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Introducing Graphical Models to Analyze GeneticProgramming Dynamics
Erik HembergMIT CSAIL
Constantin BerzanDepartment of Computer Science
Tufts [email protected]
Kalyan VeeramachaneniMIT CSAIL
Una-May O’ReillyMIT CSAIL
ABSTRACTWe propose graphical models as a new means of understand-ing genetic programming dynamics. Herein, we describehow to build an unbiased graphical model from a popula-tion of genetic programming trees. Graphical models bothexpress information about the conditional dependency re-lations among a set of random variables and they supportprobabilistic inference regarding the likelihood of a randomvariable’s outcome. We focus on the former information: bytheir structure, graphical models reveal structural depen-dencies between the nodes of genetic programming trees.We identify graphical model properties of potential inter-est in this regard – edge quantity and dependency amongnodes expressed in terms of family relations. Using a simplesymbolic regression problem we generate a graphical modelof the population each generation. Then we interpret thegraphical models with respect to conventional knowledgeabout the influence of subtree crossover and mutation upontree structure.
Categories and Subject DescriptorsD.1.2 [Programming Techniques]: Automatic Program-ming
General TermsAlgorithms
Keywordsgenetic programming, graphical models, Bayesian network
1. INTRODUCTIONWe begin by noting that the population of a GP run can be
regarded as observed stochastic samples of a set of random
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variables where the stochasticity injected into the sampleschanges over time (generations) due to the evolutionary op-erators of selection, mutation and crossover. Alternatively,but equivalently, the GP population each generation can beregarded as observed samples of a different statistical pop-ulation. Generally, a GP run is the outcomes from a set ofrandom variables, spaced in time at the interval of genera-tions.
The population dynamics of genetic programming are stillnot completely understood (15), i.e. how the dependenciesbetween the solutions vary in the population and how theychanges over time (generations). Using the insight that eachgenetic population of a GP generation is a set of samples, ouraim is to model the statistical population from which theyare drawn as a distribution of random variables. We willthen analyze the changes in each distribution as the gener-ations proceed. Previously, analyzing a GP run’s dynamicshas focused on studying the distribution of tree size, treedepth or solution fitness as probability distributions or sim-ply calculating the moments of this distribution like meanand standard deviation (17). In contrast, our method is aprobabilistic statistical approach.
We examine how to build a multivariate distribution Πwhich is represented by a set of samples (a generation) ofGP trees. We use a prototype tree which is the size andshape of the largest possible GP tree for the problem. Eachnode of the prototype tree represents a random variable. GPtrees are mapped onto the prototype tree to become samplesof the multivariate distribution, see Fig. 1. We use a prob-abilistic graphical model to represent this multivariate dis-tribution. (In a graphical model a directed acyclical graphdenotes the conditional dependences between random vari-ables.) To efficiently build Π requires accommodating GPtrees which are of variable sizes and shape, identifying thebest possible representation of the graphical model (GM)given a set of trees, and devising computational efficien-cies which allow the building of the graphical model to betractable.
In the last two decades building graphical models and as-sociated inference engines has been addressed by numerousresearchers (7; 11). We explored a variety of techniques(which lead to different graphical models) from this body ofliterature to address the challenges graphical model buildingin the GP context specifically poses. We present one tech-nique in Section 3. It is chosen because it is unbiased withrespect to any assumptions about variable dependency.
FOGA’13, January 16-20, 2013, Adelaide, Australia.Copyright 2013 ACM 978-1-4503-1990-4/13/01 ...$15.00.
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Random VariablesOutcomes
Example GP tree 1Prototype Tree Example GP tree 2
Figure 1: A prototype tree of depth d = 2, twoexample GP trees, and, as a result of alignment withthe prototype tree, their random variable outcomes.Note that we do not show # (null) nodes.
Once we describe computationally efficient means to builda probabilistic graphical model we proceed to analyze howthe generational graphical models transition over the courseof a run as a result of selection and crossover. Specifi-cally, we set up a very simple GP symbolic regression prob-lem (Pagie-2D (16)) to execute over a fixed number of gener-ations. Then we execute experiments composed of GP runswhich vary the presence or absence of different operators.For each set of runs with a particular variation operator(only crossover or only mutation), we then aggregate all thetrees of a generation and treat this as a set of samples fromwhich we build a graphical model. We analyze the changesin the resulting graphical models as we process one genera-tion after another.
The paper’s contributions are:
1. Use of probabilistic graphical models for modeling struc-tural dynamics of a population of GP trees
2. Description of an efficient technique to build graphicalmodels. The technique needs to be fast because ouraim is to to build graphical models for the populationfrom every generation. The technique further needsto be unbiased to permit the data to be as accuratelyreflected as possible.
3. Analyzes of GP tree dynamics in different variationaloperator scenarios with respect to structural depen-dencies mapped from graphical models.
We proceed in the following manner: In Section 2 wepresent an overview of current techniques that analyze GPpopulations. In Section 3 we present how we resolve theissues arising in how to build a graphical model for a GPpopulation. We also present how we have used a genetic al-gorithm to efficiently learn the structure of a Bayesian net-work from a fully observed data set. In Section 4 we presentthe results for a first set of statistical analyzes that referencegenerational graphical models. We present our conclusionsand future work in Section 5. The Appendix describes thebuilding of the graphical model in more detail.
2. BACKGROUNDMany different statistics have been used to study GP.
Widely used and reported, “conventional” analyzes includea time series plot, per generation, of mean, standard devi-ation, maximum and minimum size of structure (e.g. tree,
graph or other executable representation) or population fit-ness. Tomassini et al. (21) study fitness distance correlationas a difficulty measure in genetic programming. Langdonand Poli (12) analyze the behavior of GP on a wide rangeof problems, e.g. artificial ant and the Max Problem usingschema analysis and Price’s covariance and selection theo-rem.
