Results

The first experiment was designed to gauge the effects of different mutation rates upon the emergence of the Baldwin effect. 3-dimensional input vectors were used to train the SOM. Data on the effects of each mutation rate was collected from 20 trials of 1,000 generations each. Mutation rates (p_m) of 0.001 to 0.01 in increments of 0.001 were used. Then the average learning iterations per generation (ALI) was plotted against mutation rate. The value of ALI plotted was the average of the 20 trials for each value of p_m. In all the runs the crossover probability (p_c) was kept constant at 0.2. The results are plotted in Figure 2.

Figure 2: ALI vs. Mutation Probability

We observe in this figure that average fitness decreased as the mutation rate increased in a roughly linear fashion. Fitting the results to a linear equation using the least square method yielded the following equation:

ALI = 26778p_m + 301 (3)

A subsequent run of the simulation showed that the equation's predictions were correct to within 8% of the actual data points.

Other simulations using input vectors of higher dimensionality produced similar linear equations whose predicted values are consistently within 10% of the actual data. The equations produced are given below:

4-dimensional:

ALI = 27512p_m + 277 (4)

5-dimensional:

ALI = 27280p_m + 298 (5)

6-dimensional:

ALI = 29100p_m + 306 (6)

The rates of deterioration of fitness (the slope terms) were consistently within 2% of each other -- even when the search space was expanded from 3 to 6 dimensions. In each case, fitness was found to be a linearly decreasing function of mutation rate.

The effect of higher crossover rates on the Baldwin effect is similar to that of increasing the mutation rates. Again using the least squares approximation algorithm, we arrive at the following linear fit for the 3-dimensional vector case (p_m = 0.001):

ALI = 526p_c +262 (7)

for $0~\leq~p_{c}~\leq~1$. As in the case of mutation, increasing the probability of crossover decreases the ability of the network to learn. In fact, the individuals with the least number of learning iterations were part of an asexual population (p_c = 0). For 400 trials of 1000 generations each, by far the highest performing population operated with p_c = 0 and p_m = 0.001. In other words, most of the optimization performance of the algorithm seemed to be coming from straight selection with low mutation rates. Doing similar simulations with higher dimensional input vectors led to similar results. The linear equations produced by least square analysis are as follows:

4-dimensional:

ALI = 531p_c + 273 (8)

5-dimensional:

ALI = 554c + 293 (9)

6-dimensional:

ALI = 592c + 301 (10)

As before, the behavior of the learning process was very similar in all the cases without regard to the dimensionality of the input vectors.

We performed one further simulation of this system with a p_c = 0.2 and a p_m = 0.0. The ALI obtained was 388. This value clearly does not fall on the line pictured in Figure 2. This value of ALI lies between those obtained using a p_m of 0.003 and 0.004. It is also 50 points higher than the ALI obtained at p_c = 0.2 and a p_m = 0.001. From this we believe it can be concluded that although the Baldwin effect can operate effectively in an asexual environment its efficiency is decreased by the absence of a small quantity of mutative pressure.