Scale up Bayesian Networks Learning Dissertation Proposal

Scale Up Bayesian Networks Learning Dissertation Proposal

Xiannian Fan [email protected] March 2015

Abstract

Bayesian networks are widely used graphical models which represent uncertain relations between the random variables in a domain compactly and intuitively. The first step of applying Bayesian networks to real-word problems typically requires building the network structure. Among others, optimal structure learning via score-and-search has become an active research topic in recent years. In this context, a scoring function is used to measure the goodness of fit of a structure to the data, and the goal is to find the structure which optimizes the scoring function. The problem has been shown to be NP-hard. This proposal focuses on scaling up structure learning algorithms. Two potential directions are preliminarily explored. One is to tighten the bounds; we tighten the lower bound by using more informed variable groupings when creating the heuristics. and tighten the upper bound using anytime learning algorithm. The other is to prune the search space by extracting constraints directly from data. Our preliminary results show that the two directions can significantly improve the efficiency and scalability of heuristics search-based structure learning algorithms, which gives us strong reasons to follow these directions in the future.

Professor: Dr. Changhe Yuan Signature: Date: 03-11-2015

Professor: Dr. Liang Huang Signature: Date: 03-11-2015

Professor: Dr. Andrew Rosenberg Signature: Date: 03-11-2015

Professor: Dr.John Paisley Signature: Date: 03-11-2015 Advisor: Dr. Changhe Yuan

1 Contents

1 Introduction 3 1.1 Notation and Termimology ...... 4 1.2 Outline ...... 5

2 Bayesian Network Structure Learning 6 2.1 Scoring Functions ...... 7 2.2 Local Search Strategies ...... 8 2.2.1 Hill Climbing ...... 9 2.2.2 Max-Min Hill Climbing ...... 9 2.2.3 Stochastic Search ...... 10 2.3 Optimal Search Strategies ...... 10 2.3.1 Branch-and-Bound ...... 10 2.3.2 Dynamic Programming ...... 11 2.3.3 Integer Linear Programming ...... 12 2.3.4 Heuristic Search ...... 12

3 Exact Struture Learning 13 3.1 Integer Linear Programming ...... 13 3.2 Heuristic Search ...... 14 3.2.1 Finding the Shortest Path: Heuristic Search Algorithms ...... 16 3.3 Integer Linear Programming VS Heuristic Search ...... 17

4 Proposal I: Tightening Bounds 19 4.1 K-Cycle Conﬂict Heuristic ...... 19 4.2 Tightening Dynamic Pattern Database ...... 22 4.3 Tightening Static Pattern Database ...... 22 4.3.1 Creating the Partition Graph ...... 23 4.3.2 Performing the Partition ...... 24 4.4 Topology Grouping ...... 24 4.5 Tightening Upper Bounds ...... 25 4.6 Preliminary Results ...... 25 4.6.1 Results on Dynamic Pattern Databases ...... 26 4.6.2 Results on Upper Bounds ...... 27 4.6.3 Results on Static Pattern Databases ...... 28

1 5 Proposal II: Using Constraints Learned from Data 30 5.1 Potential Optimal Parents Sets ...... 30 5.2 Using POPS Constraints ...... 31 5.2.1 A Simple Example ...... 31 5.2.2 Ancestor Relations ...... 32 5.2.3 POPS Constraints Pruning ...... 33 5.2.4 Recursive POPS Constraints Pruning ...... 34 5.2.5 Bridge Constraints ...... 35 5.3 Preliminary Results ...... 37

6 Conclusion and Future Work 40

2 1 Introduction

Bayesian networks, also known as belief networks, are widely-used graphical models that represent uncertain relations between the random variables in a uncertain domain compactly and intuitively [40]. In particular, each node in the graph represents a random variables, while edges between nodes represent probabilistic dependencies among the corresponding random variables. These conditional dependencies are often estimated by using known statistical and computaitonal methods. Hence, Bayesian Networks combine principles from graph theory, probability theory, computer science and statistics.

A Bayesian network is a directed acyclic graph (DAG) in which nodes represent random variables. The structure of a DAG is deﬁned by two sets, the set of nodes and the set of directed

edges. Each node represents one random variable. An edge from node Xi to node Xj represents a statistical dependency between the corresponding variables. Thus, the arrow indicates that a value taken by Xj depends on the value taken by Xj, or roughly speaking that Xi inﬂuences

Xj. Xj is then referred to as a parent of Xj and, similarly, Xj is then referred to as the child of

Xi. The structure of DAG guarantees that there is no cycle in the Bayesian networks; in other words, there is no node that can be its own ancestor or its own descendent. Such a condition is of vital importance to the factorization of the joint probability of a collection of nodes.

In addition to the DAG structure, which often considered as the qualitative part of the Bayesian Networks, one needs to specify the quantitative parameters of Bayesian networks. These parameters are quantiﬁed by a set of conditional probability distributions or conditional probability tables (CPTs), one for each variable conditioning on its parents. Overall, a Bayesian network represents a joint probability distribution over the variables.

Figure 1 show an example of Bayesian Network, the qualitative part is indicated by the DAG structure, and the quantitative part is given by four CPTs.

Applying Bayesian networks to real-world problems typically requires building graphical rep- resentations of relationships among variables. When these relationships are not known a priori,

3 Figure 1: A Bayesian Network the structure of Bayesian networks must be learned. The topic of this proposal is about Bayesian networks structure learning.

The remainder of this section ﬁrst introduce the Bayesian networks as well as the notation and terminology used throughout the rest of this paper, and the outline of this proposal.

1.1 Notation and Termimology

A Bayesian network is a directed acyclic graph (DAG) G that represents a joint probability distribution over a set of random variables V = {X1,X2, ..., Xn}. A directed arc from Xi to Xj represents the dependence between the two variables; we say Xi is a parent of Xj.

We use PAj to stand for the parent set of Xj. The dependence relation between Xj and PAj are quantiﬁed using a conditional probability distribution, P (Xj|PAj). The joint probability distribution represented by G is factorized as the product of all the conditional probability Qn distributions in the network, i.e., P (X1, ..., Xn) = i=1 P (Xi|PAi).

In addition to the compact representation, Bayesian networks also provide principled approaches to solving various inference tasks, including belief updating, most probable explanation, maxi-

4 mum a Posteriori assignment [40], and most relevant explanation [54, 52, 53].

Given a dataset D = {D1, ..., DN }, where each data point Di is a vector of values over variables V, learning a Bayesian network is the task of finding a network structure that best fits D. In this work, we assume that each variable is discrete with a finite number of possible values, and no data point has missing values.

1.2 Outline

The remainder of the proposal is organized as follows: in section 2, we introduce the basic concepts of Bayesian Network structure learning and different learning algorithms. In section 3, we elaborate two most efﬁcient exact structure learning methods. We propose methods for tightening the bouds in section 5, and propose methods to prune search space by extracting constraints directly from data in section 5. We brieﬂy conclude and discuss the future work in section 6.

Related Publication: Parts of the proposed methods appeared in the following paper:

Xiannian Fan, Changhe Yuan, and Brandon Malone.Tightening Bounds for Bayesian Network Structure Learning. In Proceedings of the 28th AAAI Conference on Artiﬁcial Intelligence (AAAI), 2014.

5 2 Bayesian Network Structure Learning

Bayesesian networks Struture Learning problem takes the data as inputs and produces an directed acyclic graph as the output. Before we dive into the algorithmic details in the next section, let us review these basic concepts and brieﬂy introduce different learning methods in this section.