Nearly all visualizations of structure have been for illus-trative purposes according to Daida et al. (5), who visualizedtree structures in GP. This new way of “seeing” can afford apotentially rich way of understanding dynamics that under-pin GP. Almal et al. (2) perform a population based study ofevolutionary dynamics in GP and show that heat maps area useful tool for exploring the dynamics of genetic program-ming. The visualization allows the user to draw conclusionsfrom data. We expand on this by inducing a graphical modelto both visualize and explain the structure and content of aGP tree.
Daida et al. (6) study phase transitions in GP search tosupport a statistical mechanics approach where GP wouldbe quantitatively compared to a broad range of other sys-tems rigorously described by phase transitions. In order toidentify subtrees that carry out similar semantic tasks, Wolf-son et al. (22) perform post-evolutionary analysis of GP bydefining a functionality-based similarity score between ex-pressions. Similar sub-expressions are clustered from a num-ber of independently-evolved fit solutions, thus identifyingimportant semantic building blocks lodged within the hard-to-read GP trees.
Raidl and Gottlieb (18) empirically investigate locality,heritability and heuristic bias in Evolutionary Algorithmsby generating individuals with static measurements based onrandomly created solutions. This is done in order to quantifyand visualize specific properties, and shed light onto thecomplex behavior of the system. Here, we take a differentapproach and we investigate the structure and contents of aGP tree by creating a graphical model from a population ofGP trees.
In a paper by McPhee and Hopper (13) they analyzegenetic diversity through population history by size andshowed that the genetic diversity was low. They measurethe number of different nodes in the population based on ifthe nodes begin at the root or non-root. Our study differs byinvestigating the dependencies between the nodes in the so-lutions based on a graphical model from the solution. Thus,we do not need to identify any relationships between individ-ual solutions in the population, instead we take a statisticalapproach and learn a model of the population.
3. GRAPHICAL MODELS FOR GPOur goal is to model the multivariate distribution of the
trees in the GP population in generation t. We denote thisby Πt. The following issues are resolved in order to build amultivariate distribution for the population:
GP trees are of variable size: Because a population’s treesvary in size and structure and GP’s representation isnot homologous, the question of defining the randomvariables is open to question. We choose a prototypetree based approach in which each node in a tree witha maximum size is a random variable and its supportis all possible functions and terminals that could be atits position. We present more details in Section 3.1.
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Defining the random variables in this manner providesus with a fixed set of variables for the multivariatedistribution.
Modeling the multivariate distribution: Our represen-tation of the multivariate distribution needs to allowus to gain insight from it. We choose a Bayesian net-work based representation which allows us to identifydirected dependencies between the nodes of GP trees.We provide more details in Section 3.2.
Efficient building of the distribution: Given our choicesof a prototype tree for the random variable set and aBayesian network model, a further challenge is to buildthe distribution efficiently.
3.1 GP trees to random variablesTo define the random variables for the modeling distribu-
tion we adopt a prototype tree based approach, see Fig. 1.Prototype trees have been used to represent GP trees whendeveloping estimation of distribution based GP (EDA-GP),e.g. Salustowicz and Schmidhuber (19) and Hasegawa andIba (8). A prototype tree is the size and shape of the largest(i.e. completely full) tree in the GP experiment. Each nodein the prototype tree designates a random variable whosesupport (possible outcomes) depends on its position in thetree. That is, the root of the prototype tree, which can as-sume any function, is a random variable with its supportbeing the problem’s function set because each function is apossible outcome for the root’s random variable. Each leafof the prototype tree, which can only assume a terminal, isa random variable with support comprising the problem’sterminal set (because each terminal is a possible outcome ofa leaf’s random variable). For each interior node, supportis the set of all functions and terminals because each termi-nal or function is a possible outcome of an interior node’srandom variable. Because any GP tree may be smaller thanthe experiment’s prototype tree, an extra outcome, null, isadded to the support at every node except the root.
For example, for a symbolic regression problem with func-tion set {+,−, /, ∗} and terminal set {x1, x2}, prototype treeleaves have a support of two terminals and null, the root hasa support of four functions, and all the intermediate nodeshave a support of six outcomes (function set, terminals andnull). The number of random variables in the multivariatedistribution at time t, Πt depends on the size of the proto-type tree. The fundamental challenge in using a prototypetree based representation is supporting a large prototypetree so that every possible size or shape of a GP tree withina run can align with the prototype tree.
The GP tree population at a generation is a set of sam-ples. To transform a tree to a sample, it is aligned withthe prototype tree from the root downwards. Each functionor terminal of the sample is counted as an outcome of therandom variable onto which it maps in the prototype tree.Where the sample tree has no nodes, null outcomes are as-sumed. Figure 1 illustrates the transformation of a set ofGP trees to outcome statistics of the prototype tree’s ran-dom variables. As a graphical model for Πt is built for eachgeneration, the population of GP trees is transformed tosamples by alignment with the prototype tree, and the out-comes of the random variables are statistically accumulatedfrom each sample.
We choose a Bayesian network representation for the graph-ical model because it allows us to analyze the changes in themultivariate distribution via its graphical properties. Next,we describe this representation.
