Searching nonlinear functions for high values

摘要

Many complex systems of great interest-ecologies, economies, immune systems, etc.-can be described as adaptive nonlinear networks (ANNs), wherein the network specifies the allowed nonlinear interactions of a large number of components. With an appropriate representation, the adaptation of an ANN can be looked upon as a search in the space {;1,0};k, using a progressively biased probability distribution, p(t). Samples of this space return a value that measures the current performance of the ANN. The corresponding function u:{;1,0};k → Reals is usually badly nonlinear with multitudes of local optima. The possibilities for biasing p(t), as information accumulates, are more readily seen if {;1,0};k is treated as a k-dimensional space re-represented via a hyperplane transform. Sampling then supplies estimates of the expected value of u, under p(t), over hyperplanes of various dimensions. Though it is possible in principle, it is not feasible to calculate the estimated expectations for even a small proportion of the hyperplanes for which information is available. However, it can be proved that there is a class of procedures, called genetic algorithms, that rapidly bias p(t) to take advantage of large numbers of above-average hyperplanes. Several properties of genetic algorithms are discussed using this point of view.