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Mark Bartlett

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5 papers

2 author rows

JAIR Journal 2017 Journal Article

Bayesian Network Structure Learning with Integer Programming: Polytopes, Facets and Complexity

James Cussens
Matti Järvisalo
Janne H. Korhonen
Mark Bartlett

The challenging task of learning structures of probabilistic graphical models is an important problem within modern AI research. Recent years have witnessed several major algorithmic advances in structure learning for Bayesian networks - arguably the most central class of graphical models - especially in what is known as the score-based setting. A successful generic approach to optimal Bayesian network structure learning (BNSL), based on integer programming (IP), is implemented in the GOBNILP system. Despite the recent algorithmic advances, current understanding of foundational aspects underlying the IP based approach to BNSL is still somewhat lacking. Understanding fundamental aspects of cutting planes and the related separation problem is important not only from a purely theoretical perspective, but also since it holds out the promise of further improving the efficiency of state-of-the-art approaches to solving BNSL exactly. In this paper, we make several theoretical contributions towards these goals: (i) we study the computational complexity of the separation problem, proving that the problem is NP-hard; (ii) we formalise and analyse the relationship between three key polytopes underlying the IP-based approach to BNSL; (iii) we study the facets of the three polytopes both from the theoretical and practical perspective, providing, via exhaustive computation, a complete enumeration of facets for low-dimensional family-variable polytopes; and, furthermore, (iv) we establish a tight connection of the BNSL problem to the acyclic subgraph problem.

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IJCAI Conference 2017 Conference Paper

Bayesian Network Structure Learning with Integer Programming: Polytopes, Facets and Complexity (Extended Abstract)

James Cussens
Matti Järvisalo
Janne H. Korhonen
Mark Bartlett

Developing accurate algorithms for learning structures of probabilistic graphical models is an important problem within modern AI research. Here we focus on score-based structure learning for Bayesian networks as arguably the most central class of graphical models. A successful generic approach to optimal Bayesian network structure learning (BNSL), based on integer programming (IP), is implemented in the Gobnilp system. Despite the recent algorithmic advances, current understanding of foundational aspects underlying the IP based approach to BNSL is still somewhat lacking. In this paper, we provide theoretical contributions towards understanding fundamental aspects of cutting planes and the related separation problem in this context, ranging from NP-hardness results to analysis of polytopes and the related facets in connection to BNSL.

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AIJ Journal 2017 Journal Article

Integer Linear Programming for the Bayesian network structure learning problem

Mark Bartlett
James Cussens

Bayesian networks are a commonly used method of representing conditional probability relationships between a set of variables in the form of a directed acyclic graph (DAG). Determination of the DAG which best explains observed data is an NP-hard problem [1]. This problem can be stated as a constrained optimisation problem using Integer Linear Programming (ILP). This paper explores how the performance of ILP-based Bayesian network learning can be improved through ILP techniques and in particular through the addition of non-essential, implied constraints. There are exponentially many such constraints that can be added to the problem. This paper explores how these constraints may best be generated and added as needed. The results show that using these constraints in the best discovered configuration can lead to a significant improvement in performance and show significant improvement in speed using a state-of-the-art Bayesian network structure learner.

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UAI Conference 2013 Conference Paper

Advances in Bayesian Network Learning using Integer Programming

James Cussens
Mark Bartlett

We consider the problem of learning Bayesian networks (BNs) from complete discrete data. This problem of discrete optimisation is formulated as an integer program (IP). We describe the various steps we have taken to allow efficient solving of this IP. These are (i) efficient search for cutting planes, (ii) a fast greedy algorithm to find high-scoring (perhaps not optimal) BNs and (iii) tightening the linear relaxation of the IP. After relating this BN learning problem to set covering and the multidimensional 0-1 knapsack problem, we present our empirical results. These show improvements, sometimes dramatic, over earlier results.

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ECAI Conference 2010 Conference Paper

Instruction Cache Prediction Using Bayesian Networks

Mark Bartlett
Iain Bate
James Cussens

Storing instructions in caches has led to dramatic increases in the speed at which programs can execute. However, this has also made it harder to reason about the time needed for execution in those domains where temporal behaviour of code is important. This paper presents a novel approach to predicting which instructions will be found in the cache when required using machine learning. More specifically, we demonstrate a method in which a Bayesian network is inferred from examples of a program running and is then used to predict the presence of instructions in the cache when the same program is run with unknown inputs.

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