There are roughly three main approaches to the learning problem: score-based learning, constraint- based learning, and hybrid methods. Score-based learning methods evaluate the quality of Bayesian network structures using a scoring function and selects the one that has the best score [13, 27]. These methods basically formulate the learning problem as a combinatorial optimization problem. They work well for datasets with not too many variables, but may fail to find optimal solutions for large datasets. We will discuss this approach in more detail in the next section, as it is the approach we take. Constraint-based learning methods typically use statistical testings to identify conditional independence relations from the data and build a Bayesian network structure that best fits those independence relations [40, 46, 9, 20, 51]. Constraint-based methods mostly rely on results of local statistical testings, so they can often scale to large datasets. However, they are sensitive to the accuracy of the statistical testings and may not work well when there are insufficient or noisy data. In comparison, score-based methods work well even for datasets with relatively few data points. Hybrid methods aim to integrate the advantages of the previous two approaches and use combinations of constraint- based and/or score-based methods for solving the learning problem [17, 2, 50, 41]. One popular strategy is to use constraint-based learning to create a skeleton graph and then use score-based learning to find a high-scoring network structure that is a subgraph of the skeleton [50, 41].

In this paper, we mainly focus on the score-based approach. we do not consider Bayesian model averaging methods which aim to estimate the posterior probabilities of structural features such as edges rather than model selection [27, 23, 16].

Score-based learning methods rely on a scoring function Score(.) in evaluating the quality of a Bayesian network structure. A search strategy is used to ﬁnd a structure G∗ that optimizes the

6 score. Therefore, score-based methods have two major elements, scoring functions and search strategies. We ﬁrst introduce the scoring functions, then review two classes of search stratigies, local search strategies and optimal search strategies.

2.1 Scoring Functions

Many scoring functions can be used to measure the quality of a network structure. Some of them are Bayesian scoring functions which define a posterior probability distribution over the network structures conditioning on the data, and the structure with the highest posterior probability is presumably the best structure. These scoring functions are best represented by the Bayesian Dirichlet score (BD) [28] and its variations, e.g., K2 [13], Bayesian Dirichlet score with score equivalence (BDe) [28], and Bayesian Dirichlet score with score equivalence and uniform priors (BDeu) [8]. Other scoring functions often have the form of trading off the goodness of fit of a structure to the data and the complexity of the structure. The goodness of fit is measured by the likelihood of the structure given the data or the amount of information that can be compressed into a structure from the data. Scoring functions belonging to this category include minimum description length (MDL) (or equivalently Bayesian information criterion, BIC) [42, 47, 33], Akaike information criterion (AIC) [4, 7], (factorized) normalized maximum likelihood function (NML/fNML) [44], and the mutual information tests score (MIT) [19]. All of these scoring functions have some gerneral properties. The most important property is decomposable, that is, the score of a network can be decomposed into a sum of node scores [27].

It is worth noting that the optimal structure G∗ may not be unique because multiple Bayesian network structures may share the same optimal score1. Two network structures are said to belong to the same equivalence class [10] if they represent the same set of probability distributions with all possible parameterizations. Score-equivalent scoring functions assign the same score to structures in the same equivalence class. Most of the above scoring functions are score equivalent.

1That is why we often use “an optimal” instead of “the optimal” throughout this paper.

7 Due to the limit of the space, we mainly introduce the MDL score in this survey. Let ri be the number of states of X , N be the number of data points consistent with PA = pa , i pai i i and N be the number of data points further constrained by X = x . MDL is deﬁned as xi,pai i i follows [33].

X MDL(G) = MDL(Xi|PAi), (1) i where

Xl∈PAi

The goal is then to ﬁnd a Bayesian network that has the minimum MDL score. Any other decomposable scoring function, such as BIC, BDeu, or fNML, can be used instead without affecting the search strategy. One slight difference between MDL and the other scoring functions is that the latter scores need to be maximized in order to ﬁnd an optimal solution. But it is rather straightforward to translate between maximization and minimization problems by simply changing the sign of the scores. Also, we sometimes use costs to refer to the scores.

2.2 Local Search Strategies

Given n variables, there are O(n2n(n−1)) directed acyclic graphs (DAGs). The size of the solution space grows exponentially in the number of variables. It is not surprising that score-based structure learning has been shown to be NP-hard [11]. Due to the complexity, early research focused mainly on developing local search algorithms [27, 6]. Popular local search strategies that were used include greedy hill climbing, Min Min Hill Climbing, stochastic search, etc.

8 2.2.1 Hill Climbing

Greedy hill climbing methods have been proposed. They start from an initial solution, which typically is an empty network without any edges, or a randomly generated structure, and iteratively apply single edge operations, including addition, deletionn and reversal, looking for the choice that locally maximizing the score improvement. Extensions to this approach include tabu search with random restarts [25], limiting the number of parents or parameters for each variable [24], searching in the space of equivalence classes [12], searching in the space of variable orderings [49], and searching under the constraints extracted from data.

2.2.2 Max-Min Hill Climbing

Max-min hil-climbing (MMHC) [50], a double step algorithm, was proposed. After first exe- cuting a statistical conditional independence test to find a reduced set of candidate parents sets, it applies a greedy hill-climbing step to find a local optimal solution. MMHC is classifed as a hybrid methods between scoring-based methods and constraint based on methods.

The Max-min step, based on the Max-min parent-chilren (Tsamardinos2006), is a heuristic that looks for a set of candidates for each node in the graph. It aims to ﬁnd an undirected graph representing the skeleton of original DAG by looking for subsets of variables Z conditionally separting pairs of variables X,Y . Such test is denoted as T (X,Y |Z), and is G2 test in [50].

G MMPC works as follows. Let PCX be the set of parents and children for node X in a Bayesian

G G0 Ntwork over P that is Markov equivalent with respect to G, it holds that PCX = PCX , and we can write only PCX . Thus, the set of parents and children of a node is the same for all the Markove equivalent Bayesian Networks over the same probabiliby distribution P . MMPC performs the conditional independence test T (X,Y |Z) in a subroutine called MinAssoc, The logarithm of the so obtained p−value is called the association between variables.

The Max-min heuristic, for each varibale X, iteratively constructs a set of variables with high association with X (the CPC set), picking up at each iteration the variable Y with the largest

9 association, until such association falls under a minimum value. The set of CPCs found by the Max-min heuristic form the skeleton of the Bayesian Network.

The hill-climbing algorithm is then applied to the skeleton to perform a local search, by using three possible edge operations over DAGs (edge insertion, edge removal and edge reversal) and greedily choosing the one that increases the score the most until no improvement. The search space for edges is limited to the ones allowed by skeleton found by Max-min heuristic.

2.2.3 Stochastic Search

Stochastic search methods such as Markov Chain Monte Carlo and simulated annealing have also been applied to ﬁnd a high-scoring structure [27, 21, 36]. These methods explore the solution space using non-deterministic transitions between neighboring network structures while favoring better solutions. The stochastic moves are used in hope to escape local optima and ﬁnd better solutions.

2.3 Optimal Search Strategies

Local search methods are quite robust in the face of large learning problems with many variables. However, they do not guarantee to ﬁnd an optimal solution. What is worse, the quality of their solutions is typically unknown. Recently multiple exact algorithms have been developed for learning optimal Bayesian networks.

2.3.1 Branch-and-Bound

A branch and bound algorithm (BB) was proposed in [18] for learning Bayesian networks. The algorithm ﬁrst creates a cyclic graph by allowing each variable to obtain optimal parents from all the other variables. A best-ﬁrst search strategy is then used to break the cycles by removing one edge at a time. The algorithm uses an approximation algorithm to estimate an initial upper bound solution for pruning. The algorithm also occasionally expands the worst nodes in the

10 search frontier in hope to find better networks to update the upper bound. At completion, the algorithm finds an optimal network structure that is a subgraph of the initial cyclic graph. If the algorithm ran out of memory before finding the solution, it will switch to using a depth- first search strategy to find a suboptimal solution. Authors claim their alorithms to be optimal. Such optimality, however, is under the constraint that each variable can only select parents from optimal parents set.

