Wednesday, March 26, 2008

The Power of Meta

Evolutionary algorithms are oohhh so trendy, but they often provide very poor convergence rates. This occurs particularly when the population is in the neighborhood of a good solution, but the mutations being produced are far from that neighborhood because the mutation rate is too high.

Decreasing the mutation rate often doesn't help because then it takes forever to find the right neighborhood.

Changing the mutation rate externally can work, but you will usually get the annealing schedule wrong and have too much or too little mutation at some point.

One of the most effective ways to fix these problems is the technique known as meta-mutation. With meta-mutation, each member of the population has data about how dramatically it should be mutated. Then, when you are far from a good solution, members that take big steps can be big winners and their descendants will also take big steps. When you get close to a good solution, members with large mutation rates will step far from the best solution and will be mostly unable to improve the solution, but some descendants will have smaller mutation rates and these will able to find improved solutions. These calmer descendants will quickly dominate the population because they will be able to produce significant improvements.

Unfortunately, meta-mutation introduces the new problem of how you should mutate the mutation rate. (danger ... recursion alert)

I describe some methods that avoid all of these problems in my 1998 paper Recorded Step Mutation for Faster Convergence. The techniques are very simple and very effective. The paper is available from arxiv.

The real trick is that you can have the mutation rate encode the meta-mutation rate as well. One approach using a fixed distribution that is scaled by the current mutation rate. Another is to use the last step determine a directional component of the new mutation rate. Together, these approaches can improve convergence rate and final accuracy by many orders of magnitude relative to other approaches. Self-similar meta-mutation alone gives results that are nearly identical in symmetric Gaussian bowl problems to analytically determined optimal annealing schedules. When combined with directional mutation, results are near optimal for Gaussian bowls with low rank cross correlations.

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