Impacts of sampling strategies in tournament selection for genetic programming

Abstract

Tournament selection is one of the most commonly used parent selection schemes in genetic programming (GP). While it has a number of advantages over other selection schemes, it still has some issues that need to be thoroughly investigated. Two of the issues are associated with the sampling process from the population into the tournament. The first one is the so-called “multi-sampled” issue, where some individuals in the population are picked up (sampled) many times to form a tournament. The second one is the “not-sampled” issue, meaning that some individuals are never picked up when forming tournaments. In order to develop a more effective selection scheme for GP, it is necessary to understand the actual impacts of these issues in standard tournament selection. This paper investigates the behaviour of different sampling replacement strategies through mathematical modelling, simulations and empirical experiments. The results show that different sampling replacement strategies have little impact on selection pressure and cannot effectively tune the selection pressure in dynamic evolution. In order to conduct effective parent selection in GP, research focuses should be on developing automatic and dynamic selection pressure tuning methods instead of alternative sampling replacement strategies. Although GP is used in the empirical experiments, the findings revealed in this paper are expected to be applicable to other evolutionary algorithms.

Appendix: Proof of Eqs. 12 and 17 being equivalent

Equation 17 can be simplified to:

$$ \begin{aligned} P(W_j)&=\frac{\sum^{k}_{n=1}\frac{1}{n} \frac{(|S_j|-1)!}{(n-1)!(|S_j|-1-n+1)!} \left(\begin{array}{l} \sum^{j-1}_{i=1}|S_i|\\ k-n\end{array}\right)} {\left(\begin{array}{l} N\\ k \end{array}\right)}\\ &=\frac{\sum^{k}_{n=1}\frac{(|S_j|-1)!}{n!(|S_j|-n)!} \left(\begin{array}{l} \sum^{j-1}_{i=1}|S_i|\\ k-n\end{array}\right)} {\left(\begin{array}{c} N\\ k \end{array}\right)}\\ &= \frac{\sum^{k}_{n=1}\frac{1}{|S_j|} \frac{|S_j|!}{n!(|S_j|-n)!}\left(\begin{array}{l} \sum^{j-1}_{i=1}|S_i|\\ k-n \end{array}\right)} {\left(\begin{array}{l} N\\ k \end{array}\right)}\\ &= \frac{\sum^{k}_{n=1} \left(\begin{array}{l} |S_j|\\ n \end{array}\right) \left(\begin{array}{l} \sum^{j-1}_{i=1}|S_i|\\ k-n \end{array}\right)} {\left(\begin{array}{l} N\\ k \end{array}\right)|S_j|} \end{aligned} $$

After applying the relation $\sum^{n}_{m=0}\left(\begin{array}{l}r\\m\end{array}\right)\left(\begin{array}{l}s\\n-m\end{array}\right)=\left(\begin{array}{l}r+s\\n\end{array}\right)$ (Abramowitz and Stegun 1965), we can further simply the equation to

$$ \begin{aligned} &=\frac{\left(\begin{array}{c}|S_j|+ \sum^{j-1}_{i=1}|S_i|\\ k \end{array}\right)- \left(\begin{array}{l} |S_j|\\ 0 \end{array}\right) \left(\begin{array}{l} \sum^{j-1}_{i=1}|S_i|\\ k \end{array}\right)} {\left(\begin{array}{l} N\\ k \end{array} \right)|S_j|}\\ &=\frac{\left(\begin{array}{l} \sum^{j}_{i=1}|S_i|\\ k\end{array}\right)- \left(\begin{array}{l} \sum^{j-1}_{i=1}|S_i|\\ k \end{array}\right)} {\left(\begin{array}{c} N\\ k \end{array}\right)|S_j|} \end{aligned} $$

which is the same as Eq. 12.