3.2 Bayesian networks as a graphical modelWe can associate the population in standard GP with an
implicit probability distribution over the random variables inthe prototype tree. We want to explicitly model this distri-bution, and examine the dependency structure that emergesbetween nodes in the prototype tree. Our modeling tool ofchoice is the Bayesian network, consult (11) for an introduc-tion to Bayesian networks. A Bayesian network B = 〈G, θ〉 isa probabilistic graphical model that represents a joint prob-ability distribution over a set of random variables Y1, . . . , Yn.The Bayesian network representation has two components.A directed acyclic graph (DAG) G encodes (conditional) in-dependence relations between random variables. More pre-cisely, a missing edge encodes an independence relationship.Each random variable is a node in this graph. A set of lo-cal probability models θ defines the conditional probabilitydistribution of each node given its parents in the graph. LetPaY denote the parents of node Y in G. Then the networkB encodes the following probability distribution:
P (Y1, . . . , Yn) =
nYi=1
P (Yi | PaYi).
Learning Bayesian networks from data has received muchattention in the machine learning community (11). Thespace of all possible DAGs for n nodes is super-exponentialin n, and finding the optimal network for a given data set isin general NP-hard (11). Learning algorithms for Bayesiannetworks loosely fit in two categories: constraint-based ap-proaches, which use statistical tests to determine whetheran edge is present or not, and search-and-score techniques,which define a scoring function and then search for a highscoring network. The scoring function prefers network struc-tures that model the data set well, while penalizing struc-tures that are too complex.
3.3 Learning network structureSome search-and-score techniques, such as the K2 algo-
rithm (4), require that we prepare an edge order for learn-ing the parameters of the nodes. Given an order, an edgecan appear between nodes A and B only if A precedes Bin the order. In our case of prototype trees for GP, speci-fying an accurate node order is not obvious. For example,if we impose a left-to-right depth-first search order, then anode in the graphical model cannot have an edge from itsright sibling or its right uncle. In GP terms, this implieswe would not sample relationships among the functions andterminals which are parameters of a parent function, in aright to left order. In Fig. 2 the possible dependencies inleft-to-right depth-first order is shown, the dashed arrowsare impossible dependencies given the ordering. If we im-pose a left-to-right breadth-first search order, a node in thegraphical model cannot have an edge from any node to itsright (at the same height) or below it. K2 is commonlyused in EDA-GP (8) where the node order is assumed tobe depth-first search yet its assumptions are not necessarilyvalid.
Because we wanted to discover the true dependency struc-ture in a population’s distribution, we aimed to impose as
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Figure 2: Shows order bias, with possible depen-dencies in left-to-right depth-first order depicted bythe arrows, the dashed arrows are impossible depen-dencies given the ordering. In the square the stringrepresentation of the tree, with the same possibleand impossible dependencies, is shown.
A B C D E F ...
Figure 3: Possible structure with known node or-der. Suppose the node order is A, B, C, D, E, F, ....This means that every edge in the network must gofrom left to right. Suppose we have already foundthe best parent sets for nodes A, B, C, and D, andare now considering node E. Each of E’s parentsmust precede E in the order. We can add any ofthe edges shown in red without introducing a cycle.Furthermore, the parents we choose for E will notlimit the choice of parents for nodes F and beyond.
few restrictions as possible on the type of graphical modelstructure we could learn. Thus, we opted for a search-and-score algorithm that did not require a node order asinput (1). This algorithm uses an evolutionary algorithmin a search-and-score approach to propose and search fornode orders leading to an optimal network structure. Foreach order the EA proposes, the algorithm then identifiesan optimal network structure. This requires evaluating ev-ery structure that is consistent with that order. Since in ourapproach each individual is an order, this identification oc-curs during the evaluation of an individual’s fitness. Whilethe identification is computationally expensive, the numberof structures (DAGs) that are consistent with the order isfinite. In Section A, we present a scalable approach to sys-tematically overcome the computational burden.
When learning the order of a DAG, parents of any nodeY must precede Y in the ordering. In other words, givena node ordering R, such that for any nodes X and Y , if
XR< Y (X precedes Y in R), then any edge between X and
Y must be directed from X to Y . The ordering constraint Rallows the choice of parents for one node to be independentof the other nodes’ parents. This is because any combinationof edges consistent with the ordering R produces an acyclicgraph. Figure 3 illustrates this observation.
With a decomposable score this makes the choice of par-ents for any node X independent of the choice of parentsfor any other node Y . A decomposable score allows us toevaluate parent sets for each node independently and selectthe best one for each node and total score of the entire net-work is the total sum of all the scores for each node. It isapparent that the minimum across each node’s family scoreminimizes the overall score as well. This means that nowwe can search for each node’s parent set independently. Ifwe limit the indegree (number of parents) of every node toa constant k < n/2, then there are at most
1+
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possible parent sets for each node, where n is the number ofnodes in the network. Finding the optimal parent set for allnodes thus requires at most
n · O(nk) = O(nk+1)
parent sets to be evaluated, which is polynomial in n, for afixed k. This means that we can exhaustively search overthe space of networks that are consistent with an orderingR and have an indegree of at most k.
Of course, an O(nk+1)-time exhaustive search is only rea-sonable if k is small (up to about 3 or 4). This restric-tion is not as bad as it seems, because we cannot learnBayesian networks with large parent sets anyway, unless wehave an extremely large training set. To see why, consider aBayesian network where all variables are binary. If a variableX has no parents, then to compute the Bayesian score weonly need two counts from the training set: M [X = 0] andM [X = 1]. If X has one parent Y , then we need four counts:M [X = 0, Y = 0], M [X = 0, Y = 1], M [X = 1, Y = 0], andM [X = 1, Y = 1]. If X has two parents, we need eightcounts, and so on. In general, if X has k parents, we needa total of 2k+1 counts. As the number of parents PaX in-creases, it becomes less and less likely to find representativecounts for all possible assignments M [PaX = paX , X = x]in the training set. This phenomenon of data fragmentationmeans that we can only learn networks where each node hasa small numbers of parents. Thus, setting the maximumindegree k to a small value is justified.