2.3.2 Dynamic Programming

Several dynamic programming algorithms are proposed based on the observation that a Bayesian network has at least one leaf [37, 45]. A leaf is a variable with no child variables in a Bayesian network. In order to find an optimal Bayesian network for a set of variables V, it is sufficient to find the best leaf. For any leaf choice X, the best possible Bayesian network is constructed by

letting X choose an optimal parent set PAX from V\{X} and letting V\{X} form an optimal subnetwork. Then the best leaf choice is the one that minimizes the sum of Score(X, PAX ) and Score(V\{X}) for a scoring function Score(.). More formally, we have:

Score(V) = min{Score(V \{X}) + BestScore(X, V \{X})}, (5) X∈V

where

BestScore(X, V \{X}) = min Score(X, PAX ). (6) PAX ⊆V\{X}

Given the above recurrence relation, a dynamic programming algorithm works as follows. It first finds optimal structures for single variables, which is trivial. Starting with these base cases, the algorithm builds optimal subnetworks for increasingly larger variable sets until an optimal network is found for V. The dynamic programming algorithms can find an optimal Bayesian network in O(n2n) time and space [31, 37, 43, 45]. Recent algorithms have improved the memory complexity by either trading longer running times for reduced memory consumption [39] or taking advantage of the layered structure present within the dynamic programming lattice.

11 2.3.3 Integer Linear Programming

Integer linear programming (ILP) has been used to learn optimal Bayesian network structures [15, 30]. The BNSL problem is formulated as an integer linear program over a polytope with an exponential number of facets. An outer bound approximation to the polytope is then solved. If the solution of the relaxed problem is integral, it is guaranteed to be the optimal structure. Otherwise, cutting planes and branch and bound algorithms are subsequently applied to ﬁnd the optimal structure. More details will be covered in next section.

2.3.4 Heuristic Search

Several heuristic search algorithms have been developed for solving this problem by formu- lating the problem as a shortest path problem [57], e.g., A*, breadth-ﬁrst branch and bound (BFBnB) [35], and anytime Window A* [34]. Most of these algorithms need a lower bound function in estimating the quality of a search path so that they can prune paths that are guaranteed to lead to suboptimal solutions and focus on exploring the most promising search spaces.

A lower bound was ﬁrst computed by a simple huristic function which completely relaxes the acyclicity constraint of Bayesian networks; then a better lower bound was proposed by using static k-cycle conﬂict heuristic [55], which demonstrated excellent performance when used by the search algorithms to solve many benchmark data sets. It takes as input a partition of the random variables of a data set, and computes the heuristic value of a path by enforcing the acyclicity constraint between the variables within each group and relaxing the acyclicity between different groups. More details will be covered in next section.

12 3 Exact Struture Learning

In this proposal, we focus on the exact Bayesian network structure learning. We brieﬂy review some algorithms which adopted optimal search strategies in the last section. Previous study [56] showed that integer linear programming and heuristic search methods are more efﬁcient than branch-and-bound and dynamic programming methods. We highlight and elaborate these two methods in this section.

3.1 Integer Linear Programming

In [15], BN structure learning was encoded as an integer programming. For each variable x and candidate parent set PA , a binary variable I(PA → x) is created. I(PA → x) = 1 if and only if PA are the parents of x in an optimal BN.These variables were called f amily variables.

Table 1 shows coding of familiy variables. For each variable Xi, there are ki potential optimal P parents candidates. The total number of family variables is i ki.

I(PA1,1 → X1) I(PA1,2 → X1) ... I(PA1,k1 → X1)

I(PA2,1 → X2) I(PA2,2 → X2) ... I(PA2,k2 → X2)

I(PA3,1 → X3) I(PA3,2 → X1) ... I(PA3,k3 → X3) ......

I(PAn,1 → Xn) I(PAn,2 → Xn) ... I(PAn,kn → Xn) Table 1: coding for family variables

Then BN structure learning can be formulated as the following contrained optimisation problem:

X s(x, P A)I(PA → x) (7) x,P A

To use an tnteger linear programming approach it is necessary to use linear constraints to ensure that only valid DAGs are feasible.

Jaakkola etc. [30] use a family of cluster constraints which impose that, for every subset C ⊂ V of the nodes of the graph G, there must be at least one node whose parent set either completely

13 lies outside C, or is the empty set. Formally, the cluster constraints is deﬁned as follows: for every set of nodes C ⊂ V ,

X X I(PA → x) ≥ 1 (8) x∈C PA:PA∩C=∅

This class of constraints stems from the observation that any sebset C of nodes in a DAG must contain at least one node who has no parent in that subset. Clearly, there is an exponential number of cluster constrains. The method to this problem is to remove then, solve the linear relaxation of the problem and look for violated cosntraints.

Cussens etc. [15, 5] inherited Jaakola appraoch, but treated problem with branch-and-cut strategy: they consider the relaxated problem obtained by removing a more general version of the cluster constraints from the mode and solve it, andding the most effective cluster constraints as cutting planes when needed, getting a solution x∗. If x∗ does not violate any cluster constaints and is integer-value, the problem is solved; otherwise, x∗ is used to be branched on, creating two new subproblems.

3.2 Heuristic Search

In [57], BN structure learning was cast as a shortest path search problem. The state space graph for learning Bayesian networks is basically a Hasse diagram containing all of the subsets of the variables in a domain. Figure 2 visualizes the state space graph for a learning problem with four variables.

Figure 2 shows the implicit search graph for four variables. The top-most node with the empty set at layer 0 is the start search node, and the bottom-most node with the complete set at layer

n is the goal node, where n is the number of variables in a domain. An arc from U to U∪{Xi} represents generating a successor node by adding a new variable {Xi} to an existing set of variables U; U is called a predecessor of U ∪ {Xi}. the cost of the arc is equal to the score of the optimal parent set for Xi out of U, which is computed by considering all subsets of the

14 { }

{ 1 } { 2 } { 3 } { 4 }

{ 1,2 } { 1,3 } { 2,3 } { 1,4 } { 2,4 } { 3,4 }

{ 1,2,3 } { 1,2,4 } { 1,3,4 } { 2,3,4 }

{ 1,2,3,4 }

Figure 2: An order graph of four variables.

variables in PA ⊆ U,PA ∈ Pi, i.e.,

cost(U → U ∪ {Xi}) = BestScore(Xi, U) (9)

= min si(PAi). (10) PAi⊆U,P Ai∈Pi

For example, the arc {X1,X2} → {X1,X2,X3} has a cost equal to BestScore(X3, {X1,X2}). Each node at layer i has n − i successors as there are this many ways to add a new variable, and i predecessors as there are this many leaf choices. We deﬁne expanding a node U as generating all successors nodes of U.

With the search graph thus deﬁned, a path from the start node to the goal node is deﬁned as a sequence of nodes such that there is an arc from each of the nodes to the next node in the sequence. Each path also corresponds to an ordering of the variables in the order of their appearance. For

example, the path traversing nodes ∅, {X1}, {X1,X2}, {X1,X2,X3}, {X1,X2,X3,X4} stands

for the variable ordering X1,X2,X3,X4. That is why the greaph is called order graph. The cost of a path is deﬁned as the sum of the costs of all the arcs on the path. The shortest path is then the path with the minimum total cost in the order graph.

15 Given the shortest path, we can reconstruct a Bayesian network structure by noting that each arc on the path encodes the choice of optimal parents for one of the variables out of the preceding variables, and the complete path represents an ordering of all the variables. Therefore, putting together all the optimal parent choices generates a valid Bayesian network. By construction, the Bayesian network structure is optimal.

3.2.1 Finding the Shortest Path: Heuristic Search Algorithms

This shortest path problem has been solved using several heuristic search algorithms, including A* [57], anytime window A* (AWA*) [34] and breadth-ﬁrst branch and bound (BFBnB) [35].

# # # #! #! #!

# # #!

Figure 3: Finding Shortest Path by Heuristic Search.

