A hyper-heuristic approach to automated generation of mutation operators for evolutionary programming

Highlights

• Designing mutation operators for evolutionary programming involves much manual effort.

• Genetic programming is used to evolve mutation operators for evolutionary programming.

• A train-and-test approach is used to evaluate performance over 23 function classes.

• The evolved mutation operators outperform existing operators on classes of functions.

• Operators evolved for specific classes also outperform other evolved operators.

Abstract

Evolutionary programming can solve black-box function optimisation problems by evolving a population of numerical vectors. The variation component in the evolutionary process is supplied by a mutation operator, which is typically a Gaussian, Cauchy, or Lévy probability distribution. In this paper, we use genetic programming to automatically generate mutation operators for an evolutionary programming system, testing the proposed approach over a set of function classes, which represent a source of functions. The empirical results over a set of benchmark function classes illustrate that genetic programming can evolve mutation operators which generalise well from the training set to the test set on each function class. The proposed method is able to outperform existing human designed mutation operators with statistical significance in most cases, with competitive results observed for the rest.

Introduction

Black-box function optimisation is the task of finding the optima of an objective function for which we do not have access to an analytical form. This paper is concerned with evolutionary programming (EP) [1], which evolves a population of real-valued input vectors for a function, a technique widely applied to real-world problems [2], [3], [4]. As EP has an evolutionary basis, each vector undergoes selection, evaluation, and mutation, with the expectation that fitter and fitter vectors are obtained. Here we focus on the mutation component of EP, which in the past has been designed manually. A hyper-heuristic is a search method or learning mechanism for selecting or generating heuristics to solve computational search problems [5]. Genetic programming (GP) [6] is a population-based evolutionary computation method for evolving program trees, that has frequently been used as a generation hyper-heuristic in the literature [7].

In this paper, we use GP as an offline generation hyper-heuristic to automatically create mutation operators for EP operating on function classes. A mutation operator in EP is a probability distribution, represented as a random number generator. We present an algorithmic framework which can not only express a number of currently existing EP mutation operators, but also generate novel variants of EP mutation operators. Using a train-and-test approach, GP is used to evolve mutation operators for EP, using a training set drawn from a class of functions which is then validated on a larger set of unseen instances taken from the same class. We use the term automatically designed mutation operators (ADMs) to describe the mutation operators generated by GP. We demonstrate that the ADMs for EP are capable of comparable, and often superior, performance to existing human designed operators. An additional set of experiments that takes ADMs that are trained on one function class, but then tested on a different function class is also conducted to further examine the performance of the evolved ADMs.

This current paper builds on previous papers. Hong et al. [8] first demonstrated that GP could automatically construct random number generators which are typically used in EP. In a second paper, it was shown that ADMs could be trained on collections of functions classes, showing good performance across a broader range of functions [9], however a tradeoff between general training and specific performance was observed. This paper presents a study of the design of 23 ADMs, for 23 functions classes, and then tests each of the 23 ADMs on each of the function classes. Real-valued optimisation is an active research topic and several population-based metaheuristics have been applied to function optimisation, including differential evolution and its variants [10], [11], particle swarm optimisation [12], covariance matrix adaptation evolution strategy (CMA-ES) [13] and hybrid methods [14]. We acknowledge the existence of these algorithms. However, as this study is concerned specifically with the modification of EP, a full review of all of these algorithms is beyond the scope of this paper.

The outline of the remainder of this paper is as follows. In Section 2, we give the background to the proposed approach of automatically designing algorithms using GP-based hyper-heuristics. In Section 3 we consider the task of function optimisation and introduce the notion of a function class. Section 4 presents our experimental results, which are analysed in Section 5. In Section 6 we discuss the research presented and in Section 7 we summarise the article and outline potential further research directions.