Per Kristian Lehre

Dynamic evolutionary optimisation: an analysis of frequency and magnitude of change

In this paper, we rigorously analyse how the magnitude and frequency of change may affect the performance of the algorithm (1+1) EA_dyn on a set of artificially designed pseudo-Boolean functions, given a simple but well-defined dynamic framework. We demonstrate some counter-intuitive scenarios that allow us to gain a better understanding of how the dynamics of a function may affect the runtime of an algorithm. In particular, we present the function Magnitude, where the time it takes for the (1+1) EA_dyn to relocate the global optimum is less than <i>n</i>²log <i>n</i> (i.e., efficient) with overwhelming probability if the magnitude of change is large. For small changes of magnitude, on the other hand, the expected time to relocate the global optimum is e^Ω(<i>n</i>) (i.e., highly inefficient). Similarly, the expected runtime of the (1+1) EA_dyn on the function Balance is <i>O</i>(<i>n</i>²) (efficient) for a high frequencies of change and n^{Ω(√<i>n</i>)} (highly inefficient) for low frequencies of change. These results contribute towards a better understanding of dynamic optimisation problems in general and show how traditional analytical methods may be applied in the dynamic case.

Fitness-levels for non-elitist populations

This paper introduces an easy to use technique for deriving upper bounds on the expected runtime of non-elitist population-based evolutionary algorithms (EAs). Applications of the technique show how the efficiency of EAs is critically dependant on having a sufficiently strong selective pressure. Parameter settings that ensure sufficient selective pressure on commonly considered benchmark functions are derived for the most popular selection mechanisms. Together with a recent technique for deriving lower bounds, this paper contributes to a much-needed analytical tool-box for the analysis of evolutionary algorithms with populations.

On the effect of populations in evolutionary multi-objective optimization

Multi-objective evolutionary algorithms (MOEAs) have become increasingly popular as multi-objective problem solving techniques. An important open problem is to understand the role of populations in MOEAs. We present a simple bi-objective problem which emphasizes when populations are needed. Rigorous runtime analysis point out an exponential runtime gap between the population-based algorithm Simple Evolutionary Multi-objective Optimizer (SEMO) and several single individual-based algorithms on this problem. This means that among the algorithms considered, only the population-based MOEA is successful and all other algorithms fail.

On the Impact of Mutation-Selection Balance on the Runtime of Evolutionary Algorithms

The interplay between mutation and selection plays a fundamental role in the behavior of evolutionary algorithms (EAs). However, this interplay is still not completely understood. This paper presents a rigorous runtime analysis of a non-elitist population-based EA that uses the linear ranking selection mechanism. The analysis focuses on how the balance between parameter η, controlling the selection pressure in linear ranking, and parameter χ controlling the bit-wise mutation rate, impacts the runtime of the algorithm. The results point out situations where a correct balance between selection pressure and mutation rate is essential for finding the optimal solution in polynomial time. In particular, it is shown that there exist fitness functions which can only be solved in polynomial time if the ratio between parameters η and χ is within a narrow critical interval, and where a small change in this ratio can increase the runtime exponentially. Furthermore, it is shown quantitatively how the appropriate parameter choice depends on the characteristics of the fitness function. In addition to the original results on the runtime of EAs, this paper also introduces a very useful analytical tool, i.e., multi-type branching processes, to the runtime analysis of non-elitist population-based EAs.

Negative drift in populations

An important step in gaining a better understanding of the stochastic dynamics of evolving populations, is the development of appropriate analytical tools. We present a new drift theorem for populations that allows properties of their long-term behaviour, e. g. the runtime of evolutionary algorithms, to be derived from simple conditions on the onestep behaviour of their variation operators and selection mechanisms.

Concentrated Hitting Times of Randomized Search Heuristics with Variable Drift

Drift analysis is one of the state-of-the-art techniques for the runtime analysis of randomized search heuristics (RSHs) such as evolutionary algorithms (EAs), simulated annealing etc. The vast majority of existing drift theorems yield bounds on the expected value of the hitting time for a target state, e. g., the set of optimal solutions, without making additional statements on the distribution of this time. We address this lack by providing a general drift theorem that includes bounds on the upper and lower tail of the hitting time distribution. The new tail bounds are applied to prove very precise sharp-concentration results on the running time of a simple EA on standard benchmark problems, including the class of general linear functions. The usefulness of the theorem outside the theory of RSHs is demonstrated by deriving tail bounds on the number of cycles in random permutations. All these results handle a position-dependent (variable) drift that was not covered by previous drift theorems with tail bounds. Moreover, our theorem can be specialized into virtually all existing drift theorems with drift towards the target from the literature. Finally, user-friendly specializations of the general drift theorem are given.

Theoretical analysis of rank-based mutation - combining exploration and exploitation

Parameter setting is an important issue in the design of evolutionary algorithms. Recently, experimental work has pointed out that it is often not useful to work with a fixed mutation rate. Therefore it was proposed that the population be ranked according to fitness and the mutation rate of an individual should depend on its rank. The claim is that this allows the algorithm to explore new regions in the search space as well as progress quickly towards optimal solutions. Complementing the experimental investigations, we examine the proposed approach by presenting rigorous theoretical analyses which point out the differences of rank-based mutation compared to a standard approach using a fixed mutation rate. To this end we theoretically explain the behaviour of rank-based mutation on various fitness landscapes proposed in the experimental work and present new significant classes of functions where the use of rank-based mutation may be both beneficial or detrimental compared to fixed mutation strategies.

Unbiased Black-Box Complexity of Parallel Search

We propose a new black-box complexity model for search algorithms evaluating λ search points in parallel. The parallel unbiased black-box complexity gives lower bounds on the number of function evaluations every parallel unbiased black-box algorithm needs to optimise a given problem. It captures the inertia caused by offspring populations in evolutionary algorithms and the total computational effort in parallel metaheuristics. Our model applies to all unary variation operators such as mutation or local search. We present lower bounds for the LeadingOnes function and general lower bound for all functions with a unique optimum that depend on the problem size and the degree of parallelism, λ. The latter is tight for OneMax; we prove that a (1+λ) EA with adaptive mutation rates is an optimal parallel unbiased black-box algorithm.

A runtime analysis of simple hyper-heuristics: to mix or not to mix operators

There is a growing body of work in the field of hyper-heuristics. Hyper-heuristics are high level search methodologies that operate on the space of heuristics to solve hard computational problems. A frequently used hyper-heuristic framework mixes a predefined set of low level heuristics during the search process. While most of the work on such selection hyper-heuristics in the literature are empirical, we analyse the runtime of hyper-heuristics rigorously. Our initial analysis shows that mixing heuristics could lead to exponentially faster search than individual (deterministically chosen) heuristics on chosen problems. Both mixing of variation operators and mixing of acceptance criteria are investigated on some selected problems. It is shown that mixing operators is only efficient with the right mixing distribution (parameter setting). Additionally, some of the existing adaptation mechanisms for mixing operators are also evaluated.

Crossover Can Be Constructive When Computing Unique Input Output Sequences

Unique input output (UIO) sequences have important applications in conformance testing of finite state machines (FSMs). Previous experimental and theoretical research has shown that evolutionary algorithms (EAs) can compute UIOs efficiently on many FSM instance classes, but fail on others. However, it has been unclear how and to what degree EA parameter settings influence the runtime on the UIO problem. This paper investigates the choice of acceptance criterion in the (1+1) EA and the use of crossover in the (μ +1) Steady State Genetic Algorithm. It is rigorously proved that changing these parameters can reduce the runtime from exponential to polynomial for some instance classes.