Generalized Heavy-Tailed Mutation for Evolutionary Algorithms

Anton Eremeev, Valentin Topchii, Silaev Dmitry Vyacheslavovich

Keywords: Evolutionary algorithms, regularly varying functions, heavy-tailed mutation, optimization time

Abstract:

Abstract. The heavy-tailed mutation operator, proposed by Doerr et al (2017) for evolutionary algorithms, is based on the power-law assumption of mutation rate distribution. Here we generalize the power-law assumption using a regularly varying constraint on the distribution function of mutation rate. In this setting, we generalize the upper bounds on the expected optimization time of the $(1+(\lambda,\lambda))$ genetic algorithm obtained by Antipov, Buzdalov and Doerr (2022) for the OneMax function class parametrized by the problem dimension $n$. In particular, it is shown that, on this function class, the sufficient conditions of Antipov, Buzdalov and Doerr (2022) on the heavy-tailed mutation, ensuring the $O(n)$ optimization time in expectation, may be generalized as well. This optimization time is known to be asymptotically faster than what can be achieved by the $(1+(\lambda,\lambda))$ genetic algorithm with any static mutation rate.
A new version of the heavy-tailed mutation operator is proposed, satisfying the generalized conditions, and promising results of computational experiments are presented.

Introduction

In evolutionary algorithms (EA), individuals correspond to tentative solutions in the solution space of an optimization problem, and the fitness of individuals is determined by the values of the objective function, taking into account penalties for violating the problem's constraints, if any. The construction of new trial points in EA is accomplished using mutation and crossover operators. When using the crossover, EAs are commonly referred to as genetic algorithms. When solving unconstrained pseudo-Boolean maximization problems $\max(f(x) : x\in {0,1}^n)$, or minimization problems $\min(f(x) : x\in {0,1}^n)$, one of the most frequently used mutation operators is the standard mutation (Goldberg, 1989), where each bit of the given string $x$ independently changes its value with a given probability $p$. In this paper, we will assume that in the case of standard mutation, at each iteration of an EA, the number of mutated bits $\ell$ is selected with distribution Bin$(n, p)$, and the next descendant is obtained from the parent solution by making changes to $\ell$ randomly selected bits. The optimization time is defined as the number of times the fitness function is evaluated until the optimum is reached for the first time. In this paper, we study the optimization time of a genetic algorithm from [1] when maximizing the fitness function OneMax$(x)= \sum_{i=1}^n x_i$.

An upper bound for the average optimization time of order $O(n)$ for $(1+(\lambda,\lambda))$ GA with fast mutation on OneMax, under a specific choice of the distributions of the random variables $\lambda$ and $p$ is proved in [1]. This is less than the average optimization time of $(1+(\lambda, \lambda))$ GA with any fixed probability of mutations. In [1], both the population size $\lambda$ and the mutation parameter $p$ have a truncated power-law distribution with upper bounds $\lambda\le u_n$ and $p\le u_n/n$, respectively. The linear bound from [1] holds when the power-law exponent $\beta$ satisfies the inequalities $2 <\beta< 3$ and $u_n \ge \ln^{1/(3-\beta)} n$.

Obtained Results

We show that upper bounds on the expected optimization time of $(1 + (\lambda, \lambda))$ GA, similar to those from [1], are valid not only for truncated power-law distributions of random variables $\lambda$ and $p$, but also for a broader class of distributions described in terms of regularly varying constraints on the distribution function of $\lambda$.

A computational experiment was carried out with the $(1+(\lambda, \lambda))$ GA, where the $\lambda(n)$ was generated as specified in our generalized conditions and algorithm $(1+(\lambda,\lambda))$ GA, where $\lambda(n)$ is chosen according to a power law with support ${0,1,2,\dots,n}$, parameter $\beta=2.75$ and $u_n=n$. The experiment showed encouraging results, implying that the proposed method for randomly selecting the population size $\lambda(n)$ will prove useful in practice.

Conclusion

The OneMax function considered here has only one local optimum, which is also global. From the theoretical results [2] for $(1+(\lambda, \lambda))$ GA on the Jump benchmark function with multiple local optima, the usage of fast mutation on this function eliminates the problem of precisely selecting the population size and mutation parameter. Therefore, in further research, it makes sense to consider relaxation of the requirements on the distribution of parameters in the fast mutation operator when optimizing multi-extremal functions, in particular, when solving NP-hard pseudo-Boolean optimization problems.

References

1. D. Antipov, M. Buzdalov, B. Doerr. Fast mutation in crossover-based algorithms. Algorithmica, 84:6 (2022), 1724-1761.
2. B. Doerr, H. P. Le, R. Makhmara, and T. D. Nguyen. Fast genetic algorithms. In Proc. of the Genetic and Evolutionary Computation Conference, 2017, 777-784.

Submission Number: 6