In A* [26], an admissible heuristic function is used to calculate a lower bound on the cost from a node U in the order graph to goal. An f-cost is calculated for U by summing the cost from start to U (called g(U)) and the lower bound from U to goal (called h(U)). Figure 3 illustrates the basics of the heuristic search methods for ﬁnding shortest path of BNSL problem. g(U) corresponds to the score of the subnetwork over the variables U, and h(U) estimates the score of the remaining variables. So f(U) = g(U) + h(U). The f-cost provides an optimistic estimation on how good a path through U can be. The search maintains a list of nodes to be expanded sorted by f-costs called open and a list of already-expanded nodes called closed.

16 Initially, open contains just start, and closed is empty. Nodes are then expanded from open in best-ﬁrst order according to f-costs. Expanded nodes are added to closed. As better paths to nodes are discovered, they are added to open. Upon expanding goal, the shortest path from start to goal has been found.

In AWA* [3], a sliding window search strategy is used to explore the order graph over a number of iterations. During each iteration, the algorithm uses a fixed window size, w, and tracks the layer l of the deepest node expanded. For the order graph, the layer of a node corresponds to the number of variables in its subnetwork. Nodes are expanded in best-first order as usual by A*; however, nodes selected for expansion in a layer less that l − w are instead frozen. A path to goal is found in each iteration, which gives an upper bound solution. After finding the path to goal, the window size is increased by 1 and the frozen nodes become open. The iterative process continues until no nodes are frozen during an iteration, which means the upper bound solution is optimal. Alternatively, the search can be stopped early if a resource bound, such as running time, is exceeded; the best solution found so far is output.

In BFBnB [58], nodes are expanded one layer at a time. Before beginning the BFBnB search, a quick search strategy, such as AWA* for a few iterations or greedy hill climbing, is used to ﬁnd a “good” network and its score. The score is used as an upper bound. During the BFBnB search, any node with an f-cost greater than the upper bound can safely be pruned.

3.3 Integer Linear Programming VS Heuristic Search

Integer linear programming approach formulates the BNSL problem as an integer linear program whose variables correspond to the potential optimal parent sets of all variables (called famility variables), thus the efﬁciency depends of the number of family varaibles, while the heuristic search approach formulates the BNSL problem as the graph search problem who ef- ﬁciency depends on complexity of the search space, which is 2n, where n is the number of variables.

Yuan etc. [56] compared the two approaches. They compared GOBNILP (a public software

17 using integer linear programming method) to A* algorithm ( which guarantees to expand the mininum number of nodes in heuristic search algorithms). The comparison between GOBNILP and A* showed that they each has its own advantages. A* was able to find optimal Bayesian networks for all the datasets well within the time limit. GOBNILP failed to learn optimal Bayesian networks for some benchmark datasets. The reason is that GOBNILP formulates the learning problem as an integer linear program whose variables correspond to the optimal parent sets of all variables. Even though these datasets do not have many variables, they have many optimal parent sets, so the integer programs for them have too many variables to be solvable within the time limit. On the other hand, the results also showed that GOBNILP was quite efficient on many benchmark datasets. Even though a dataset may have many variables, GOBNILP can solve it efficiently as long as the number of optimal parent sets is small.

These insights are quite important, as they provide a guideline for choosing a suitable algorithm given the characteristic of a dataset. If there are many optimal parent sets but not many variables, heuristic search is the better algorithm; if the other way around is true, integer linear programming is better. In comparison to integer linear programming, heuristic search is less sensitive to the number of optimal parent sets, number of data points, or scoring functions, but is more sensitive to the number of variables in the datasets.

18 4 Proposal I: Tightening Bounds

For the rest of this paper, we focus on heuristic search methods. Malone et al. [35] proposed to use breadth-ﬁrst search to solve the shortest path problem. They observed that the order graph has a regular layered structure, and the successors of any node only appear in the very next layer. By searching the graph layer by layer, the algorithm only needs the layer being generated in RAM. All nodes from earlier layers are stored on disk, and nodes from the layer being expanded are read in from disk one by one. Furthermore, nodes from the layer being generated are written to disk as RAM ﬁlls; delayed duplicate detection methods are used to remove unnecessary copies of nodes [32].

The basic breadth-ﬁrst search can be enhanced with branch and bound, resulting in a breadth- ﬁrst branch and bound algorithm (BFBnB). An upper bound ub is found in the beginning of the search using a hill climbing search. A lower bound called f-cost is calculated for each node U by summing two costs, g-cost and h-cost, where g-cost stands for the shortest distance from the start node to U, and h-cost stands for an optimistic estimation on how far away U is from the goal node. The h-cost is calculated from a heuristic function. Whenever f > ub, all the paths extending U are guaranteed to lead to suboptimal solutions and are discarded immediately. The algorithm terminates when reaching the goal node. Clearly the tightness of the lower and upper bounds has a high impact on the performance of the algorithm.

In this section, we proposes methods for tightening the lower bounds of BFBnB. We ﬁrst provide a brief review of the k-cycle conﬂict heuristic, and then discuss how it can be tightened.

4.1 K-Cycle Conﬂict Heuristic

The effectiveness of the heuristic function is a critical factor in the effectiveness of the search.

The following simple heuristic function was introduced in [57] for computing lower bounds for A* search.

19 Deﬁnition 1. Let U be a node in the order graph, its heuristic value is

X h(U) = BestScore(X, V\{X}). (11) X∈V\U

The above heuristic function allows each remaining variable to choose optimal parents from all of the other variables. Therefore it completely relaxes the acyclicity constraint of Bayesian networks in the estimation. The heuristic was proven to be admissible, meaning it never over- estimates the future distance [57]. Admissible heuristics guarantee the optimality. However, because of the complete relaxation of the acyclicity constraint, the simple heuristic may generate loose lower bounds.

In [55], an improved heuristic function called k-cycle conﬂict heuristic was proposed by reduc- ing the amount of relaxation. The idea is to divide the variables into multiple groups with a size up to k and enforce acyclicity within each group while still allowing cycles between the groups. There are two major approaches to dividing the groups. One is to enumerate all of the subsets with a size up to k (each subset is called a pattern); a set of mutually exclusive patterns covering V \U can be selected to produce a lower bound for node U as the heuristic is additive [22]. This approach is called dynamic pattern database. In [55], the dynamic pattern database was created by performing a reverse breadth-ﬁrst search for the last k layers in the order graph. The search began with the goal node V whose reverse g cost is 0. A reverse arc from U0 ∪ {X} to U0 corresponds to selecting the best parents for X from among U0 and has a cost of BestScore(X, U0).

The other approach is to divide the variables V into several static groups Vi (typically two). S The idea is to partition the variables V into multiple groups Vi (typically two), i.e. V = i Vi, and enforce acyclicity within each group while still allowing cycles between the groups. For

the partition, we need to compute a pattern database for each group Vi. For this particular

problem, a pattern database for group Vi is basically a full order graph containing all subsets

of Vi. We will use a backward breadth ﬁrst search to create the graph layer by layer starting

from the node Vi. The cost for any reverse arc from U ∪ {X} to U in this order graph will be

Figure 4: Two pattern databases for a 8 variables problem. 8 variables are partitioned into two groups, V1 = {X1,X2,X3,X4} and V2 = {X5,X6,X7,X8}. (Left ) The pattern database for group V1; bold arrows show the path corresponding to the score P1 for pattern {X2,X3}, where P1 = BestScore(X3, {X1,X2,X4}∪V2)+BestScore(X2, {X1,X4}∪V2). (Right ) The pattern database for group V2; bold arrows show the path corresponding to the score P2 for pattern {X5,X7}, where P2 = BestScore(X7, {X5,X6,X8}∪V1)+BestScore(X5, {X6,X8}∪V1) . The heuristic value for pattern {X2,X3,X5,X7} is P1 + P2.

