Ralf Salomon

Re-evaluating genetic algorithm performance under coordinate rotation of benchmark functions. A survey of some theoretical and practical aspects of genetic algorithms.

In recent years, genetic algorithms (GAs) have become increasingly robust and easy to use. Current knowledge and many successful experiments suggest that the application of GAs is not limited to easy-to-optimize unimodal functions. Several results and GA theory give the impression that GAs easily escape from millions of local optima and reliably converge to a single global optimum. The theoretical analysis presented in this paper shows that most of the widely-used test functions have n independent parameters and that, when optimizing such functions, many GAs scale with an O(n ln n) complexity. Furthermore, it is shown that the current design of GAs and its parameter settings are optimal with respect to independent parameters. Both analysis and results show that a rotation of the coordinate system causes a severe performance loss to GAs that use a small mutation rate. In case of a rotation, the GAs complexity can increase up to O(nn) = O(exp(n ln n)). Future work should find new GA designs that solve this performance loss. As long as these problems have not been solved, the application of GAs will be limited to the optimization of easy-to-optimize functions.

Evolutionary algorithms and gradient search: similarities and differences

Classical gradient methods and evolutionary algorithms represent two very different classes of optimization techniques that seem to have very different properties. This paper discusses some aspects of some obvious differences and explores to what extent a hybrid method, the evolutionary-gradient-search procedure, can be used beneficially in the field of continuous parameter optimization. Simulation experiments show that on some test functions, the hybrid method yields faster convergence than pure evolution strategies, but that on other test functions, the procedure exhibits the same deficiencies as steepest-descent methods.

Adaptation on the Evolutionary Time Scale: A Working Hypothesis and Basic Experiments

In the pertinent literature, an ongoing discussion can be found about whether evolutionary algorithms are better suited for optimization or adaptation. Unfortunately, the pertinent literature does not offer a definition of the difference between adaptation and optimization. As a working hypothesis, this paper proposes adaptation as tracking the moving optimum of a dynamically changing fitness function as opposed to optimization as finding the optimum of a static fitness function. The results presented in this paper suggest that providing enough variation among the population members and applying a selection scheme is sufficient for adaptation. The resulting performance, however, depends on the problem, the selection scheme, the variation operators, as well as possibly other factors.

The Influence of Different Coding Schemes on the Computational Complexity of Genetic Algorithms in Function Optimization

Function optimization is a typical application domain for genetic algorithms (GAs). Traditionally, GAs work on bit strings of fixed total length l. Significant research has been done on designing and analyzing different coding schemes, of which Gray coding is one of the most used forms. Surprisingly little attention has been devoted to directly encoding the parameters by floating-point values provided by the programming language. This form of coding has been in favor in evolution strategy. This paper discusses several coding schemes and derives the resulting complexity when optimizing functions with n independent continuous parameters. It turns out that the direct use of real-valued parameters has certain advantages. First of all, it speeds up convergence by a factor of up to l q −1, where q denotes the number of bits per parameter. Furthermore, the use of real-valued parameters allows for more flexibility in designing the mutation operator and eases many implementation issues. The theoretical analysis presented here strongly suggests that real-valued parameters (implemented by floating point values provided by the programming language) should be the best choice when applying a GA in the field of function optimization.

Raising Theoretical Questions About the Utility of Genetic Algorithms

Genetic algorithms are believed by some to be very efficient optimization and adaptation tools. So far, the efficacy of genetic algorithms has been described by empirical results, and yet theoretical approaches are far behind. This paper aims at raising fundamental theoretical questions about the utility of genetic algorithms. These questions originate from various existing theories and the no-free-lunch theorem, a theory that compares all possible optimization procedure with respect to an equal distribution of all possible objective functions. While these questions are open at least in part, they all indicate that genetic algorithms yield worse performance than any other (deterministic) optimization algorithm. Consequently, future research should answer the question of whether the real world (or another application domain) imposes a non-equal distribution for which genetic algorithms yield advantageous performance, or whether genetic algorithms should apply operators in a deterministic fashion.

Some comments on evolutionary algorithm theory

The development of a sound theory that predicts and verifies existing evolutionary algorithms (EA) is one of the most important research issues in the field today. In mathematical proofs, the assumption of spherical symmetry is probably one of the most widely used simplifications. This paper discusses the extent to which spherical symmetry is appropriate for certain EAs. It turns out that spherical symmetry leads to simplifications in (self-adaptive) EAs but seems inappropriate for certain genetic algorithm variants, since small mutation rates bias a search algorithm toward the coordinate axes. This paper also argues that current test suites are weak in that they do not provide problems with significant epistasis that describes the interaction between different parameters. Consequently, when using an empirical test for pushing existing theory beyond its limits, benchmark functions should include more epistatic interaction or at least should use coordinate rotations.

The evolution of different neuronal control structures for autonomous agents

The use of evolutionary methods to generate controllers for real-world autonomous agents has attracted recent attention. Most of the pertinent research has employed genetic algorithms or variations thereof. Recent research has indicated that the presence of epistasis drastically slows down genetic algorithms. For this reason, this paper uses a different evolutionary method, evolution strategies, for the evolution of various (complex) neuronal control architectures for mobile robots inspired by Braitenberg vehicles. In these experiments, the evolution strategy accelerates the development process by more than an order of magnitude (a few hours compared to more than two days). Furthermore, the evolution strategy yields the same efficacy when applied to receptive-field controllers that require many more parameters than Braitenberg controllers. This dramatic speedup is very important, since the development process is to be done in real robots.

Accelerating the Evolutionary-Gradient-Search Procedure: Individual Step Sizes

Recent research has proposed the evolutionary-gradient-search procedure that uses the evolutionary scheme to estimate a gradient direction and that performs the parameter updates in a steepest-descent form. On several test functions, the procedure has shown faster convergence than other evolutionary algorithms. However, the procedure also exhibits similar deficiencies as steepest-descent methods. This paper explores to which extent the adoption of individual step sizes, as known from evolution strategies, can be beneficially used. It turns out that they considerably accelerate convergence.

Short Notes on the Schema Theorem and the Building Block Hypothesis in Genetic Algorithms

After decades of success, research on evolutionary algorithms aims at developing a sound theory that describes and predict the behavior of these algorithms. One research topic of interest is the analysis of the role of crossover and recombination in genetic algorithms, especially since various papers come to different conclusions. The goals of this paper are to revisit some well-known concepts and to discuss some new aspects that might be helpful for further clarification.

Achieving robust behavior by using proprioceptive activity patterns

This paper proposes a new self-organizing, biologically-inspired control architecture for mobile robots consisting of a controller and a value system. The controller uses activity patterns of visual sensors to determine the motor commands, whereas the value system receives stimuli from proprioceptive sensors. This design decision is justified by the following arguments: (1) the feedback of proprioceptive sensory patterns is omnipresent in biological systems and has been widely neglected in control systems, (2) both components are significantly decoupled by using different sensory modalities, and (3) proprioceptive sensors operate more reliably and can be used more efficiently than visual sensors, such as pixels in a CCD camera. Practical experiments with the Khepera robot show that by using proprioceptive sensor values, the control architecture can adapt to different environments and yield very robust behavior with respect to, for example, sensor failures. Furthermore, the new control architecture can be easily enhanced by further components.