4. EXPERIMENTSIn this section we start by describing our choice of demon-
stration problem (Section 4.1), our procedure for transform-ing population data into graphical models (Section 4.2) andhow we set up a number of experiments (Section 4.3) forwhich we build a graphical model every at generation. Bytheir structure, graphical models reveal structural depen-dencies between the nodes of genetic programming trees. InSection 4.2 we identify graphical model properties of poten-tial interest in this regard – edge quantity and dependencyamong nodes expressed in terms of family relations. Wethen interpret each experiment in light of the operator ithas used and the information the graphical model conveys.This section contains an extended analysis of experimentsinitially done in (9) and (10).
4.1 Demonstration ProblemBecause our goal is to figure out the nature of insights
a graphical model offers in terms of its dependency struc-
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Table 1: Parameters for the GP runs in ECJ. XO iscrossover, Mut is mutation, No ops is selection with-out variational operators and No Sel is crossoverwith random selection.
Parameter XO Mut No ops No SelCrossover 0.9 0.0 0.0 0.9Mutation 0.0 1.0 0.0 0.0Selection Tournament size 7 RandomPopulation size 10,000Generations 40Initialization Ramped Half-HalfMax Depth 5Language Symbolic RegressionFunctions (F) {+,−, ∗, %}Terminal (T ) {x,y, 0.1, 1.0}
ture, for our experiments we choose Pagie-2D (16) which isa simple but sufficiently realistic symbolic regression prob-lem with a modestly sized set of functions and terminals.The problem is to regress an expression matching the targetfunction 1/(1 + x−4) + 1/(1 + y−4) over 676 fitness cases in[−5, 5]2, and fitness is calculated as the mean squared er-ror. The GP parameters are summarized in Table 1, and% is protected division. We run standard GP on the Pagie-2D problem using tournament selection with a tournamentsize of 7 and ramped half-half initialization. We employ aprototype tree of depth 5, which has 63 nodes implying anequivalent maximum tree size for the Pagie-2D runs. Ourruns execute for 40 generations, using ECJ1, with a popula-tion size of 10,000.
4.2 Graphical Model Building ProcedureAfter each generation, we store the entire population to
disk so we can build its corresponding graphical model. Westart by converting each population into a data set D withone row for every tree and 63 columns, one per outcomeof the 63 random variables. We represent each tree as arow of 63 outcomes, by aligning it to the prototype tree,starting from the root, with a depth-first order traversaland substituting null for any node of the tree not in theprototype tree.
The root can take any outcome from the set of func-tions: Sroot = {+,−, ∗, %}. The leaves can take any out-come from the set of terminals, as well as null: Sleaf ={x, y, 0.1, 1.0, null}. Intermediary nodes can take any out-come from the set of functions and terminals, as well asnull: Sintermediary = {+,−, ∗, %, x, y, 0.1, 1.0, null}. Thus,our data set D’s columns are 63 random variables Y ={Y1, Y2, . . . Y63}. Each variable is discrete, and its supportis either Sroot, Sleaf, or Sintermediary. We then learn, as de-scribed in Section 3, a graphical model for the distributionP (Y1, Y2 . . . Y63|D).
4.3 Experiment Definition and MethodWe define three experiments which vary in terms of GP
operator in order to detect differences in this respect:
No variational operators Neither crossover nor mutationis used, selection only acts on the population.
1http://cs.gmu.edu/~eclab/projects/ecj/
Crossover Single point subtree crossover is the only vari-ational operator and it is used in combination withselection.
Mutation Subtree mutation is the only variational opera-tor and it is used in combination with selection.
No selection Single point subtree crossover is the only vari-ational operator and selection is random.
Each experiment consists of 30 independent runs. For eachrun we build a graphical model each generation. This resultsin 1200 graphical models per experiment and 4800 graphicalmodels overall.
4.4 Study of Graphical Model PropertiesThis section outlines the properties of the graphical model
which is studied as well as the results from the differentexperimental setups. The aim is to identify properties whichcan explain the dynamics of GP populations.
4.4.1 Edge QuantityFig. 4 shows an example graphical model for a GP popula-
tion at generation 25 (G25), taken from a run using crossover.In addition to studying graphical models at different gener-ations directly, a graphical model structural property wor-thy of attention is the number of edges in the graph. Ingraphical model terms, because an edge from Node A toNode B implies that the likelihood of random variable Bdepends on random variable A, edge quantity reveals thedegree of variable inter-dependence. Edge quantity conveysdifferent information from previous work in GP where, bysimple counting, the frequency of symbols is determined oreven the frequency of symbols in specific tree locations isdetermined. Simple counting does not reflect dependencieswhich the graphical modeling mines from the population bytreating it as samples and using max-likelihood estimation.Given that a graphical model supplies dependency informa-tion, these dependencies should then be mapped back tothe prototype tree to provide interpretation in a GP con-text. That is, each random variable of a graphical modelmaps to a node of a prototype tree which in turn maps tothe symbol set, i.e. the function and terminal sets, of a GPexperiment.