S BestScore(X, ( j6=i Vj)∪U). We then enumerate all subsets of each group Vi as the patterns, which can be done by a reverse breadth-ﬁrst search in an order graph containing only Vi [55]. The patterns from different groups are guaranteed to be mutually exclusive, so we simply pick out the maximum-size pattern for each group that is a subset of V \ U and add them together as the lower bound. Figure 4 shows two pattern databases for a 8-variable problem, as well as the procedure of calculating the heuristic value of node {X2,X3,X5,X7}.

The tightness of the static k-cycle conﬂict heuristic depends highly on the partition being used. The heuristic can avoid directed cycles for the patterns within the same group, but cannot avoid cycles between different groups. In the example shown in Figure 4, X3 selects parents

{X1,X5} as parents (subset of {X1,X4} ∪ V2 ) and X5 selects {X3,X6} as parents; there is a cycle between {X3} and {X5}. In current works so far, now simple partition are used, e.g, ﬁrst half variables fall into one group and second half variables fall into the other group. How to form a good partition could be one future research direction.

21 4.2 Tightening Dynamic Pattern Database

To use the dynamic pattern database to calculate the tightest-possible lower bound for a search node U, we select the set of mutually exclusive patterns which covers all of the variables in V \ U and has the maximum sum of costs. This can be shown to be equivalent to the maximum weighted matching problem [22], which is NP-hard [38] for k > 2. Consequently, the previous work [55] used a greedy approximation to solve the matching problem; its idea is to greedily choose patterns with the maximum differential scores.

Solving the matching problem exactly improves the tightness of dynamic pattern database heuristic. To achieve that, we formulate the problem as an integer linear program [?] and solve it exactly. A set of binary variables Ai,j are created which indicate if Xi is in pattern j. A second set of variables pj are created which indicate if pattern j is selected. The linear constraints that A · p = e, where e is a vector with 1s for each variable in V \ U, are also added to the program. A standard integer linear program solver, such as SCIP [1], is used to maximize the cost of the selected patterns subject to the constraints. The resulting value is the lower bound which is guaranteed to be at least as tight as that found by the greedy algorithm. Solving the integer linear program introduces overhead compared to the simple greedy algorithm, though.

4.3 Tightening Static Pattern Database

The tightness of the static pattern database heuristic depends highly on the static grouping used during its construction. A very simple grouping (SG) method was used in [55]. Let X1, ..., Xn be the ordering of the variables in a dataset. SG divides the variables into two balanced groups,

{X , ..., X n } and {X n , ..., X }. Even though SG exhibited excellent performance on 1 d 2 e d 2 e+1 n some datasets in [55], there is potential to develop more informed grouping strategies by taking into account of the correlation between the variables.

A good grouping method should reduce the number of directed cycles between the variables and enforce acyclicity as much as possible. Since no cycles are allowed within each group,

22 we should maximize the correlation between the variables in each group. Also, because cycles are allowed between groups, we should minimize the correlation between the groups. Consider two variables X1 and X2. If they have no correlation, there will be no arc between these two variables in the optimal Bayesian network, so there is no need to put the two variables in the same group. On the other hand, if they have strong correlation, and if they are put into different groups G1 and G2, X1 is likely to select X2 from G2 as a parent, and vice versa. Then a cycle will be introduced between the two groups, resulting in a loose bound. It is better to put these variables in one group and disallow cycles between them.

The above problem can be naturally formulated as a graph partition problem. Given a weighted undirected graph G = (V,E) with V vertices and E edges, which we call the partition graph, the graph partition problem cuts the graph into 2 or more components while mini- mizing the weight of edges between different components. We are interested in uniform or balanced partitions as it has been shown that such partitions typically work better in static pattern databases [22]. Two issues remain to be addressed in the formulation: creating the partition graph and performing the partition.

4.3.1 Creating the Partition Graph

We propose to use two methods to create the partition graph. One is to use constraint-based learning methods such as the Max-Min Parent Children (MMPC) algorithm [50] to create the graph. The MMPC algorithm uses independence tests to ﬁnd a set called candidate parent and children (CPC) for each variable Xi. The CPC set contains all parent and child candidates of

Xi. The CPC sets for all variables together create an undirected graph. Then MMPC assigns a weight to each edge of the undirected graph by independent tests, which indicate the strength of correlation with p-values. Small p-values indicate high correlation, so we use the negative p-values as the weights. We name this approach family grouping (FG).

The second method works as follows. The simple heuristic in Eqn. 11 considers the optimal parent set out of all of the other variables for each variable X, denoted as PA(X). Those

23 PA(X) sets together create a directed cyclic graph; by simply ignoring the directions of the arcs, we again obtain an undirected graph. We then use the same independence tests as in MMPC to obtain the edge weights. We name this approach parents grouping (PG).

4.3.2 Performing the Partition

Many existing partition algorithms can be used to perform the graph partition. Since we prefer balanced partitions, we select the METIS algorithm. METIS is a multilevel graph partitioning method: it first coarsens the graph by collapsing adjacent vertices and edges, then partitions the reduced graph, and finally refines the partitions back into the original graph. Studies have shown that the METIS algorithm has very good performance at creating balanced partitions across many domains. Also, since the partition is done on the reduced graph, the algorithm is quite efficient. This is important for our purpose of using it only as a preprocessing step.

4.4 Topology Grouping

Besides the above grouping methods based on partitioning undirected graphs, we also propose another method based on topologies of directed acyclic graphs found by approximate Bayesian network learning algorithms. Even though these algorithms cannot guarantee to ﬁnd the optimal solutions, some of them can ﬁnd Bayesian networks that are close to optimal. We assume that the suboptimal networks capture many of the dependence/independence relations of the optimal solution. So we simply divide the topological ordering of the suboptimal network into two groups, and those are the grouping for the static pattern database. Many approximate learning algorithms can be used to learn the suboptimal Bayesian networks. We select the recent anytime window A* algorithm [3, 34] (introduced for obtaining upper bounds in the next section) for this purpose. We name this approach as topology grouping (TG).

24 4.5 Tightening Upper Bounds

Breadth-first branch and bound improves search efficiency by pruning nodes which have an f-cost worse than some known upper bound, ub. In the best case, when ub is equal to the optimal cost, BFBnB provably performs the minimal amount of work required to find the shortest path [58]. As the quality of ub decreases, though, it may perform exponentially more work. Consequently, a tight upper bound is pivotal for good search behavior.

A beam search-based hill climbing algorithm was used in [55] to find an upper bound. Recently, anytime window A* (AWA*) [3, 34] was shown to find high quality, often optimal, solutions very quickly. Briefly, AWA* uses a sliding window search strategy to explore the order graph over a number of iterations. During each iteration, the algorithm uses a fixed window size, w, and tracks the layer l of the deepest node expanded. Nodes are expanded in best-first order as usual by A*; however, nodes selected for expansion in a layer less that l − w are instead frozen. A path to the goal is found in each iteration, which gives an upper bound solution. After finding the path to the goal, the window size is increased by 1 and the frozen nodes become the open list. The iterative process continues until no nodes are frozen during an iteration, which means the upper bound solution is optimal, or a resource bound, such as running time, is exceeded. Due to its ability to often find tight upper bounds quickly, we use AWA* in this work.

4.6 Preliminary Results

We empirically tested our new proposed tightened bounds using the BFBnB algorithm. We use benchmark datasets from the UCI machine learning repository and Bayesian Network Repos- itory. The experiments were performed on an IBM System x3850 X5 with 16 core 2.67GHz Intel Xeon Processors and 512G RAM; 1.7TB disk space was used.

25 Figure 5: The number of expanded nodes and running time needed by BFBnB to solve Parkin- sons based on dynamic pattern database strategies: matching (“Exact” or “Approximate”) and k (“3” or “4”).

