Regular Paper
Improved sampling using loopy belief propagation for probabilistic model building genetic programming

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Abstract

In recent years, probabilistic model building genetic programming (PMBGP) for program optimization has attracted considerable interest. PMBGPs generally use probabilistic logic sampling (PLS) to generate new individuals. However, the generation of the most probable solutions (MPSs), i.e., solutions with the highest probability, is not guaranteed. In the present paper, we introduce loopy belief propagation (LBP) for PMBGPs to generate MPSs during the sampling process. We selected program optimization with linkage estimation (POLE) as the foundation of our approach and we refer to our proposed method as POLE-BP. We apply POLE-BP and existing methods to three benchmark problems to investigate the effectiveness of LBP in the context of PMBGPs, and we describe detailed examinations of the behaviors of LBP. We find that POLE-BP shows better search performance with some problems because LBP boosts the generation of building blocks.

Introduction

In the present paper, we introduce loopy belief propagation (LBP) in probabilistic model building GPs (PMBGPs) in order to generate the most probable solutions in sampling process. We call our novel method as POLE-BP.

Estimation of distribution algorithms (EDAs) are promising evolutionary algorithms and attract much attention from a lot of practical fields. EDAs optimize solution candidates represented by one dimensional arrays as well as Genetic Algorithms (GAs). Although EDA and GA employ the same chromosome representation, EDAs are different from GAs in the sense that EDAs generate new individuals by estimation of probabilistic models and sampling, whereas GAs generate them using genetic operators. EDAs can solve deceptive problems more efficiently than GAs by estimating dependencies between loci [27], which is one of the notable features of EDAs. Because of their effectiveness, many EDAs have been devised by incorporating many distinct statistical and machine learning approaches. Recently, EDAs using loopy belief propagation (LBP) as sampling were proposed in order to improve the sampling process [24], [21]. LBP approximately infers marginal and the highest joint probabilities with configurations, and has been applied to a wide range of real world problems [7], [6]. In EDAs, the individual with the highest joint probability in learned probabilistic models describes the models most and is often called as most probable solution (MPS). MPS is the individual which most reflects the learned models, and generation of it is important to take advantage of the models efficiently. However, traditional sampling methods used in EDAs, e.g. probabilistic logic sampling (PLS) [15] and Gibbs sampling, do not always generate MPS, and EDAs using only those samplings cannot make the best use of the models. In order to solve this problem, [24], [21] generate MPS by LBP in addition to traditional sampling and showed better search performance than existing methods using only traditional samplings (PLS or Gibbs sampling) in benchmark problems.

The estimation of distribution concept employed in EDAs has been applied to the optimization of tree structures, which is traditionally addressed using GP. GP optimizes tree structures using operators, such as crossover and mutation, as well as GA. Numerous improved genetic operators have been proposed because it is difficult to deal with tree structures using only these simple operators. EDAs for tree structures are often called as Genetic Programming-EDAs (GP-EDAs) [11] or Probabilistic Model Building GPs (PMBGPs) [32], and the present paper adopts the latter abbreviation throughout the paper. PMBGPs are broadly classified into two types. One type uses probabilistic context free grammar (PCFG) to represent distributions of promising solutions and learns production rule probabilities. The other type is a prototype tree based method, which converts trees to one dimensional arrays and applies EDAs to them. From the viewpoint of probabilistic models, the prototype tree-based method is essentially equivalent to EDAs and hence it can easily incorporate techniques devised in the field of EDA.

We propose POLE-BP [34], the novel prototype tree-based PMBGP with LBP. POLE-BP generates MPS at every generation in addition to normal samplings (i.e. PLS) and makes the optimal use of the learned probabilistic model. We compare our proposed method against existing methods on three benchmark problems: the problem with no dependencies between nodes (MAX problem), the deceptive problem (Deceptive MAX problem) and the problem with dependencies between nodes (Royal Tree Problem). From results of the experiments, we show that the proposed method competes with the existing method in the deceptive problem and beats the existing method in the problems with no deceptiveness from the point of the number of fitness evaluations to get an optimum solution. Moreover, we investigate behaviors of LBP in the context of PMBGP by observing fitness values and structures generated by LBP, and show reasons why the proposed method does not exhibit search performance improvement in deceptive problems whereas it does in other benchmark problems.

The present paper extends our prior work [34] by studying the effectiveness of LBP in detail. The remainder of the paper is organized as follows. Section 2 introduces related work. Section 3 explains details of the proposed method. Section 4 presents the experimental condition and results, which is followed by the discussion in Section 5. Finally Section 6 concludes the paper.

Section snippets

Related work

We introduce existing PMBGPs and methods using loopy belief propagation as sampling in this section.

The proposed method: POLE-BP

We briefly describe POLE-BP [34] in this section. POLE-BP is the first approach combining PMBGP and LBP. POLE-BP introduces LBP to the sampling process of Program Optimization with Linkage estimation (POLE) [12] and guarantees that population includes MPS, which is an individual having the highest joint probability, in each generation.

Prototype tree-based PMBGPs use more symbols than EDAs, which causes the following problems:

  • 1.

    require more population size and the number of evaluations;

  • 2.

    difficult

Experiments

In order to evaluate the search performance of the proposed method, we apply POLE-BP, POLE and simple GP (SGP) to three benchmark problems, MAX problem, deceptive MAX (DMAX) problem and royal tree problem and compare their performances. Common parameters in POLE-BP and POLE are described in Table 1, and parameters of SGP are described in Table 2. In POLE-BP, message passing schedule of loopy max-sum is that messages are sent from all factor nodes and variable nodes by turns. A termination

Discussion

In the previous section, for respective problems, we displayed the reduction of the average number of fitness evaluations by LBP and the behaviors of LBP in a PMBGP. Up to this point, we summarize the function of LBP in a PMBGP as the following: although LBP stimulates generation of building blocks, degree of the effectiveness differs dependent on the nature of problems. In the MAX problem, the best individual at each generation is apt to be generated by LBP. The MAX problem has no dependencies

Conclusion

We proposed POLE-BP, a variant of PMBGP that uses LBP for sampling to generate MPS at each generation. We compared the proposed POLE-BP against existing methods using the number of fitness function evaluations required as the evaluation criterion. Our experimental results showed that POLE-BPB׳s performance is statistically comparable with that of the existing methods for deceptive problems. Moreover, POLE-BP significantly outperformed other methods in non-deceptive problems. We analyzed the

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