On the efficiency of non-elitist evolutionary algorithms in presence of local optima

Anton Eremeev

Keywords: evolutionary algorithm, local optimum, optimization time, density, sparsity

TL;DR: This talk is aimed at better theoretical understanding when the non-elitist evolutionary algorithms (EAs) efficiently solve the problems with local optima, which are hard for the elitist EAs.

Abstract:

Many optimization problems, including those emerging in the field of artificial
intelligence, are solved approximately by the means of evolutionary algorithms
(EAs), and most of the well-known EAs implement some form of elitism, meaning
that they make the biologically implausible assumption that the fittest individuals
never die. Although the elitism favours exploitation and ensures that the
best seen solutions are not lost, there are a number of theoretical and experimantal
results which show that non-elitism is necessary to explore promising areas of the
search space without getting stuck in local optima. This talk is aimed at better
theoretical understanding when the non-elitist EAs efficiently solve the problems,
which are hard for the elitist EAs. One of the standard approaches to analyzing the
efficiency of evolutionary algorithms is based on dividing the solution space into
subsets (level sets) indexed in the expected order of their visit by the population
of the evolutionary algorithm. Here we consider the class SparseLocalOptα,ε of
pseudo-Boolean optimization problems in which the union of the family of level
sets that are in some sense inconsistent with respect to the objective function is an
ε-sparse set, and the solution sets where the objective function is greater than in
inconsistent level sets have density at least α. The main result is a new polynomial
upper bound for the mathematical expectation of the time in which nonelitist evolutionary
algorithms first reach the global optimum; this bound holds for problems
from SparseLocalOptα,ε, where elitist evolutionary algorithms are inefficient,
i.e., reach the optimum in exponential time on average. In addition, the efficiency
of nonelitist evolutionary algorithms is shown on a wider class of problems. The
values of adjustable parameters that guarantee the polynomial boundedness of the
optimization time for some α and ε are found for evolutionary algorithms with
tournament and linear ranking selection. An example of using the obtained results
for the family of vertex cover problems on star graphs is given, and the advantage
of nonelitist evolutionary algorithms is demonstrated compared to the simplest
algorithm with one elite individual.

Submission Number: 41