Semantics in Multi-objective Genetic Programming

Highlights

• Developed a new semantic-distance based approach for Multi-objective GP named SDO.

• Show how this new method naturally promotes semantic diversity.

• Results are significantly better compared to canonical EMO approaches (NSGA-II and SPEA2).

• New method outperforms two other semantic based methods.

Abstract

Semantics has become a key topic of research in Genetic Programming (GP). Semantics refers to the outputs (behaviour) of a GP individual when this is run on a dataset. The majority of works that focus on semantic diversity in single-objective GP indicates that it is highly beneficial in evolutionary search. Surprisingly, there is minuscule research conducted in semantics in Multi-objective GP (MOGP). In this work we make a leap beyond our understanding of semantics in MOGP and propose SDO: Semantic-based Distance as an additional criteriOn. This naturally encourages semantic diversity in MOGP. To do so, we find a pivot in the less dense region of the first Pareto front (most promising front). This is then used to compute a distance between the pivot and every individual in the population. The resulting distance is then used as an additional criterion to be optimised to favour semantic diversity. We also use two other semantic-based methods as baselines, called Semantic Similarity-based Crossover and Semantic-based Crowding Distance. Furthermore, we also use the Non-dominated Sorting Genetic Algorithm II and the Strength Pareto Evolutionary Algorithm 2 for comparison too. We use highly unbalanced binary classification problems and consistently show how our proposed SDO approach produces more non-dominated solutions and better diversity, leading to better statistically significant results, using the hypervolume results as evaluation measure, compared to the rest of the other four methods.

Introduction

Genetic Programming [1], one of the four canonical Evolutionary Algorithms paradigms, was popularised by Koza in the early 1990s. Over the years, researchers have been interested in making GP more amenable to evolutionary search. A key element that has been proven to make GP more robust is semantics. The latter has become a key topic of research in GP. Semantics can be seen as the behaviour of a GP program. This behaviour is the output of a GP program when executed on a set of fitness cases.

The number of scientific publications in GP semantics has increased significantly thanks to promising results found by the research community. We discuss in Section 2 some relevant works of semantics in GP. Interestingly, the vast majority of these work have concentrated on Single-objective GP (SOGP), with minuscule progress in Multi-objective GP (MOGP), with the exception of  [2], [3], [4], [5]. Thus, this scientific work extends significantly this line of research and uses three forms of semantics in a MOGP setting. Each of these are compared independently against two well-established Evolutionary Multi-objective Optimisation (EMO) approaches: the Non-dominated Sorting Genetic Algorithm II (NSGA-II) [6] and the Strength Pareto Evolutionary Algorithm (SPEA2) [7]. The semantic-based MOGP approaches used in this study are:

Semantic Similarity-based Crossover. (SSC).

This is motivated by the SOGP approach presented in [8]. This approach was one of the early methods in SOGP semantics where the authors were able to promote it in continuous search spaces. We extended this well-known method in MOGP.

Semantic-based Crowding Distance. (SCD).

Here, the main idea is to replace the crowding distance, commonly used in EMO algorithms, by a semantic-based distance, originally studied in the first author’s MOGP works [3], [4].

Semantic-based Distance as an additional criteriOn. (SDO).

This approach draws from SCD and uses the resulting semantic distance as another component to optimise by an EMO algorithm, briefly studied in [5].

Using these three semantic-based methods allow us to show the following:

Firstly,

by using SSC, we show how the semantic distance computed in the crossover operator and used to successfully promote semantic diversity in single-objective GP does not have the same positive impact in MOGP.

Secondly,

by using SCD, inspired by the crowding distance commonly used in EMO, we show that a semantic distance can be naturally computed between every GP tree in the population and a “pivot”. The latter is an individual in the sparsest region of the first Pareto front. SCD then moves away from SSC, which tries to promote semantic diversity by forcing diversity to emerge by using crossover repeatedly, as proposed in [8].

Finally,

we build from our understanding drawn from SSC and SCD in semantics and propose a robust mechanism for the emergence of semantic diversity in MOGP. Particularly, we use the semantic distance value as an additional indicator to evolve the population. This naturally promotes semantic diversity in MOGP leading to better, statistically significant results based on the average hyper-volume of the evolved Pareto approximations with respect to the other four methods (two semantic-based methods and two EMO methods) in a range of highly imbalanced datasets.

In our previous work [5], we carried out an initial limited study on semantics in Multi-objective Genetic Programming (MOGP). Specifically, we initially proposed and used three semantic-based methods, named Semantic Similarity-based Crossover (SSC), Semantic-based Distance as an additional criteriOn (SDO) and Pivot Similarity Semantic-based Distance as an additional criteriOn (PSDO).

The main conclusions from our initial investigation is that the use of a semantic-based distance value as computed in either SDO or PSDO to be used as another objective to be optimised in an EMO setting is robust enough to outperform the results yield by the well-known NSGA-II and SPEA2 approaches. Furthermore, in our initial research, we found out that the distance computed from a pivot, which is the furthest point in the search space, to every individual in the population and used as an additional criterion to be optimised in a EMO setting tends to improve the performance of our semantic-based approaches. Moreover, we were able to fine-tune how this distance can be computed to significantly improve the evolutionary search. This is attained using the SDO approach, which is used again in this work. It is, however, worth saying that these conclusions were drawn from an initial limited study including a restricted statistical analysis impeding drawing general conclusions, limited results as well as a lack of explanation that help us to clearly indicate why SDO yields better results compared to their respective canonical methods as well as the other two semantic-based approaches. Furthermore, in our initial study [5], we omitted to discuss the limitations of SDO.

In this work, we have addressed all these issues. More specifically, the main contributions of this scientific study are as follows:

• We consistently show how Semantic Similarity-based Crossover (SSC) used in single-objective GP and widely reported to be beneficial in GP does not have the same positive impact in a multi-objective GP (MOGP) setting.

• From this, we show how a semantic-based distance approach can enhance the evolutionary search in MOGP. To this end we use two semantic-based approaches: Semantic-based Crowding Distance (SCD) and a Semantic-base Distance as an additional criteriOn (SDO).

• We demonstrate how SDO yields better results against all the approaches used in this work, including the semantic-based methods and canonical EMO approaches.

• Another major contribution of this scientific study is to include detailed results using two well-established EMO approaches NSGA-II and SPEA2. By doing so, as opposed to the limited results reported in [5], we are now in a position to draw sound conclusions by carrying out a systematic statistical analysis, explained in detail in Section 6.

• Another important contribution in this work is that we are able to explain why the semantic-based technique employed in SDO tends to improve evolutionary search. We do so by extensively analysing the behaviour of the SDO in terms of number of unique solutions, duplicate frequency of solutions over generations, etc.

• Finally, another major contribution of this work is the discussion of the limitations of this work by using a Multi-objective Evolutionary Algorithm Based on Decomposition  [9].

This work is organised and presented as follows. Relevant studies to this work are presented in Section 2. The fundamental background in semantics and in MOPG is discussed in Section 3. Section 4 presents the MOGP semantic methods proposed and used in this work. The setup of experiments is presented in Section 5. Section 6 presents in detail the results yield by all the MOGP semantic approaches (SSC, SDO and SCD) and by the EMO methods (NSGA-II and SPEA2). It also offers an explanation as to why SDO finds better results compared to all the other algorithms. Section ?? discusses the limitations of SDO in Multi-Objective Evolutionary Algorithms based on Decomposition. In Section 8, we draw some conclusions.