Fuzzy Speciation in Genetic Algorithms using k-d Trees

You would need to know the basics of Genetic Algorithms to understand this post. AI Junkie’s tutorial is a good place to get to upto speed. You would also need to know what k-D Trees are, for which the Wiki article is a good reference.

allopatric_speciation_med

What is speciation?

Speciation, in simple terms, is an evolutionary process by which new biological species arise. This concept is used in the implementation of evolutionary algorithms (such as GAs) as a form of optimization. To understand speciation, its necessary to understand what we mean by ‘species’.

A species is a group of organisms that – 1) possess some common prominent traits, and 2) can breed/mate with one-another. For example, consider cats and leopards. They share many characteristics, and hence exist in the same ‘family’ of animals. However, since they can’t really mate to provide a cat+leopard offspring, they don’t belong to the same species. The part 2) of the above definition may not be applicable in all cases (such as asexual plants). Nevertheless, it is what is important to us in the context of evolutionary algorithms.

Basically, a genetic algorithm (that implements speciation) will divide the population of candidates into a number of species. Essentially, each species will be a group of solutions that are allowed to crossover with each other. Candidates belonging to different species rarely mate.

Why do we need speciation?

A Genetic Algorithm chiefly deals with two genetic operations: Crossover and Mutation. Mutation works on a single candidate solution by perturbing it in some small way. Crossover, on the other hand, takes two candidate ‘parents’ (selected based on their fitness) and produces offspring(s) that share parts of their genome with the parents. This seems good enough, but it has some serious pitfalls.

Many times, crossover has a pretty high rate of failure (that is, it produces very inferior solutions). The reason for this, in simple terms, is: though both parents may individually have very high levels of fitness, the result of combining random parts of their genome together can be sub-optimal. This tends to happen when the parents come from distant parts of the search space, causing the offspring to lie in some entirely new region that has no optima in the vicinity. For example, imagine you are trying to evolve the fastest mammal. During a crossover, you select a deer (a fast mammal), and a cheetah(the fastest indeed). In this case, even if you could breed their offspring, it may have a random mixture of the anatomical features of its parents thats disastrous from the point of view of speed. In other words, mating fit parents does not guarantee a fit offspring, if the properties of the parents are vastly different.

In cases like these, speciation comes to the rescue. By restricting mating between greatly dissimilar candidates, evolution of sub-par candidates is minimized.

How is speciation implemented?

Usually, the two most common ways of implementing speciation in evolutionary algorithms are: Threshold Speciation and K-Means/Medoids clustering. Both are explained pretty well in Jeff Heaton’s book on Nature-Inspired algorithms in Artificial Intelligence.

The key feature of speciation is the fact that it prevents mating among candidates with highly differing characteristics. This can be thought of as promoting breeding between candidates that lie close to each other in the search-space. To achieve the same effect, I have been using k-d trees to perform nearest-neighbour searches to provide a form of speciation to my implementation of GAs. Its pretty simple(and intuitive) in terms of its approach, doesn’t degrade performance by too much and has been providing good results so far. So, here’s a basic outline of it for you all.

Implementation using k-D Trees

I will outline the algorithm first, and point out the nuances later.

First off, you must have a way of computing the ‘distance’/dissimilarity between candidate solutions. If your genome has a fixed length, you can use Euclidean distance.

Extra attributes required: 1. Starting Temperature (T), and 2. Neighbourhood Value(V). T must be in the range (0. 1). The higher this value, the greater is the tendency to neglect speciation (especially in the earlier iterations). The Neighbourhood Value V denotes the number of nearest candidates that will be considered in searches.

Algorithm:

I) The following selection process is repeated in each iteration until the new population size does not reach the required value:

Initial Values: Temperature t = T

i. Pick a random candidate parent based on fitness (using something like Roulette-Wheel selection, or Tournament selection)

ii.a. Pick a random number in (0, 1). If it is lesser than the current value of t, pick another parent randomly using the same selection process used in step i. If not, goto step ii.b.

ii.b. Using the k-D Tree trained over candidates from the current iteration, find the V nearest candidates to the parent selected in step i. Choose one randomly (using the process employed in step i), based on fitness.

iii. Use crossover with parents produced in steps i and ii. Mutate as/when appropriate. Add offspring to the pool of candidates of the next generation.

II) Once the new population grows to the required size, a new k-D Tree is generated using the set of all new candidates for the next iteration.

III) Decrease t for the next iteration.

Pointers

1) First of all, you need to have a way to compute dissimilarity between candidate solutions (apart from their fitness). As mentioned, if your genome size is constant, you could always use good-old Euclidean distance. If not, you might have to be creative. If you are using Scikit-Learn’s implementation of k-D Trees like me, you could check out the metrics it supports by doing kd_tree.valid_metrics over an instance.

2) Because the selection of the first parent in step i. depends only by the global measure for fitness, parts of the search space that generate good solutions will have higher numbers of candidates in subsequent iterations.

3) Unlike using K-Medoids for speciation, you don’t need to have a strict definition of the species that a given candidate belongs to. Nearest-neighbour searching ensures that a candidate usually mates only with other candidates that are similar to it.

4) As you might have understood, the Temperature parameter only adds some amount of exploratory tendencies to the algorithm. The higher the temperature, the greater is the tendency to ignore speciation. The decrease in t could most likely be done linearly from the user-provided starting point T, to a pre-defined minimum value(like 0.05), across iterations.

Effects on Performance of the GA

As mentioned in the discussion here, it doesn’t make much sense to analytically define the time-complexity of GAs. However, let me give you a rough idea of how things will change. Assume the total size of the population is n. First off, Roulette-Wheel selection (done in step i) takes O(n) time if done in a naive way (going through all candidates in some order one by one). If you do it using the stochastic acceptance method, it takes O(1) time. Usually, this selection will be done twice during every crossover. In case of the algorithm outlined above, for approximately (1-t) fraction of the second crossovers, the Roulette-wheel selection will be replaced by an O(log(n)) search using the k-D Tree and a much smaller (almost O(1)) Roulette Wheel selection amongst the nearest V candidates. Hence, unless your population size is very, very large, this won’t affect time that drastically.

The only significant increase in time will be seen as a result of the k-D Tree learning at the end of each iteration, which would add an O(nlog(n)) time to the time required for generating the next-gen pool of candidates (which is O(n^2) for naive implementations anyways). Since O(n^2 + nlog n) is equivalent to O(n^2), the performance decrease isn’t that significant, provided the size of your population isn’t really high. Even if you were using a highly efficient selection procedure during crossovers, for reasonable population sizes, your total run-time would only be multiplied by a factor less than 2 (mainly because crossover and mutation by themselves take some time too).

Thats all for now! Keep reading and cheers 🙂

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