Multiobjective genetic programming with partial sampling and its extension to manyobjective
Abstract
This paper describes a technique on an optimization of treestructure data by of multiobjective evolutionary algorithm, or multiobjective genetic programming. GP induces bloat of the tree structure as one of the major problem. The cause of bloat is that the tree structure obtained by the crossover operator grows bigger and bigger but its evaluation does not improve. To avoid the risk of bloat, a partial sampling operator is proposed as a mating operator. The size of the tree and a structural distance are introduced into the measure of the treestructure data as the objective functions in addition to the index of the goodness of tree structure. GP is defined as a threeobjective optimization problem. SD is also applied for the ranking of parent individuals instead to the crowding distance of the conventional NSGAII. When the index of the goodness of treestructure data is two or more, the number of objective functions in the above problem becomes four or more. We also propose an effective manyobjective EA applicable to such the manyobjective GP. We focus on NSGAII based on Pareto partial dominance (NSGAIIPPD). NSGAIIPPD requires beforehand a combination list of the number of objective functions to be used for Pareto partial dominance (PPD). The contents of the combination list greatly influence the optimization result. We propose to schedule a parameter r meaning the subset size of objective functions for PPD and to eliminate individuals created by the mating having the same contents as the individual of the archive set.
Keywords
Manyobjective genetic programming Partial sampling Tree structural distance Pareto partial dominance Subset size scheduling Elimination of duplicates1 Introduction
A technique of genetic programming (GP) [17, 18] is an algorithm to optimize structured data based on a evolutionary algorithm (EA) [11, 25]. GP is applied to various fields such as program synthesis [5], function generations [14] and rule set discoveries [30]. Although GP is very effective for optimizing structured data, it has several problems such as getting into a bloat, inadequate optimization of constant nodes, being easily captured in local optimal solution area when applied to complicated problems. The main cause of the bloat is a crossover operator which exchanges partial trees of parent individuals [2, 3, 7, 27], where this paper focuses on the optimization of treestructure data by means of GP. Several techniques to reduce the bloat have been proposed by improving the simple crossover operation [6, 10, 13, 18, 19, 26]. Although these methods have successfully inhibited bloat to a certain extent, effective search has not necessarily been performed. Moreover, there is no theoretical basis that crossover is effective for optimizing the treestructure data.
Apart from reduction of the bloat, a search method for optimizing the graph structure has been proposed [15]. Although this method is suitable for searching various size of the structured data consisting of nodes and branches, the algorithm is complicated and the computation cost is very high. In this paper, we exclude the crossover operator which is the cause of the bloat in GP, and propose a partial sampling (PS) operator [29] as a new mating operator. In PS operator, first of all, a partial sample of a partial tree structure is extracted from several individuals of a parent individual group by a procedure of a proliferation. Next, the partial tree structure obtained by the proliferation is combined with a new tree structure by a metastasis. In this paper, two types of metastasis are prepared for GP, one that depends on the original upper node and the other that does not. Repeating the proliferation and the metastasis regenerates a new treestructure data for the next generation.
Moreover, a multiobjective EA (MOEA) technique for suppressing the bloat problem and acquire many kinds of various treestructure data is applied for GP by adding two more objective functions. One of the newly added objective functions is the size of the treestructure data. Furthermore, the relative position of the target individual in the population in terms of the structural distance (SD) is also evaluated as an objective function. Then, the optimization of the treestructure data is formulated as a multiobjective optimization problem (MOP) based on these three objective functions. NSGAII [8, 9] is applied to this MOP. In the conventional NSGAII, a crowding distance (CD) is applied for ranking the front set overflowing from the parent group. Because the conventional NSGAII sorts such the individuals of the overflowing front set with CD and selects extreme solution, diversity about tree structure is not maintained. In this paper, SD is applied, instead of CD, for ranking the overflowing front set from the parent group.
