3 Random Polynomials

It can be seen in Figure 6.6 that increasing the degree of the random polynomials results in a behaviour that is characteristic of increased difficulty. If one looks at the rate of improvement, it is decreasing as the degree increases. The higher degree instances cause a higher rate of code growth and populations with deeper trees. Again, the easier instance (degree 3) has higher edit distance diversity than the harder instance (degree 11). As noted before, the degree 7 polynomial tends to perform with less uniformity than the Binomial-3 instances. A higher rate of code growth and a lower edit distance diversity are seen in the degree 7 polynomial results.

There is one major difference between the results of the Binomial-3 and the polynomial experiments: the initial decrease in entropy of the easier and harder instances. While the initial fluctuation of entropy is not very important for the final analysis, it demonstrates an interesting difference between the problems. In Figure 6.5, an increase in instance difficulty results in less of an initial decrease in entropy. However, in Figure 6.6 a smaller decrease in initial entropy can be observed for the lower degree polynomial. As the ERC range is increased in the Binomial-3 problem, it is likely that initial populations will be able to represent more unique fitness values than with smaller ERC ranges. While each ERC range does contain the same number of usable constants, some of these values become functionally identical, especially small numbers that may cause division by zero or be similar because of precision representation. In the polynomial experiments, the initial greater loss of entropy by the degree 11 polynomial shows the initial difficulty genetic programming has in representing different initial solutions when the function closely represents a straight line. These observations refer to the populations that undergo the initial rounds of selection and that have higher or lower decrease in entropy.