Chapter 2. TOURNAMENT SELECTION USING
2.2. Tournament Selection Strategies
Selection is a key factor that affects the performance of Evolutionary Algorithms (EAs) [1, 24]. Commonly used parent selection strategies in EAs include fitness proportionate selection, rank selection, and tour- nament selection [10]. The most popular selection method in GP is tournament selection [123]. In standard tournament selection, a number of individuals (tournament size) are randomly selected from the popu- lation. These individuals are compared with each other and the winner (in terms of better fitness) is selected to go to the mating pool. This process is then repeated n times where n is the population size [10].
The advantage of tournament selection is that it allows the adjustment of the selection pressure by tuning the tournament size. A small value of the tournament size leads to a low selection pressure while a large one results in a high selection pressure. Moreover, this method does not require a comparison of the fitness between all individuals leading to saving a large amount of the processing time [125].
Since tournament selection is the most popular selection method in GP, there have been many studies to analyse its behaviour and improve its effectiveness. As the selection process of standard tournament selec- tion consists of sampling and selecting, the majority of the early studies have focused on sampling and selecting [26].
2.2.1. Sampling strategies
Gathercole et al. [32] analysed the selection frequency of each in- dividual and the likelihood of not-selected and not-sampled individ- uals in tournament selection with different values of the tournament size. Sokolov and Whitley proposed unbiased tournament selection [108]
where all individuals have a fair chance to participate in a tournament.
Later, Xie and his colleagues conducted a series of studies to investi- gate tournament selection in GP. In [122], Xie indicated that standard tournament selection can lead to the result in which the individuals with bad fitness could be selected multiple times while the individuals with good fitness not selected any time. Thus, he proposed the fully covered tournament selection method [122] which excludes the sampled individ- uals in the next tournament to ensure that each individual has an equal
chance to participate into tournaments. Next, Xie et al. [123, 126] anal- ysed the performance of no-replacement tournament selection in GP.
In the no-replacement strategy, no individual can be sampled multiple times into the same tournament. Another problem in tournament selec- tion is that some individuals are not sampled at all when using small values of the tournament size. However, Xie at al. [127] showed that the not-sampled issue does not seriously affect the selection performance in standard tournament selection.
Overall, previous research has shown that sampling strategies have a minor impact on GP performance. Consequently, researchers have paid more attention to the second step in tournament selection: selection.
2.2.2. Selecting strategies
Goldberg and Deb [34] introduced binary tournament selection in which two individuals are selected at random, and the individual with better fitness could be selected with probability p, 0.5 < p ≤ 1.0.
Back [5] ranked the individuals in the population in which the best individual was ranked 1st. After that, the selection probability of an individual of rank j is calculated by:
n−k((n−j + 1)k −(n−j)k) (2.1) where k is size of tournament and n is size of population. Conversely, Blickle and Thiele [10] defined the worst individuals to be ranked 1st and introduced the cumulative fitness distribution, s(fj), which denotes the number of individuals with fitness value fj or worse. Finally, the
selection probability of individuals with rank j is calculated as:
(s(fj)
n )k −(s(fj−1)
n )k. (2.2)
Next, Julstrom and Robinson proposed weighted k-tournaments method [42]
in which a parameter w between 0 and 1 is chosen, and the probability that the ith worst contestant is selected is proportional to wk−i. More precisely, the selection probability of individuals with rankj is calculated by the formula:
k(1−w)
nk(1−wk)((j −1) +w(n−j))k−1 (2.3) Later, Hingee and Hutter [39] introduced the polynomial rank scheme of degree d for calculating the probability of an individual with rank j as follows:
P(I = j) =
d+1
X
t=1
atjt−1 (2.4)
They also showed that every probabilistic tournament is equivalent to a unique polynomial rank scheme.
Recently, some researchers have focused on adapting the selection pressure. Xie and Zhang [124] proposed a method for automatically tuning the selection pressure during evolution based on the fitness rank distribution of the population. In each generation, they cluster the pop- ulation into s clusters. Next, they sample k clusters from s clusters with replacement and selected the winner among k sampled clusters. Finally, a random individual is returned from the winning cluster. Their experi- mental results showed that the proposed approach is able to tune parent selection pressure automatically and dynamically along the evolution.
In an approach to control semantic locality and preserve semantic diversity during selection in grammatical evolution, Forstenlechner et al. [27] introduced semantic clustering selection. First, the individu- als in a GP population are clustered based on the similarity of their error vectors. Then, parents are drawn from within the same cluster to improve semantic locality. Moreover, semantic diversity is managed through the preservation of the existence of multiple clusters.
Another method proposed by Trujillo et al. [112] that allows to elim- inate the concept of tournament size is neatGP. In fact, neatGP was a recent method with the main objective is to reduce code bloat in GP.
This method was inspired by neuro evolution of augmenting topologies algorithm in neural networks [113] and the flat operator equalization bloat control method in GP [21]. The detailed description of neatGP could be found in [112].
More recently, Lexicase Selection was proposed for solving uncompro- mising problems [38, 68]. The idea is to evaluate the goodness of an individual based on part of fitness cases instead of all fitness cases. Each time a parent must be selected, Lexicase Selection first shuffles the list of fitness cases into a random order. Next it removes any individual that did not achieve the best error value on the first fitness case. If more than one individual remains in the population, the first fitness case is removed and this process is repeated with the next fitness case. This technique was then extended to the real-valued regression problem by La Cava et al. [56] and was proved to maintain better diversity compared to standard tournament selection [37, 55, 83].
In this chapter, the dissertation introduces a new proposed method for selecting the winner in tournament selection that is based on the statistical analysis of the semantics of GP programs. Specifically, we focus our attention on the error vectors produced in symbolic regression problems. The most similar approach in the literature was Semantic in Selection (SiS) technique proposed by Galvan-Lopez et al [29] which cal- culates the semantic similarity of parents and selects parents which are semantically dissimilar (i.e., have large differences between their seman- tic vectors). Rather than computing differences in semantic vectors, we perform a statistical analysis based on error vectors to ascertain seman- tic diversity of the individuals competing in a tournament. A detailed description of our method will be presented in the next section.