EL ALUMNADO Y LA EFYC

Another technique, first introduced by Naemura et al. [89], was also studied by Jackson: incremental evolution where the early stages are evaluated on a simplified version of the goal problem. Both the Japanese group and Jackson studied the technique on even-n-parity.

Naemura et al. found that they could beat the performance of GP with ADFs if even-3 was used as an initial stage for an even-4-parity goal. An efficiency ratio (based on their final success proportions) of approximately 1.2 was observed and my analysis of their results show that, with 95% confidence, there was at least a slight improvement in performance. They also used even-3 as an initial stage for a goal of even-6 and demonstrated an efficiency ratio of three (95% confidence of at least 1.6). Finally they experimented with a three-stage setup: the first was evaluated on even-3, the second on even-6 and the final, goal stage was even-9. 40% of their 30 runs found a solution compared to zero when GP with ADFs were used. It is unclear, but it appears that the cost of the non-goal stages were not included in their measures. Although that made for an unfair comparison, Jackson repeated this technique explicitly including the cost of executing the initial stages.

On the even-4, -5, and -6 problems, Jackson showed an efficiency ratio at 95% confidence ofat least 1.5 [67]. His experiments used either even-2 or even-3 as the initial stage. Similar results were obtained on the majority-on problem domains with five and seven inputs using three inputs as an initial stage [67]. Even greater improvements were demonstrated on the two- and three-bit half- and full-adders in Jackson’s third paper on incremental evolution [66].

8.1.4

Summary

Jackson looked at three topics of incremental evolution: fitness-based incremen- tal evolution, parameterless functions, and incremental evolution with simplified problems.

Jackson’s hand-coded “select” function and evolved parameterless functions were not considered a reasonable approach for the development of GP; it would be more effective to use a lookup table.

On the other hand his use of simplified problems in the intermediate stages was very successful and holds significant potential benefit for the scaling up of GP to more difficult problems. However, it relies on the user’s ability to usefully simplify the problem.

Fitness-based incremental evolution, although unsuccessful in Jackson’s study, holds the most general potential benefits. There are a vast number of problem domains that lend themselves to this technique and if we understood how to ensure the technique was beneficial, it could be even more useful than the use of simplified problems.

8.2

Method

In this chapter we will use the methods developed in Part I to take a more thorough look at Jackson’s original form of fitness-based incremental evolution. Given the evidence, we avoided his saturation suggestion and instead moved from one stage to the next immediately after a solution was found.

We looked at five decisions that may have an impact on the performance of fitness-based incremental evolution:

• the allocation of generations to each stage,

• the number of fitness cases in stage one,

• the difficulty of the problem domain,

• and the weighting placed on the cost of a generation.

For this study we experimented with the even-4, -5, -6, and -7 problems up to 50 generations. The generations were allocated between two stages, always totalling 50 generations. The first stage evaluated the population on a subset of the full set of fitness cases. If a solution to the subset of cases was found in the first stage then the second stage would begin immediately and the unused generations from the first stage’s allocation would be available to the second stage. If a solution was not found in the allocated number of generations then the first stage would terminate and then the second stage would begin. The final population from the first stage was used unchanged in the second. The second stage always evaluated the population on the full set of fitness cases for the given problem. If ADFs were used then they were configured in the same way as the direct evolution runs (see table 7.1). The minor parameters were specified in the same way as was done with direct evolution (see section 7.2).

The even-4 problem was considered with a first stage of either the first four or the first eight of the sixteen fitness cases. When four fitness cases were used in the first stage, the probability of solving that subset within a few generations was very high (see figure 8.3), as a result ten experiments were executed with the first stage allocated one to ten generations. When eight fitness cases were used in the first stage, ten experiments were executed, with the first stage allocated from 5 through to 50 generations in steps of 5.

For the even-5, -6, and -7 problems, ten experiments were executed with the first stage allocated from 5 to 50 generations in steps of 5. For even-5, eight and 16 fitness cases (of the 32) were used in the first stage. For even-6, eight, 16 and 32 fitness cases (of the 64) were used in the first stage. For even-7, 16, 32, and 64 fitness cases (of the 128) were used in the first stage.

The fitness cases in the first stage were selected in the same order that Jackson specified; they represented the lower-order even-parity problems. Thus, for even- 4-parity, when the first stage consisted of the first four fitness cases, the inputs were 0000, 0001, 0010, 0011. When the first stage used eight fitness cases, the inputs were 0000,0001, 0010, 0011, 0100, 0101, 0110, 0111.

Even-4, -5, and -6 experiments were replicated with and without ADFs. Even- 7 was executed only with ADFs (performance of even-5 and -6 without ADFs was so poor we held no hope for even-7).

0 1 2 3 4 5 6 7 0.0 0.2 0.4 0.6 0.8 1.0

First 4 fitness cases of even−4−parity with ADFs

M=500 N=500

Generation

Cumulative probability of success

E=2,870 P(j)=0.982 R(j,z)=1.15 j=4 P(7)= 1 0 2 4 6 8 10

Individuals to be processed (thousands)

Probability of success (P) Computational effort (I)

Figure 8.3: Computational effort and success proportion curves for the first four of the 16 fitness cases of even-4-parity. A solution in this instance is a score of four out of the four fitness cases.

A total of 17 experimental configurations were considered. For each configu- ration ten variations were used in the number of generations allocated to stage one, giving a total of 170 experiments. All experiments were executed to 500 runs.