The proposed technique and the conventional techniques are applied to a double spiral problem [6, 38] for verification. This problem is a classification problem containing two classes of point sets arranged on a spiral shape to be classified with a function. This problem is well known as one of difficult problem to solve with a neural network.
The number of the index of the goodness of the treestructure data When the index of the goodness of treestructure data is two or more, the number of objective functions in the above problem becomes four or more. We also propose an effective manyobjective EA (MaOEA) applicable to such the manyobjective GP. Manyobjective optimization problems (MaOPs) are difficult to solve and is tackled by many researchers [4, 8, 9, 39, 40, 41]. Although SPEA2 [39, 40, 41] and NSGAII [8, 9] are well known as powerful algorithm for MOPs, they do not work so effectively for MaOPs [1, 12, 33]. In this paper, we handle the case of solving an MaOP by NSGAII based algorithm.
When applying NSGAII or SPEA2 to MaOP, as the objective number increases, most of the solutions in the solution set, or population, become a relation that is not superior or inferior to each other. This relation is called nondominated (ND) relationship. As a result, the convergence of the obtained set of Pareto Optimal Solutions (\({\mathcal{POS}}\)) to the optimum Pareto front remarkably decreases. Sato et al. [34] have proposed a concept of Pareto partial dominance that makes it easier to determine the superiority/inferiority relationship between solutions by using several objective functions instead of all objective functions as an algorithm for such MaOP. Since NSGAII based on Pareto partial dominance (NSGAIIPPD) focuses on a relatively small number of objectives, solutions are easy to decide superiority/inferiority even on MaOP, and an effective selection pressure can be expected.
SPEA2 with a shiftbased density estimation (SDE) strategy [20, 21, 23] is also very strong algorithm to solve multiobjective optimization problems. This method requires a lot of computational cost to forcibly rank individual subsets in nondominant relationships. Also when optimizing the tree structure by SPEA2 technique with SDE, it has been difficult to suppress bloat. Therefore, this research focuses on CD which is advantageous in terms of simplicity and less computational cost. And this paper proposes SD for the purpose of suppressing the bloat.
NSGAIIPPD has the following three problems. The first problem is that a combination list of the number of objects to be used for Pareto partial dominance must be specified before the optimization. The second one is that an appropriate number of selected objectives according to the complexity of the problem in undecided. Moreover, the contents of the combination list greatly influence the optimization result. NSGAIIPPD performs ND sorting using all objective functions at a specific generation cycle, and preserves parents as an archive set for the next generation. This process generates child individuals having the same contents as the already existing individual in the archive set in some cases. As a result, the same individuals increases in the first front set, which disturbs effective ranking in the front selection. This is the third problem. By consideration of these problems, this paper proposes a simple scheduling technique of partial objective set used for Pareto partial dominance and a technique of killing individuals having the same contents in preserving the archive set [28]. In order to verify the effectiveness of the proposed techniques, we examine a manyobjective 0/1 knapsack problem [41].
2 Partial sampling operator for mating
One of the main causes of the bloat is the crossover operator generally applied in the conventional GPs, used for regenerating a new treestructure data. This paper proposes to exclude the crossover operator from the conventional GP and to apply PS operator for regeneration of a new treestructure data instead of the crossover operator. The PS operator creates a new treestructure data by partially sampling tree structures from a parent individual and joining them together. This procedure is called a proliferation. The proliferation is terminated according to the probability, \(p_{\mathrm{t}}\). Partially sampled subtree structures by the proliferation are combined together by a metastasis. Two types of the metastasis are prepared, one that depends on the original upper node and the other that does not. We call the the former as an upper node depend metastasis and the latter as a random metastasis.
\({\mathrm{AverageSize}}({\mathbf{R}}^{g})\) denotes a function returning the average size of each tree structure of the population, and \({\mathrm{Succ}}( \cdot )\) denotes a function returning the average size of the partial tree structure that the argument set takes over from the previous generation.