We can formulate one simple hypothesis: a random pop-ulation of GP trees would result in a graphical model wherethere are arbitrary dependencies (i.e. edges) between nodescompared to a population evolved later in a run. The laterpopulation has been “shaped” by selection and variationso it will be reflected by a structure in the dependencies.These dependencies will reflect the complex mapping be-tween genotype and phenotype, i.e. the direct representationof the solution and its behavior. Fig. 5(a), with data aggre-gated over each of the four experiments, shows the change inedge quantity over generations. It provides one unit of evi-dence for the hypothesis. In the initial populations’ graph-ical model there are few edges, then, due to selection, thenumber sharply rises regardless of whether variation is em-ployed. When a variational operator is employed, subse-quently the edge quantity diminishes and eventually tapers.This can also be seen for the run without selection, wherethe rise is significantly smaller. We speculate that this dy-namic will generally remain the same with other problemsdue to the macro-influences of the operators but will vary indetail due to the unique fitness landscape of every problem.
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Figure 5: Edge quantity plots
Edge quantity can be studied further. Fig. 5(b) indicateshow many edges there are at each depth level of the proto-type tree by showing the average number of edges per node,
with error bars, for a run which used crossover. The nodesare ordered depth-first left-right. The coloring shows thedepth level of the node and the number of edges are nor-malized according to the possible number of edges, 62. Weobserve more edges at lower depth levels.
4.4.2 Edge TypeWhile a random population’s graphical model edges will
generally span, in GP terms, between a node and its parent,we would hypothesize that an evolved population’s edgeswould span between a node and its sibling and even, tohigher ancestral generations (in tree structure terms), to itsuncles and grandparents or farther. This leads us to inves-tigate the ancestral property of edges.
We therefore also perform a basic enumeration of differ-ent types of edges. We define parent, sibling, grand-parent,grand-grand-parent, grand-grand-grand-parent, grand-grand-grand-grand-parent, uncle, and unclassified types. Siblingedges are one direction only: a sibling is the rightmost childof a parent but not the leftmost child of a parent. The edgetypes are illustrated in Fig. 6(a) using different dashes todenote each type. In Fig. 6(b) an example tree with a countof parent, left sibling, nephew and grandparent edge typesis shown.
The interpretation of edges according to this structuralancestry typing is interesting. To see this, it is necessary toconceptually map between a prototype tree’s (or graphicalmodel’s) edges to GP context in the following manner: A GPtree is a representation for an (executable) expression thatis a nested set of functions where arguments of a functionare recursively themselves functions until the recursion“bot-toms out” where arguments are non-functions (variables orconstants, a.k.a terminals) represented as leaves. A GP treeis evaluated (a.k.a executed or interpreted) by a preorderparse where successive arguments are recursively evaluatedin preorder. The dependency edges to a leaf from nodesabove it in the tree indicate which functions in the expres-sion depend on that argument’s value. If the dependencyedge is reversed in direction, i.e. from a leaf to a node aboveit, it indicates that a function elsewhere in the tree (comingeither before or after the evaluation of the terminal whenthe nested expression is evaluated) influences the value ofthe terminal (e.g. whether it is 0.1 or 1.0 in Pagie-2D).
We now proceed to analyze each experiment wherein vari-ation differs. We start with a baseline where the GP run usesselection but no crossover or mutation.
4.4.3 No Variational Operators ExperimentWhen GP is run without any variation operators, selection
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Figure 6: Possible edge types by ancestry. In Fig. 6(a) the edges (dependencies) in the GM which are studiedare shown. Fig. 6(b) there is an example of how the edge types are counted.
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(c) Generation 40, no variational operators
Figure 7: Selection Only Experiment: graphicalmodels for one randomly selected run at generation0, 13 and 40.
eventually hones in on a completely homogeneous popula-tion. We find in our runs that this usually happens by gen-eration ten. For one run, Fig. 7(a) shows a graphical modelof the random population at generation 0. It has relativelyfewer edges than subsequent generations. Fig. 7(b) shows agraphical model of generation 13 when the number of edgeshas increased. After the generation when the populationhas converged to a single point, any variation we observe ingenerational graphical models is due to the stochasticity ofthe EA we use to build the model.
Recall that our model building algorithm limits the num-ber of dependency edges from a node to three. This impliesthat the maximum number of edges, given a prototype treeof size 63, equals 183. Which is the case at generation 40,when the population has completely converged to one GPsolution, per Fig. 7(c) the graphical model has the maximumnumber of edges.
Next we consider edge types. The prevailing edge typeis parent, as shown in Fig. 8(a). This figure is a time se-ries plot of edge type quantity expressed as the ratio of thenumber of edges in the graph of the type to the total possi-ble number of edges of the type. For example, for an edgeof type parent there are 62 possible edges, a ratio of 0.5means that there are 31 edges in the graphical model of this
type. There are many different types of edges in a homoge-neous population and they vary more between each genera-tion compared to the experiments using operators. Half ofthe edges are unclassified, the other half is almost uniformlydistributed between the edge types, except for grand grandparent. Moreover, the sibling relations are not as frequentcompared to the experiments using operators. This is againlikely due to the fact that the population is homogeneousand the structure finding algorithm is stochastic.
4.4.4 Mutation Only ExperimentIn the experiment where GP was run only with a subtree
mutation operator for variation, we examine three resultinggraphical models at different generations, the quantity ofedges and their ancestry type.
Mutation randomizes the population, but selection pro-motes the edge types associated with fit solutions. Thismakes later generation graphical models distinguishable fromthe initial population. The graphical models in Fig. 9 showa selective pruning of edges over generations.
Consulting Fig. 5(a) as to the number of edges, we observethe quantity rapidly increasing (in approximately the first7 generations) to reach close to the maximum number ofpossible edges in the graphical model, before decreasing toalmost the same number of edges as nodes. The edge ratiosshown in Fig. 8(b) indicate the most frequent type of edge isfrom a parent to a child. As the number of edges decrease,the most frequent edge type is parent, then sibling and someuncles, with hardly any grand-parent dependencies. Our in-terpretation is that by adding new random subtrees, muta-tion creates new local dependence which is confined to thesubtree. It disrupts any dependencies extending out of thesubtree which it excises. This would imply that mutationruns have more parent edges.