4.6.1 Results on Dynamic Pattern Databases

We compared the dynamic pattern database with exact matching against with the previous greedy matching in [55] on the Parkinsons dataset. We used AWA* upper bounds in these experiments. The exact matching method is guaranteed to produce a tighter bound than the greedy method. However, an exact matching problem needs to be solved at each search step; the total amount of time needed may become too prohibitive. This concern is conﬁrmed in the experiments. Figure 5 compares the performance of BFBnB with four dynamic pattern databases (k = 3/4, and matching=exact/approximate). Solving the matching problems exactly did reduce the number of expanded nodes by up to 2.5 times. However, solving the many exact matching problems necessary to calculate the h-costs of the generated nodes in the order graph far outweighed the beneﬁt of marginally fewer expanded nodes. For example, BFBnB with the exact matching required up to 400k seconds to solve Parkinsons, compared to only 4 seconds using the approximate matching.

26 4.6.2 Results on Upper Bounds

We ﬁrst tested the effect of the upper bounds generated by AWA* on BFBnB on two datasets: Parkinsons and Steel Plates. Since AWA* is an anytime algorithm, it produces multiple upper bounds during its execution. We recorded each upper bound plus the corresponding running time of AWA*. Each upper bound is tested on BFBnB. In this experiment, we use static pattern database with family grouping (FG) as the lower bound. Figure 6 plots the running time of AWA* versus the running time and number of expanded nodes of BFBnB.

6 x 10 8 8 20 x 10 4 1000

Expanded Nodes Expanded Nodes Running Time of BFBnB Running Time of BFBnB

6 15

4 10 2 500 Running Time of BFBnB Running Time of BFBnB Number of Expanded Nodes Number of Expanded Nodes 2 5

0 0 0 0 −2 −1 0 1 2 −1 0 1 2 3 4 10 10 10 10 10 10 10 10 10 10 10 Running Time of AWA* Running Time of AWA* (a)P arkinsons (b)SteelP lates Figure 6: The effect of upper bounds generated by running AWA* for different amount of time (in seconds) on the performance of BFBnB on Parkinsons and Steel Plates.

The experiments show that the upper bounds had a huge impact on the performance of BFBnB. Parkinsons is a small dataset with only 23 variables. AWA* finds upper bounds within several hundredths of seconds and solves the dataset optimally around 10 seconds. The BFBnB was able to solve Parkinsons in 18 seconds using the first upper bound found by AWA* at 0.03s, and in 2 seconds using the upper bound found at 0.06s. Subsequent upper bounds bring additional but marginal improvements to BFBnB. This confirms the results in [34] that AWA* often finds excellent solutions early on and spends the rest of time finding marginally better solutions, or just proving the optimality of the early solution.

Steel Plates is a slightly larger dataset. AWA* needs more than 4,000 seconds to solve it. We tested all the upper bounds found by AWA*. The ﬁrst upper bound found at 0.1s enabled BF-

27 (a)P arkinsons (b)SteelP lates Figure 7: The effect of different grouping strategies on the number of expanded nodes and running time of BFBnB on Parkinsons and Steel Plates. The four grouping methods are the simple grouping (SG), family grouping (FG), parents grouping (PG), and topology grouping (TG).

BnB to solve the dataset within 1,000 seconds, and the third upper bound found at 0.5s solved the dataset within 400 seconds. Again, all the subsequent upper bounds only brought marginal improvements, even for the optimal bound found by AWA*. For much larger datasets, we be- lieve that running AWA* for several hundreds of seconds should already generate sufﬁciently tight upper bounds; the time is minimal when compared to the amount of time needed to prove the optimality of Bayesian network structures. In all these experiments, the results on the number of expanded nodes show similar patterns as the running time.

4.6.3 Results on Static Pattern Databases

We compared the new grouping strategies, including family grouping (FG), parents grouping (PG), and topology grouping (TG), against the simple grouping (SG) on Parkinsons and Steel Plates. We again used AWA* upper bounds. Figure 7 shows the results. On Parkinsons, FG and PG had the best performance; both FG and PG enabled BFBnB to solve the dataset three times faster. TG was even worse than the SG. On Steel Plates, however, TG was slightly better than FG. PG was again the best performing method on this dataset; it allowed BFBnB to solve the dataset 10 times faster than SG. These results suggest that family grouping (FG) and parents

28 grouping (PG) are both reliably better than the simple grouping (SG). We therefore use FG and PG as the lower bounds in our subsequent experiments.

29 5 Proposal II: Using Constraints Learned from Data

In this section, we propose methods using the constaints extracted from data to improve learning efficiency. The methods makes the observation that there is useful information implicit in the potential optimal parents sets (POPS). Specifically, the POPS of a variable constrain its parent candidates. Moreover, the parent candidates of all variables together give a directed cyclic graph, which often decomposes into a set of strongly connected components (SCCs). Each SCC corresponds to a smaller subproblem which can be solved independently of the others. Since potential optimal parents sets (POPS) is the basis of proposed methods, we first give the details about POPS.

5.1 Potential Optimal Parents Sets

Recall our structure learning problem. The goal is to ﬁnd a structure which optimizes the score. We only require that the scoring function is decomposable [27]; that is, the score of a network P s(N) = i si(PAi). The si(PAi) values are often called local scores. Many commonly used scoring functions, such as MDL [33] and BDe [8, 29], are decomposable.

While the local scores are deﬁned for all 2n−1 possible parent sets for each variable, this number is greatly reduced by pruning parent sets that are provably never optimal [18]. We refer to this as lossless score pruning because it is guaranteed to not remove the optimal network from consideration. We refer to the scores remaining after pruning as potentially optimal parent sets (POPS).

Other pruning strategies, such as restricting the cardinality of parent sets, are also possible, but these techniques could eliminate parent sets which are in the globally optimal network; we refer to pruning strategies which might remove the optimal network from consideration as lossy score pruning. Of course, these, and any other, score pruning strategies can be combined.

Regardless of the score pruning strategies used, we still refer to the set of unpruned local scores as POPS and denote the set of POPS for Xi as Pi. The POPS are given as input to the learning

30 problem. We deﬁne the Bayesian network structure learning problem (BNSL) as follows.

The BNSL Problem

INPUT: A set V = {X1,...,Xn} of variables and a set of POPS Pi for each Xi.

TASK: Find a DAG N ∗ such that

n ∗ X N ∈ arg min si(PAi), N i=1

where PAi is the parent set of Xi in N and PAi ∈ Pi.

5.2 Using POPS Constraints

POPS implicitly encodes some information, which can beneﬁt the search process. We will ﬁrst motivate our approach using a simple example and then describe the technical details.

5.2.1 A Simple Example

variable POPS X1 {X2} {} X2 {X1} {} X3 {X1,X2}{X2,X6}{X1,X6}{X2}{X6} {} X4 {X1,X3}{X1}{X3} {} X5 {X4}{X2} {} X6 {X2,X5}{X2} {}

Table 2: The POPS for six variables. The ith row shows Pi.

Table 2 shows the POPS for six variables. Based on these sets, we can see that not all variables can select all other variables as parents. For example, X1 can only select X2 as its parent (due to score pruning). We collect all of the potential parents for Xi by taking the union of all

PA ∈ Pi. Figure 8 shows the resulting parent relation graph for the POPS in Table 2. The parent relation graph includes an edge from Xj to Xi if Xj is a potential parent of Xi.

31 1 2

4 6

Figure 8: The parent relation graph constructed by aggregating the POPS in Table 2. The strongly connected components are surrounded by shaded shapes.

Naively, the complete order graph for six variables contains 26 nodes. However, from the

parent relation graph, we see that none of {X3,X4,X5,X6} can be a parent of X1 or X2. Consequently, we can split the problem into two subproblems as shown in Figure 9: ﬁrst,

ﬁnding the shortest path from start to {X1,X2}, and then, ﬁnding the shortest path from

2 4 {X1,X2} to goal. Thus, the size of the search space is reduced to 2 + 2 .