A new node is selected from the parent group, \({\mathbf{P}}^{g}\), according to the decided metastasis type. This node is not necessarily a root node. The proliferation is started from the selected node again.
By repeating the proliferation and the metastasis, new treestructure data is generated as shown in Fig. 2. However, when the metastasis applied to only one parent individual, or when a parent individual having the same structure as the generated tree structure, the generated tree structure is eliminated and PS operator is performed again. The terminal nodes are given as a random number in a low probability, where this is based on the conventional mutation idea.
3 Multiobjective GP with structural distance
Optimizing the treestructure data based only on the index of its own goodness brings problems that causes the bloat but also that the optimization is caught in a local optimum region. Depending on the structure of the local optimum region, the optimization stagnates, causing an illusion as if the obtained solution(s) were ultimate optimal. To avoid the risk of such the problems, this paper, therefor, proposes a technique to optimize the treestructure data based on the size of the tree structure and SD in the population in addition to the index of the goodness of tree structure.

(Step 1) Give weight 1 to the root node.

(Step 2) Assume that W is a weight given to the current node.

(Step 3) Equally distribute weights to the lower nodes of the current node so that the total is W/2.
4 Manyobjective evolutionary algorithm for MaOGP
Several effective studies [4, 8, 9, 39, 40, 41] have been made on MOP as defined by Eq.(6). NSGAII shown in Fig. 4 is a powerful multiobjective optimization scheme as a method proposed on one of these studies. NSGAII applies nondominated sorting (ND sorting) to the population \({\mathbf{Q}}\), and the individuals are classified to several ranked subsets, \({\mathbf{F}}_1,\) \({\mathbf{F}}_2,\) \({\mathbf{F}}_3,\) \(\ldots\). While not exceeding the size of the parent set \({\mathbf{P}}\), the individuals of each subset are moved to the parent set in order. Individuals of the subset that exceeds the size of the parent set is sorted using crowding distance (CD sorting) and moved to the parent set. The individuals not selected are culled. The mating operators generates the child set \({\mathbf{C}}\) from the parent set \({\mathbf{P}}\) by using the crossover and mutation operators.
5 Improvement of NSGAII based on Pareto partial dominance
In NSGAIIPPD, several individuals having the same contents as an individual already existing in the children, \({\mathbf{C}}_{t}\), or the archive set, \({\mathbf{A}}\), are generated and stored by the mating. If the optimization proceeds while sustaining such the individuals having relatively good evaluation, duplicates of them increases within the population. If the problem to be optimized is relatively simple, individuals with the same content are frequently generated during the optimization. The second improvement is elimination of such the individuals having the same contents of an individual already existing in the children, \({\mathbf{C}}_{g}\), and the archive set, \({\mathbf{A}}\), after the mating. In other words, the duplicates created by the mating is eliminated, we call this elimination of duplicates (EoD). Since the optimization problem treated in this paper is the maximizing problem, by setting the value of all objective functions of such the individual to 0, the individual are eliminated. The same content individual become the worst individual. After EoD, the mating does not reproduce the individual.
6 Verification of the proposed techniques
6.1 Double spiral problem
In order to verify the effectiveness, the following four combinations are applied to the double spiral problem, combination of the conventional operators and CD (expressed as “’\({\mathrm{CO}}+{\mathrm{MU}}\) & CD”), combination of the conventional operators and SD (expressed as “CO+MU & SD”), combination of PS operator and CD (expressed as “PS & CD”) and combination of PS operator and SD (expressed as “PS & SD”). The conventional operators denotes the conventional crossover and the conventional mutations [13, 17, 18, 36]. The size of the population, \(N_{\mathrm{pop}}\), the running generations and the number of points in the double spiral problem, \(\left {\mathbf{D}}_1\cup {\mathbf{D}}_2\right\), are defined as 100, 1, 000, 000 and 190 respectively. The probability, \(p_{\mathrm{met}}\), for selecting the type of the metastasis is tried to 0.5, 0.25 and 0.75.