4.4.5 Crossover Only ExperimentConsidering the experiment where GP was run only with
the subtree crossover operator for variation, we examinethree resulting graphical models from different generations,the quantity of edges and their ancestry type.
The graphical models in Fig. 8(c) differ by generation and,in unshown detailed graphical model data (animated gener-ation by generation), we observe that the change from onegeneration to the next is very gradual, more so than withmutation. Consulting Fig. 5(a), we observe that initiallythere are approximately 70 edges, while there is a peak ofapproximately 180 edges around generation six. Then the
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Figure 8: Plot of the ratios of edges in different types. For Pagie 2D without crossover and mutation 8(a),mutation 8(b), crossover 8(c) and no selection 8(d).
number of edges tails off and stabilizes at approximately 130.When we align our animation with the fitness time series wesee the tail off synchronizes with approximately the samegeneration when fitness stops improving and the populationhas started to converge.
Considering the type of edges, per Fig. 8(b), we observethat, after generation 20, almost all nodes in the graphicalmodel have an edge with their ancestor and half have edgesto their sibling. After 25 generations the ratios of the otheredges have somewhat stabilized. Fig. 5(b) is an exampleof crossover only experiment. We see a consistency in thenumber of edges given the depth level of the node in thetree. As expected, the nodes at the lower depth levels havefewer edges.
Crossover is a subtree swap. This implies that it preservesmore short and long range, i.e. ancestral dependencies suchas parent, sibling plus grandparent and uncle. Because ofthe exchange, rather than the new material, it results in only
a minor net change in dependency, at the structural level.Of course, the issue with crossover in GP is its semanticconsequences while its structural behavior directly followsfrom its definition.
4.4.6 No Selection ExperimentFor the experiment where GP was run with random selec-
tion and only the subtree crossover operator for variation,we examine the quantity of edges and their ancestry type.
The graphical models for some generation are not shown,but they have very few edges and the connections are mainlybetween parent and sibling, we verify this by consultingFig. 8(d). In addition, in Fig. 5(a), we observe that initiallythere are approximately 70 edges and within 6 generationsthe number drops to the the minimum number of edges.This effect is most likely due to the effect of initialization.
The next section will further discuss the results from thedifferent experiments.
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4 7 11 14 19 22 26 29 35 38 42 45 50 53 57 60
5 6 8 9 12 13 15 16 20 21 23 24 27 28 30 31 36 37 39 40 43 44 46 47 51 52 54 55 58 59 61 62
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12 36 40 51
2 17 33 48
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3 10 18 25 34 41
43
49 56
4 7 11 14 19 22
24
26 29 35 38 42 45
4647
50 53 57 60
5 6 8 9 2313 15 16 20 21 27 28 30 31 37 39 44 52 54 55 58 59 62
(b) Generation 13, mutation
0
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2 17 33 48
3 10 18 25 34 41 49 56
4 7 11 14 19 22 26 29 35 38 42 45 50 53 57 60
5 6 8 9 12 13 15 16 20 21 23 24 27 28 3031 36 37 39 40 43 44 46 47 51 52 54 55 58 59 61 62
(c) Generation 40, mutation
Figure 9: Mutation Only Experiment: graphicalmodels for one randomly selected run at generation0, 13 and 40.
0
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(c) Generation 40, crossover
Figure 10: Crossover Only Experiment: graphicalmodels for one randomly selected run at at genera-tion 0, 13, and 40
4.5 Cross Experiment RemarksOur understanding of mutation and crossover would lead
us to hypothesize that the dependencies being studied, dueto their structural nature, would be different. ComparingFig. 9 to Fig. 10 bears this out. In the crossover only ex-periment we can observe that crossover’s graphical model inthe final generation has different dependencies and more de-pendencies than mutation. This is interpretable: crossoveris an exchange with less net disruption while mutation intro-duces new material while excising existing material. Bothoperators culminate with approximately the same quantity
of parent edge types but reach this endpoint with differenttrajectories (compare Fig. 8(b) to Fig. 8(c)). Crossover hasmore siblings, and longer range dependencies with unclesand grand-parents. This is consistent with the differencebetween the two operators in terms of different extents ofdependence preservation and disruption.
From the different experiments which differ by variationoperator, we are able to distinguish distinctly different graph-ical models, both regarding number of edges and type ofedges. For all runs, across the variation operator experi-ments (i.e. excluding selection only), when inspecting thefitness, the best fitness is only slightly improving duringthe runs. The operators manipulate the population in sucha way that the dependencies between experiments differ.When comparing the number of edges and the type of edgesfrom each experiment it can be seen that the number ofedges in the crossover only experiment is higher than for arandom population. The crossover operator helps to createa population that has many parent dependencies, sibling de-pendencies and some longer grand-parent dependencies. Incontrast, for the experiments with the mutation operator thenumber of edges is the same as for a initial random popula-tion, but the structure is quite different from initial random.The types of edges are not the same as when crossover is usedeither, there are hardly any longer dependencies, such asgrand-parents edges. When random selection is used thereare no long distance dependencies and the only the min-imum number of edges after a the effects of initializationhave disappeared. Finally, the number of edges in a homo-geneous setup is very high, reaching the maximum limit ofedges and the variation in edges is most likely due to thestochastic learning of the graphical model.
