Friday, 10 June 2016

Using Genetic Algorithms to find global extremes in arbitrary functions

Genetic algorithms offer the possibility to find global maxima and minima in arbitrary functions as the inputs to the functions are varied. RapidMiner has a number of evolutionary operators and one in particular is "Optimize Parameters (Evolutionary)" that, on the face of it, allows the parameters of operators contained within it to be varied as a performance vector is calculated based on how the varied parameters cause the inner operators to behave. There is a slight difficulty because not all parameters can be exposed for control by the Optimize operator so a work-around is needed. We'll come back to that shortly.

Firstly, let's choose a function to optimize. A good candidate is the Rastrigin function that, in two dimensions, has the following form.

f(i1, i2) = 20 + i1*i1 + i2*i2 - 10*(cos(2*pi*i1) + cos(2*pi*i2))

Within the range -5.12 to 5.12 for both i1 and i2, the function has many local minima which makes finding the lowest a challenging problem for some techniques. The following graph shows the function.

By inspection and exploration we can find that the global minimum is 0 when i1 and i2 are both 0. The process to generate this data is here.

Now let's see if we can find this minimum using a RapidMiner process. The process is here.

The first part of the process shown above uses a "Generate Data by User Specification" operator to generate a small example set that needs a label and a regular attribute in order for the "Optimize Parameters (Evolutionary)" operator to work.

The inner operators inside the Optimize operator are shown below.

The two operators labelled i1 and i2 are there purely to allow parameters to be passed from the control part of the Optimize operator. This is the work around I mentioned earlier. Basically, operators i1 and i2 expose parameters that can be seen from the Optimize operator and can be varied. The parameter settings for the Optimize operator are shown in the following.

This shows that the i1.constant value is allowed to vary between -5.12 and 5.12; the Optimize operator chooses the values as it proceeds.

Returning to the inner operators, the values of the parameters are accessed using a "Generate Attributes" operator. The first two attributes show how to get the values from the i1 and i2 operators. The third attribute calculates the Rastrigin value and the fourth attribute copies this into a result attribute (not strictly necessary - this process used to have other functions which I deleted to make the cut down process for this blog post).

From this point, the "Optimize" operator needs a performance vector to work on. The simplest thing to use is the "Extract Performance" operator to extract a specific value from the example set containing the Rastrigin result. This is shown below and the optimization direction is set to minimize since we are looking for a minimum.

The final "Log" operator records what happens.

If we run this process the Log result is shown in the following plot.

This shows the performance tending to 0, the global minimum, as the process proceeds and this corresponds to i1 and i2 both being 0. Plotting the log result as a scatter plot shows clusters of blue points which show how the genetic algorithm successfully keeps to areas where there are minima.
Genetic algorithms are not guaranteed to find the global minimum and in fact, even the process in this blog occasionally does not find it. If you set the parameter "specify population size" in "Optimize Parameters (Evolutionary)" to a small number like 5, you should observe this because fewer individuals are used to test the solution space and so the chance of being near the global minimum is low. Note that a little known feature of the "Process" operator is that setting "random seed" to -1 causes a new random seed to be selected for each run thereby ensuring that each run produces different results.

Of course, there is no way to detect when the global extreme is not found in the more normal case when the answer is not known beforehand. As usual, be sceptical and alive to the possibility that the result is not the best and try different parameter settings to see how results change. As mentioned, the population size affects this as do the "crossover prob" and "tournament fraction" parameters. There is no free lunch of course and an exhaustive search will always take longer.

In summary, we can see that RapidMiner can, with a bit of a work around, be used to optimize arbitrary functions. My experiments show that it seems to do a good job in most cases. A future post will show the same thing using R.