6.2 Manyobjective 0/1 knapsack problem
The combination list for NSGAIIPPD
Generation range  

0–500k  500k–900k  900k–1M  
m  r  
4  2  3  4  
6  3  5  6 
Generation range  

0–300k  300k–600k  600k–900k  900k–1M  
m  r  
8  3  5  7  8 
10  3  6  8  10 
7 Conclusion
In this paper, multiobjective optimization of treestructure data, or MOGP, has been proposed, where the tree structure size and the structural distance (SD) are additionally introduced into the measure of the goodness of the tree structure as the objective functions. Furthermore, the partial sampling (PS) operator has been proposed to effectively search tree structure while avoiding the bloat. In order to verify the effectiveness of the proposed techniques, they have applied to the double spiral problem. By means of the multiobjective optimization of treestructure data, we have found that more diverse and better tree structures are acquired. The proposed method incorporating PS operator and SD in NSGAII has given relatively good results. However, since PS operator has low ability to numerically optimize constant nodes on the tree structure, it has not well worked effectively for the function optimization. In addition, since ranking with SD in NSGAII has low ability to preserve extreme solutions in the objective function space, solutions not have been effectively selected.
When the index of the goodness of treestructure data becomes two or more, the number of objective functions in MOGP becomes four or more, MaOGP. The improved NSGAIIPPD applicable to such the MaOGP has been also proposed in this paper. In the improvement, we have proposed SSS and EoD.
The improved NSGAIIPPD with SSS and EoD and other conventional techniques are applied to the manyobjective 0/1 knapsack problem for verification of the effectiveness. The improved NSGAIIPPD has given the higher diversity than other techniques as the number of the objective functions of the problem increases. On the other hand, the improved NSGAIIPPD has given the convergence equal to or higher than the other techniques even when the number of the objective functions becomes large. By means of the proposed simple scheduling of the parameter r, sufficient convergence has been obtained in the early generations with the smaller r, and the diversity has been supplemented in the generations with the larger r at the end of the optimization.
In the future, a technique to incorporate numerical optimization ability such as a particle swarm optimization [16] and the mutation to PS operator and the ranking selection technique combining SD and CD should be considered in the future. The PS operator proposed in this paper has a mechanism to terminate the proliferation, but does not have no mechanism to forcibly exit from the PS operator. Such the mechanism to forcibly exit from the PS operator should be considered.
Since the improved NSGAIIPPD still has given insufficient results in terms of the diversity, we need to improve this point while maintaining the current convergence. Although each technique has been applied to the relatively simple manyobjective 0/1 knapsack problem in this paper, we need to apply to more complicated problems and verify the effectiveness. We also need to pursue the combination list and to compare the further improved NSGAIIPPD and the conventional NSGAIIPPD with the optimal combination list. And then, we need to propose an effective MaOGP by combining these improved techniques in the future.
In this paper, the quality indicator MS is applied to assess the diversity of the final solutions. However, MS is able to simply be affected by the convergence of the solutions, in favor of poorlyconverged solutions. In this sense, MS is not necessarily effective for the assessment of the diversity. In the future research, it is necessary to consider techniques such as a diversity comparison indicator (DCI) [22] to assess the diversity of the solutions. On the other hand, the convergence of the solutions has been evaluated only using Norm. In this regard, the future research needs to visualize solutions in multiobjective optimization with parallel coordinates [24], which can partially reflect the convergence, spread and uniformity.
Notes
Acknowledgements
This research work has been supported by JSPS KAKENHI Grant No. JP17K00339. The author would like to thank to her families, the late Miss Blackin’, Miss Blanc, Miss Caramel, Mr. Civita, Miss Marron, Miss Markin’, Mr. Yukichi and Mr. Ojarumaru, for bringing her daily healing and good research environment.
Compliance with ethical standards
Conflict of interest
The author declares that she has no conflicts of interest.
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