Working Abstract of my Next Paper
This one’s to catch the attention of folks in machine learning, and theoretical computer science.
We recently proposed a new explanation for the adaptive capacity of simple recombinative genetic algorithms. This explanation proceeds from evidence that the simple genetic algorithm with uniform crossover (UGA) can implement a stochastic non-local search heuristic called hyperclimbing extraordinarily efficiently. To showcase the core computational efficiency involved we take up the problem of learning a classifier for the attributes of an unknown parity function over n attributes, k of which are effective. We consider the case where the learning algorithm can make adaptive queries against a membership query oracle. Given a bitstring of length n, the oracle returns a boolean value indicating the parity of the bitstring under the unknown parity function. For certain small, but otherwise arbitrarily chosen, values of k, we “show” that a UGA that uses the oracle as its fitness function can learn a classifier that classifies any attribute of the parity function—as effective or non-effective—with arbitrary accuracy; the learning occurs in time that is linear in n, and with query complexity that is constant in n, even when the oracle is “moderately” noisy.
Related blog post: Red Dots, Blue Dots
Update (June 11, 2010): Had a back and forth with Vitaly Feldman about the “angle” I take in this paper. He suggested that it may not be the best. For small values of k, and particular regimes of the noise parameter, a GA based learning algorithm performs at par (in an asymptotic sense) with the best known algorithms for solving the learning parities problem. Vitaly cautioned, however, that the problem of learning parities with an adaptive memebership oracle is currently not of practical interest. And since the GA based learning algorithm does not improve upon an existing computational bound, he thinks that from a pure computational learning perspective, this result it is unlikely to be of interest.
So, back to the drawing board. Taking it from the top, the goal is to draw the attention of the machine learning community to the hyperclimbing heuristic, and the GA’s ability to implement this heuristic extraordinarily efficiently. One way to do this is to showcase the core computational efficiency at play in this implementation . And one way to do this is by showing how this core efficiency can be used to efficiently solve a problem that members of the computational learning community care about—presuming that such a problem currently exists.
Red Dots, Blue Dots
In this blog entry I’d like to showcase just one of a number of remarkable findings that comprise the basis for the generative fixation hypothesis—a new explanation for the adaptive capacity of recombinative genetic algorithms.
Consider the following stochastic function which takes a bitstring of length 
fitness(bitstring)
accum = 0
for i = 1 to 4
accum = accum + bitstring[pivotalLoci[i]]
end
if accum is odd
return a random value from normal distribution N(+0.25,1)
else
return a random value from normal distribution N(-0.25,1)
end
The variable pivotalLoci is an array of four distinct integers between 1and
What Are GAs Good For?
Researchers studying the foundations of genetic algorithms have not, to the best of my knowledge, identified a non-trivial computational problem that a simple GA can solve robustly and scaleably (I’ve previously raised this issue here) . In my opinion, this singular fact is the most clear evidence for the inadequacy of current paradigm within which we understand/study the adaptive capacity of GAs—the question of what GAs are good for is, after all, intimately related to the question of how GAs work.
In a draft of one of my dissertation chapters I identify a hard computational problem and show that a GA can solve it robustly and scalably. Remarkably, this problem is closely related to a hairy statistical problem in computational biology. How might a GA leverage this kind of computational ability to perform adaptation? I’ll be presenting my theory about this in future chapters. The idea behind this theory is delightfully simple. Presenting it formally, however, is a another story. Stay tuned.