A graphical model also provides a way to generate a GPpopulation from its parsimonious representation. It is anapproximate form of compression where a very importantproperty of the original population is preserved - variabledependency. Further information from the graphical modelcould be taken into account when designing operators inGP, or reduced Bayesian network for EDA-style GP, as inHemberg et al (9).
5. CONCLUSIONS & FUTURE WORKWe have introduced graphical models as a new means of
investigating the dynamics of genetic programming popula-tions in general and,in this work specifically, GP’s operators.Graphical model based analysis is complementary to con-ventional statistics that are tracked across generations. Thegraphical model shows the dependencies between nodes inGP trees when treated as a multivariate distribution repre-sented as prototype trees. We presented an unbiased meansof deriving a graphical model and introduced an EA basedalgorithm for efficiently finding graphical models. Moreover,algorithm wise we only search for order of the nodes ina graphical model, unlike K2 which searches for a graphi-cal model given an order. We exhaustively search for bestgraphical model for each order. For the purpose to overcomethe cost of exhaustive search, we initially build a cache andADTrees. These techniques speed up the fitness evaluationin the creation of the graphical model.
It is important to recognize that alternate kinds of graph-ical models could be built. Graphical models different fromthe one we present have the potential to emphasize differentproperties about GP’s population distribution. We envision
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future investigations that would answer which are most ap-propriate and insightful for modeling GP dynamics. Ourmotivation herein was to investigate whether insight into arun’s dynamics at the population, tree structure level, couldbe gained by observing the dependency structure graphicalmodels reveal about the multivariate distribution associatedwith functions and terminals in GP trees. The graphicalmodel information is in this form structural rather than syn-tax oriented.
Using Pagie-2D problems as exemplars, we take first stepsin analyzing generation and inter-generation dynamics of GPruns in terms of changing graphical model structure. Theexperiments show that there is a distinct difference in thegraphical model of a GP population when different operatorsare used, both in the number of edges in the graphical modeland the types of edges.
For future work further problems and operators will beinvestigated. Other work can be combinations of edge typesas well as doing inference of the outcome of the randomvariables from the generated graphical models. Moreover, ifwe correlate fitness with the graphical model we can:
• Pool all samples and take top fitness slice and learn agraphical model.
• Use fitness value as a conditional value for the graph-ical model.
• Align the data based on fitness to get the dependencystructure of populations with similar fitness distribu-tion
AcknowledgmentWe thank the support received from General Electric GlobalResearch Center and Li Ka Shing Foundation.
APPENDIXA. FAST LEARNING OF THE MODEL
We use an evolutionary algorithm for learning the struc-ture of a Bayesian network from a fully observed data set.This algorithm searches over the space of node orders, ratherthan searching over the space of DAGs. The genetic algo-rithm proposes an order and an optimal DAG is identifiedby exhaustively searching through different possible DAGsgiven this order. Each DAG is scored using a Bayesian scorefunction.
A.1 Representation and OperatorsIn our GA over orders, we use the node order R as the
genotype of an individual. We use swap mutation and cyclecrossover as our operators (3). There are many networksconsistent with the order R. We define the phenotype of anindividual as the best network consistent with R, subject toa maximum indegree k, which is fixed at the start of thealgorithm. This follows the setup that Teyssier and Koller(20) used for their hill-climbing algorithm over orders. TheGA needs to find the best DAG for R and k.
A.1.1 The Bayesian ScoreWe want our scoring function to measure how well a struc-
ture fits the data. We also want it to penalize complexstructures, because a simpler model makes inference more
tractable. The Bayesian score is one of several scoring func-tions satisfying these criteria. We define a prior P (G) overgraph structures, and another prior P (θG | G) over parame-ters given a graph. The posterior over structures for a dataset D is given by Bayes’ rule:
P (G | D) =P (D | G) P (G)
P (D)
where the denominator is a normalizing factor that does notdepend on the structure we are trying to evaluate. TheBayesian score is then given by:
scoreB(G : D) = log P (D | G) + log P (G). (1)
The second term is usually less important, because it doesnot grow with the size of D. The first term is the logarithmof the marginal likelihood of the data given the structure,which averages over the choices of parameters for G:
P (D | G) =
ZθG
P (D | θG ,G) P (θG | G) dθG ,
where P (D | θG ,G) is the likelihood of the data set given thenetwork 〈G, θ〉, and P (θG | G) is our prior over parametersgiven a structure. The marginal likelihood is also known asthe evidence function.
If the prior over parameters is a Dirichlet distributionwhere P (θYi|paYi
| G) has hyperparameters {αGy
ji |ui
} for j
from 1 to |Yi|, which satisfies global and local parameterindependence. With αG
Yi|ui=P
j αGy
ji |ui
, then the marginal
likelihood can be written as follows:
P (D | G) =
Yi
Yui∈Val(PaG
Yi)
24 Γ(αGYi|ui
)
Γ(αGYi|ui
+ M [ui])·
·Y
yji∈Val(Yi)
Γ(αGy
ji |ui
+ M [yji ,ui])
Γ(αGy
ji |ui
)
375(2)
This formula is simplified by choosing a reasonable prior.Priors and Decomposability
It is desirable to have a scoring function that decomposes:
score(G : D) =X
i
FamScore(Yi | PaGi : D)
where the family score FamScore(Yi | U : D) measureshow well the variables U serve as parents of Y . With adecomposable score, local changes to the structure affectonly a small number of families, and so we do not have torecompute the score of the entire network. For example, ifwe change the order such that we only swap the last twopositions in a node ordering R we only have to search forthe best parent sets for the last two nodes and only evaluatethe family scores for these two nodes. Additionally, we cansearch for best parent sets of individual nodes independently.