5.2.2 Ancestor Relations

This simple example shows that the parent relation graph can be used to prune the order graph without bounds. In general, we must consider ancestor relations to prune the order graph. In

particular, if Xi can be an ancestor of Xj, and Xj cannot be an ancestor of Xi (due to local score

pruning), then no node in the order graph which contains Xj but not Xi needs to be generated.

As a proof sketch, we can consider a node U which includes neither Xi nor Xj. If we add Xi and then Xj, then the cost from U to U∪{Xi,Xj} is BestScore(Xi, U)+BestScore(Xj, U∪

{Xi}). On the other hand, if we add Xj ﬁrst, then the cost from U to U ∪ {Xi,Xj} is

BestScore(Xj, U) + BestScore(Xi, U ∪ {Xj}). However, due to the ancestor relations, we know that BestScore(Xi, U ∪ {Xj}) = BestScore(Xi, U). So, regardless of the order we

32 add the two variables, Xi will have the same parent choices. If we add Xj ﬁrst, though, then

Xj will have fewer choices. Therefore, adding Xj as a leaf ﬁrst can never be better than adding

Xi ﬁrst [56].

5.2.3 POPS Constraints Pruning

We ﬁnd the ancestor relations by constructing the parent relation graph and extracting its strongly connected components (SCCs). The SCCs of the parent relation graph form the com-

ponent graph, which is a DAG [14]; each component graph node ci corresponds to an SCC scci from the parent relation graph (which in turn corresponds to a set of variables in the Bayesian

network). The component graph includes a directed edge from ci to cj if the parent relation

graph includes an edge from a variable Xi ∈ scci to Xj ∈ sccj.

The component graph gives the ancestor constraints: if cj is a descendent of ci in the compo-

nent graph, then variables in sccj cannot be ancestors of variables in scci. Consequently, the component graph gives POPS constraints which allow the order graph to be pruned without considering bounds. In particular, the POPS constraints allow us to prune nodes in the order graph which do not respect the ancestor relations.

Tarjan’s algorithm [48] extracts the SCCs from directed graphs, like the parent relation graph. We chose to use it because, in addition to its polynomial complexity, it extracts the SCCs from the parent relation graph consistent with their topological order in the component graph.

i 2 Consequently, all of the parent candidates of X ∈ scci appear in PCi = ∪k=1scck . After extracting the m SCCs, the search can be split into m indepedent subproblems: one for each

th SCC where starti is PCi−1 and goali is PCi. That is, during the i subproblem, we select the optimal parents for the variables in scci. Of course, start0 = ∅ and goalm = V. The worst- case complexity of subproblem i is then O(2|scci|). Figure 9(a) shows the pruned order graph resulting from the parent relation graph in Figure 8. In particular, it shows the ﬁrst subproblem, from PC0 = ∅ to PC1 = {X1,X2}, and the second subproblem, from PC1 to PC2 = V.

2Depending on the structure of the component graph, this may be a superset of the parent candidates for X.

33 { } { }

{ 1 } { 2 } { 1 } { 2 } 1st Subproblem { 1,2 } { 1,2 } 2nd Subproblem

{ 1,2,3 } { 1,2,4 } { 1,2,5 } { 1,2,6 } { 1,2,3 } { 1,2,4 } { 1,2,5 } { 1,2,6 }

{ 1,2,3,4 } { 1,2,3,5 } { 1,2,4,5 } { 1,2,3,6 } { 1,2,4,6 } { 1,2,5,6 } { 1,2,3,4 } { 1,2,3,5 } { 1,2,4,5 } { 1,2,3,6 } { 1,2,4,6 } { 1,2,5,6 }

{ 1,2,3,4,5 } { 1,2,3,4,6 } { 1,2,3,5,6 } { 1,2,4,5,6 } { 1,2,3,4,5 } { 1,2,3,4,6 } { 1,2,3,5,6 } { 1,2,4,5,6 }

{ 1,2,3,4,5,6 } { 1,2,3,4,5,6 }

(a) (b)

Figure 9: Order graphs after applying the POPS constraints. (a) The order graph after applying the POPS constraints once. (b) The order graph after recursively applying the POPS constraints on the second subproblem.

The (worst-case) size of the original order graph for n variables is as follows.

O(2|scc1|+...+|sccm|) = O(2n) (12)

The worst-case size of the search space after splitting into subproblems using the SCCs is as follows.

O(2|scc1| + ... + 2|sccm|) = O(m · max 2|scci|) (13) |i|

That is, the complexity is at worst exponential in the size of the largest SCC. Consequently, our method can scale to datasets with many variables if the largest SCC is of manageable size.

5.2.4 Recursive POPS Constraints Pruning

As described in Section 5.2.3, the size of the search space for the ith subproblem is O(2|scci|), which can still be intractable for large SCCs. However, recursive application of the POPS

34 1 2 3 4

Figure 10: The toy example. There is a bridge between variable x2 and x3.

constraints can further reduce the size of the search space. We refer to the constraints added by this strategy as recursive POPS constraints.

The intuition is the same as that behind the POPS constraints. As an example, consider the

subproblem associated with scc2 in Figure 9, which includes variables X3, X4, X5 and X6. Naively, the order graph associated with this subproblem has O(24) nodes. However, suppose we add variable X3 as a leaf ﬁrst. Then, the remaining variables split into three SCCs, and their order is completely determined. Similarly, selecting any of the other variables as the ﬁrst to add as a leaf completely determines the order of the rest. Figure 9(b) shows the order graph after applying recursive POPS constraints.

In general, selecting the parents for one of the variables has the effect of removing that variable from the parent relation graph. After removing it, the remaining variables may split into smaller SCCs, and the resulting smaller subproblems can be solved recursively. These SCC checks can be implemented efficiently by again appealing to Tarjan’s algorithm. In particular, after adding variable Xi as a leaf from U, we remove all of those variables from the parent relation graph. We then find the topologically first SCC and expand just the variables in that component. As we recursively explore the remaining variables, they will all eventually appear in the first SCC of the updated parent relation graph.

5.2.5 Bridge Constraints

We have shown the strongly connected components of the parent child relation graph can help to reduce the search search. Now let us consider the following toy example in Figure 10.

Using applying POPS constraints described prevously, we can reduce some search space, as

35 Ԅ

1 2 3 4

1,2 1,3 1,4 2,4 3,4

1,2,3 1,2,4 1,3,4 2,3,4

1,2,3,4

Figure 11: The toy example. The search space pruned by recursive POPS. shown in Figure 11. The usage of POPS constraints to get strongly connected components did not quite efﬁcient, Though that number of the total paths searched is reduced a lot, only one node {x2, x3} is saved from expanding.

Let us use the ancestor relation information in another way. Consider the relations between x2 and x3, if components. We will further show that we can explore more useful information from POPS constraints. Observe the following toy example in Figure 10. There is 3 bridges in the graph. In graph theory, a bridge, or cut arc, is an edge of a graph whose deletion increases its connected components. Note that in the directed graph, the bridge contains both directions.

For the bridge between x2 and x3, there are three possibalities in a optimal network:

1. there is neither edge x2 → x3 nor x3 → x2.

2. there is only edge x2 → x3

3. there is only edge x3 → x2

In each of the above three cases, there are 2 strongly connected componnets. Thus we can to search all three cases, then combine the results. The correponding order graph, needed to

36 Ԅ

1 2 3 4

1,2 3,4

1,2,4 1,2,3 1,3,4 2,3,4

1,2,3,4

Figure 12: The toy example. The search space pruned by cutting the bridge between x2 and x3. serach, is illuslated in Figure 12. More nodes are saved from expanding, and more paths are saved from searching.

5.3 Preliminary Results

We empirically tested our new proposed prune methods using the A* algorithm We use benchmark datasets from the UCI machine learning repository and Bayesian Network Repository. The experiments were performed on an IBM System x3850 X5 with 16 core 2.67GHz Intel Xeon Processors and 512G RAM; 1.7TB disk space was used.

