), a previously proposed approximate inference algorithm for statistical relational learning (SRL) models. LIS achieves substantial variance reduction over conventional importance sampling by using various lifting rules that take advantage of the symmetry in the relational representation. However, it suffers from two drawbacks. First, it does not take advantage of some important symmetries in the relational representation and may exhibit needlessly high variance on models having these symmetries. Second, it uses an uninformative proposal distribution which adversely affects its accuracy. We propose two improvements to LIS that address these limitations. First, we identify a new symmetry in SRL models and define a lifting rule for taking advantage of this symmetry. The lifting rule reduces the variance of LIS. Second, we propose a new, structured approach for constructing and dynamically updating the proposal distribution via adaptive sampling. We demonstrate experimentally that our new, improved LIS algorithm is substantially more accurate than the LIS algorithm.
Lifted Linear Programming
"... Lifted inference approaches have rendered large, previously intractable probabilistic inference problems quickly solvable by handling whole sets of indistinguishable objects together. Triggered by this success, we show that another important AI technique is liftable, too, namely linear programming. ..."
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Cited by 9 (4 self)
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Lifted inference approaches have rendered large, previously intractable probabilistic inference problems quickly solvable by handling whole sets of indistinguishable objects together. Triggered by this success, we show that another important AI technique is liftable, too, namely linear programming. Intuitively, given a linear program (LP), we employ a lifted variant of Gaussian belief propagation (GaBP) to solve the systems of linear equations arising when running an interiorpoint method to solve the LP. However, this naïve solution cannot make use of standard solvers for linear equations and is doomed to construct lifted networks in each iteration of the interiorpoint method again, an operation that can itself be quite costly. To address both issues, we show how to read off an equivalent LP from the lifted GaBP computations that can be solved using any offtheshelf LP solver. We prove the correctness of this compilation approac and experimentally demonstrate that it can greatly reduce the cost of solving LPs. 1