To ensure that the Bayesian score decomposes, our struc-ture and parameter priors should satisfy some conditions.The structure prior should satisfy structure modularity:
P (G) ∝Y
i
P (PaYi = PaGYi),
where P (PaYi = PaGYi) is the probability of choosing that
specific set of parents for Xi. The parameter prior should
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satisfy global parameter independence and parameter mod-ularity:
P (θYi|U | G) = P (θYi|U | G′)for any pair of structures G, G′.
When we have no prior knowledge about the structure weare trying to learn, we can use a structure prior that assignsequal probability to every possible structure. In this case,we can ignore the second term (logP (G)) in the Bayesianscore (equation 1), and evaluate structures solely by theirmarginal log-likelihood log P (D | G).
In the case when we have no prior knowledge about theparameters, we can use the BDeu prior. (BDeu stands foruniform Bayesian Dirichlet prior satisfying likelihood equiv-alence (11).) For this prior, we only need to specify anequivalent sample size (or strength) α. Given a node Yi andits parents PaYi , the BDeu prior assigns equal probabilitiesto all values of the node and parents:
αyi|paYi= α · 1
|Val(Yi, PaYi)|, (3)
where |Val(Yi, PaYi)| is the number of possible assignmentsto Xi and PaYi . Let γ(Y) = α · 1
|V al(Y)| for any set of nodes
Y. Then the marginal log-likelihood score decomposes intoa sum of terms for each family:
FamScore(Yi, PaYi) =X
ui∈Val(PaGYi
)
"(4)
Xy
ji∈Val(yi)
lnΓ(γ(Yi, PaYi) + M [yj
i ,ui])
Γ(γ(Yi, PaYi))
lnΓ(γ(PaGYi
))
Γ(γ(PaGYi)) + M [ui])
+
Xy
ji∈Val(yi)
lnΓ(γ(Yi, PaYi) + M [yj
i ,ui])
Γ(γ(Yi, PaYi))
#
A.2 Caching for Efficient Fitness EvaluationEvaluating the fitness of an individual in our GA involves
computing the Bayesian score of a network, which is an ex-pensive operation. We use two levels of caching to speed upthis computation. The first level caches sufficient statistics(counts) of the data set. The second level caches set scores.
Suppose that we had a contingency table for a node Yi andits parents PaYi . This table contains one row for each possi-ble assignment to Yi and PaYi , and the number of times thatassignment occurs in the training set. If we have this con-tingency table, then we can evaluate FamScore(Yi, PaYi) intime linear to the size of the contingency table, simply byperforming the summation in Equation (4). We can com-pute contingency tables naively as follows:
• We start with a table with all zeros.
• From each row in the data set, we extract the values ofYi and PaYi , and denote them yi and ui respectively.
• We then increment the row corresponding to yi,ui inthe contingency table.
With this approach, computing each contingency table re-quires one pass over the entire data set.
data set +AD-tree parameters
AD-tree
set ofvariables
contingencytable
query result
build
Figure 11: An AD-tree is built once and queriedmany times.
Our first level of caching uses the AD-tree data struc-ture of Moore and Lee (14) to speed up the computationof contingency tables. The AD-tree saves time at the ex-pense of using more memory. It is a tree data structure,where each node “splits” the data set according to all pos-sible assignments to a variable. Contingency tables can becomputed from the AD-tree in time that is independent ofthe total size of the data set. Thus, we no longer requireone full pass through the data set to compute a contingencytable. Figure 11 illustrates the usage of the AD-tree. Fordetails see Moore and Lee (14).
Traditional hill-climbing methods cache family scores sothat they can be retrieved again and again without being re-computed. We improve on this slightly by noticing that withthe BDeu prior, the family score (equation (4)) decomposesinto two set scores:
FamScore(Yi, PaYi) = SetScore(PaYi∪{Yi})− SetScore(PaYi)
where
SetScore(Y) =X
vi∈Val(Y)
lnΓ(αY)
Γ(αY + M [vi])
for any set of variables Y, and αY = α· 1|Val(Y)| . Computing a
family score seems to require the contingency tables of PaYi∪{Yi} and PaYi . But in fact, the latter contingency table canbe obtained from the former by marginalizing the columnYi. This means that only one AD-tree query is required forcomputing the score of a family.
Our second level of caching stores set scores insteadof family scores. This results in more cache hits, becausemultiple families during the lifetime of the algorithm mayhave the same set of parents.
Additional improvements We cache sufficient statisticsand set scores, as we did for our GA over graphs (sectionA.2). Note that we no longer need to build the full AD-tree.If the maximum indegree is k, we will never need contingencytables for more than k+1 variables, so we only need to buildthe AD-tree up to depth k + 1.
From Teyssier and Koller (20) we use the following. Sup-pose our maximum indegree is k. For each node X, wecompute the score of all possible parent sets of size k orless. We rank these parent sets in decreasing order by score.We then prune the set of possible parent sets as follows: ifU′ ⊂ U and FamScore(X | U′) > FamScore(X | U), thenwe remove the parent set U from consideration. For anyorder R, if U is a valid choice of parents for X, then U′ isalso valid, and has a better score. Therefore, we can safelyeliminate U, because we would never prefer it over U′. Wesay that U is dominated by U′.
To evaluate an order R, we go through every node from
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first to last. For node X, we go through the ranked listof its possible parent sets, stopping at the first parent setconsistent with the order R. This is the optimal parent setfor X, subject to the order R and the indegree constraint.We continue until we have found the optimal parent set foreach node. At this point, as we previously discussed, wehave found the optimal network for the order R and thegiven maximum indegree.
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