We considered three variants of A*: a basic version not using POPS constraints; a version using the POPS constraints but not applying them recursively as described in Section 5.2.4; and a version which uses the recursive POPS constraints. The result is illustrated in Figure 13.

As the ﬁgure shows, the versions of A* augmented with the POPS constraints always outper- form the basic version. The improvement in running time ranges from two times on several of the datasets to over an order of magnitude on three of the datasets. Additionally, the basic version is unable to solve Soybean and Barley within the time limit (30 minutes); however,

37 Graduate Center/City University of New York University of Helsinki FINDING OPTIMAL BAYESIAN NETWORK STRUCTURES WITH CONSTRAINTS LEARNED FROM DATA Xiannian Fan, Brandon Malone and Changhe Yuan Several recent algorithms for learning Bayesian network structures first calculate potentially optimal parent sets (POPS) for all variables and then use various optimization techniques to find a set of POPS, one for each variable, that constitutes an optimal network structure. This paper makes the observation that there is useful information implicit in the POPS. Specifically, the POPS of a variable constrain its parent candidates. Moreover, the parent candidates of all variables together give a directed cyclic graph, which often decomposes into a set of strongly connected components (SCCs). Each SCC corresponds to a smaller subproblem which can be solved independently of the others. Our results show that solving the constrained subproblems significantly improves the efficiency and scalability of heuristic search-based structure learning algorithms. Further, we show that by considering only the top p POPS of each variable, we quickly find provably very high quality networks for large datasets. Bayesian Network Structure Learning with Graph Search Bayesian Network Structure Learning Graph Search Formulation Potentially Optimal Parent Sets (POPS) Representation. Joint probability distribution over a set of variables. The dynamic programming can be visualized as a search Structure. DAG storing conditional dependencies. through an order graph. While the local scores are defined for all 2n-1 possible parent sets for •Vertices correspond to variables. each variable, this number is greatly reduced by pruning parent sets •Edges indicate relationships among variables. The Order Graph that are provably never optimal (Tian, J. 2000; de Campos, C. P., et al Parameters. Conditional probability distributions. Calculation. Score(U), 2011.). Learning. Find the network with the minimal score for complete best subnetwork for U. dataset D. We often omit D for brevity. Node. Score(U) for U. We refer to the above pruning as lossless score pruning because it is Successor. Add X as a guaranteed to not remove the optimal network from consideration. leaf to U. We refer to the scores remaining after pruning as potentially optimal Path. Induces an parent sets (POPS). Denote the set of POPS for variable Xi as Pi . ordering on variables. where Score(Xi | PAi) is called local score Size. 2n nodes, one Dynamic Programming for each subset. Figure 1: Order Graph Intuition. All DAGs must have a leaf. Optimal networks for a single for 4 variables variable are trivial. Recursively add new leaves and select optimal parents until adding all variables. All orderings have to be considered. Admissible Heuristic Search Formulation Start Node. Top node, {}. Goal Node. Bottom

Begin with a Pick one variable Pick another leaf. Continue picking leaves node, V. single variable. as leaf. Find its Find its optimal and finding optimal Shortest Path. Corresponds optimal parents. parents from parents. to optimal structure. current. Recurrences. g(U). Score(U). h(U). Relax acyclicity. Table 1: The POPS for six variables problem. The ith row

shows the Pi .

POPS Constraints Pruning Recursive POPS Constraints Pruning Motivation: Not all variables can possibly be ancestors of the others. Selecting the parents for one of the variables has the effect Previous technique: Consider all variable orderings anyway. of removing that variable from the parent relation graph. Shortcomings: Exponential increase in the number of paths in the After removing it, the remaining variables may split into search space. smaller SCCs, and the resulting smaller subproblems Contribution: Construct the parent relation graph and find its SCCs; can be solved recursively. Figure 3(b) shows the example. divide the problem into independent subproblems based on the SCCs. Top-p POPS Constraints Motivation: Despite POPS constraints pruning, some We collected all the potential problems remain difficult. parents–children relation from Previous technique: AWA* has been used to find bounded POPS, and get the resulting optimality solutions. parent relation graph. Shortcomings: AWA* does not give any explicit tradeoff between complexity and optimality. Contribution: Lossy score pruning gives a more principled way to control the tradeoff; we create the parent relation Figure 2: The parent graph using only the best p POPS of each variable and relation graph. discard POPS not compatible with this graph. We extracted its strongly connected components (SCCs) from parent relation graph. The SCCs form the component graph (Cormen et al. Rather than constructing the parent relation graph by 2001), giving the ancestor constraints (which we call POPS aggregating all of the POPS, we can instead create the graph Figure 5: The behavior of dataset Hailfinder under the top-p constraint). Each SCC corresponds to a smaller subproblem which can by considering only the best p POPS for each variable. POPS constraint as p varies. be solved independently of the others. Experiment Result Observation: The constraints seem to help benchmark network datasets more than UCI. FAIL FAIL Observation: Even for very small values of p, the top-p 600 No Constraint POPS constraint results in networks provably very close 511.55 to the globally optimal solution. 500 POPS Constraint 435.65

400 Recursive POPS Software: http://url.cs.qc.cuny.edu/software/URLearning.html Constraint

300 Selected References 1. de Campos, C. P.; and Ji, Q. Efficient learning of Bayesian networks using constraints. JMLR 2011. 200 2. Tian, J. A branch-and-bound algorithm for MDL learning Bayesian networks. In UAI’00, 2000. 3. Yuan, C.; Malone, B.; and Wu, X. 2011. Learning optimal Bayesian networks 100 76.51 using A* search. In IJCAI ‘11, 2011. 46.62 26.76 20.93 2.51 4. Yuan, C.,; and Maline, B. 6.47 4.06 1.28 An Improved Admissible Heuristic for Learning Optimal 0 Bayesian Networks. In UAI’12, 2012.. Figure 3: Order graphs after applying the POPS constraints. (a) Autos Soybean Alarm Barley The order graph after applying the POPS constraints once. (b) The Figure 4: Running Time (in seconds) of A* with three different Figure 13: The results on dataset Autos, Soybean, Alarm, Barly on three versions of A*.Acknowledgements order graph after recursively applying the POPS constraints on constraint settings: No Constraint, POPS Constraint and This research was supported by NSF grants IIS-0953723, IIS-1219114 and the Academy the second subproblem. Recursive POPS Constraint. of Finland (COIN, 251170). with the POPS constraints, all of the datasets are easily solved within the limit. The number of nodes expanded, and, hence, memory requirements, are similarly reduced.

The recursive POPS constraints always reduce the number of nodes expanded. However, it sometimes increases the running time. The overhead of Tarjan’s algorithm to recursively look for SCCs is small; however, in some cases, such as when the parent relation graph is dense, the additional work yields minimal savings. In these cases, despite the reduction in nodes expanded, the running time may increase.

On the other hand, when the parent relation graph is sparse, the advantages of the recursive POPS constraints are sometimes more pronounced. For example, the running time of Alarm and Barley is reduced in half by recursively applying POPS constraints. Most networks constructed by domain experts, including those evaluated in this study, are sparse. Our analysis shows that these datasets also yield sparse parent relation graphs. Thus, our results suggest that the recursive constraints are sometimes effective when the generative process of the data

38 is sparse. The overhead of looking for the recursive POPS constraints is minimal, and it sometimes offers substantial improvement for sparse generative processes. So we always use it in the remaining experiments.

39 6 Conclusion and Future Work

In this paper, we propose various methods for scaling up the Bayesian networks structure learning. Two directions are preliminarily explored. One is to tignthen the bouds; we tighten the lower boound by using more informed variable groupings when creating the heuristics, and tighten the upper bound using anytime algorithm learning algorithm. The other is to prune the search space by extracting contraints directly from data.

As future work, we plan to investigate integration of domain knowledge into heurstics building. Combining prior knowledge with information from suboptimal solutions could further scale Bayesian networks learning to big data.

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