How to Escape Local Optima in Black Box Optimisation: When Non-elitism Outperforms Elitism

Escaping local optima is one of the major obstacles to function optimisation. Using the metaphor of a fitness landscape, local optima correspond to hills separated by fitness valleys that have to be overcome. We define a class of fitness valleys of tunable difficulty by considering their length, representing the Hamming path between the two optima and their depth, the drop in fitness. For this function class we present a runtime comparison between stochastic search algorithms using different search strategies. The ((Formula presented.)) EA is a simple and well-studied evolutionary algorithm that has to jump across the valley to a point of higher fitness because it does not accept worsening moves (elitism). In contrast, the Metropolis algorithm and the Strong Selection Weak Mutation (SSWM) algorithm, a famous process in population genetics, are both able to cross the fitness valley by accepting worsening moves. We show that the runtime of the ((Formula presented.)) EA depends critically on the length of the valley while the runtimes of the non-elitist algorithms depend crucially on the depth of the valley. Moreover, we show that both SSWM and Metropolis can also efficiently optimise a rugged function consisting of consecutive valleys

Towards a Runtime Comparison of Natural and Artificial Evolution

Evolutionary algorithms (EAs) form a popular optimisation paradigm inspired by natural evolution. In recent years the field of evolutionary computation has developed a rigorous analytical theory to analyse the runtimes of EAs on many illustrative problems. Here we apply this theory to a simple model of natural evolution. In the Strong Selection Weak Mutation (SSWM) evolutionary regime the time between occurrences of new mutations is much longer than the time it takes for a mutated genotype to take over the population. In this situation, the population only contains copies of one genotype and evolution can be modelled as a stochastic process evolving one genotype by means of mutation and selection between the resident and the mutated genotype. The probability of accepting the mutated genotype then depends on the change in fitness. We study this process, SSWM, from an algorithmic perspective, quantifying its expected optimisation time for various parameters and investigating differences to a similar evolutionary algorithm, the well-known (1+1) EA. We show that SSWM can have a moderate advantage over the (1+1) EA at crossing fitness valleys and study an example where SSWM outperforms the (1+1) EA by taking advantage of information on the fitness gradient

First-Hitting Times Under Additive Drift

For the last ten years, almost every theoretical result concerning the expected run time of a randomized search heuristic used drift theory, making it the arguably most important tool in this domain. Its success is due to its ease of use and its powerful result: drift theory allows the user to derive bounds on the expected first-hitting time of a random process by bounding expected local changes of the process -- the drift. This is usually far easier than bounding the expected first-hitting time directly. Due to the widespread use of drift theory, it is of utmost importance to have the best drift theorems possible. We improve the fundamental additive, multiplicative, and variable drift theorems by stating them in a form as general as possible and providing examples of why the restrictions we keep are still necessary. Our additive drift theorem for upper bounds only requires the process to be nonnegative, that is, we remove unnecessary restrictions like a finite, discrete, or bounded search space. As corollaries, the same is true for our upper bounds in the case of variable and multiplicative drift

A Runtime Analysis of Parallel Evolutionary Algorithms in Dynamic Optimization

A simple island model with λλ islands and migration occurring after every ττ iterations is studied on the dynamic fitness function Maze. This model is equivalent to a (1+λ)(1+λ) EA if τ=1τ=1 , i. e., migration occurs during every iteration. It is proved that even for an increased offspring population size up to λ=O(n1−ϵ)λ=O(n1−ϵ) , the (1+λ)(1+λ) EA is still not able to track the optimum of Maze. If the migration interval is chosen carefully, the algorithm is able to track the optimum even for logarithmic λλ . The relationship of τ,λτ,λ , and the ability of the island model to track the optimum is then investigated more closely. Finally, experiments are performed to supplement the asymptotic results, and investigate the impact of the migration topology

On the Analysis of Simple Genetic Programming for Evolving Boolean Functions

This work presents a first step towards a systematic time and space complexity analysis of genetic programming (GP) for evolving functions with desired input/output behaviour. Two simple GP algorithms, called (1+1) GP and (1+1) GP*, equipped with minimal function (F) and terminal (L) sets are considered for evolving two standard classes of Boolean functions. It is rigorously proved that both algorithms are efficient for the easy problem of evolving conjunctions of Boolean variables with the minimal sets. However, if an extra function (i.e. NOT) is added to F, then the algorithms require at least exponential time to evolve the conjunction of n variables. On the other hand, it is proved that both algorithms fail at evolving the difficult parity function in polynomial time with probability at least exponentially close to 1. Concerning generalisation, it is shown how the quality of the evolved conjunctions depends on the size of the training set s while the evolved exclusive disjunctions generalize equally badly independent of s

Analysis of Solution Quality of a Multiobjective Optimization-based Evolutionary Algorithm for Knapsack Problem

Multi-objective optimisation is regarded as one of the most promising ways for dealing with constrained optimisation problems in evolutionary optimisation. This paper presents a theoretical investigation of a multi-objective optimisation evolutionary algorithm for solving the 0-1 knapsack problem. Two initialisation methods are considered in the algorithm: local search initialisation and greedy search initialisation. Then the solution quality of the algorithm is analysed in terms of the approximation ratio

On the choice of the update strength in estimation-of-distribution algorithms and ant colony optimization

Probabilistic model-building Genetic Algorithms (PMBGAs) are a class of metaheuristics that evolve probability distributions favoring optimal solutions in the underlying search space by repeatedly sampling from the distribution and updating it according to promising samples. We provide a rigorous runtime analysis concerning the update strength, a vital parameter in PMBGAs such as the step size 1 / K in the so-called compact Genetic Algorithm (cGA) and the evaporation factor ρ in ant colony optimizers (ACO). While a large update strength is desirable for exploitation, there is a general trade-off: too strong updates can lead to unstable behavior and possibly poor performance. We demonstrate this trade-off for the cGA and a simple ACO algorithm on the well-known OneMax function. More precisely, we obtain lower bounds on the expected runtime of Ω(Kn−−√+nlogn) and Ω(n−−√/ρ+nlogn), respectively, suggesting that the update strength should be limited to 1/K,ρ=O(1/(n−−√logn)). In fact, choosing 1/K,ρ∼1/(n−−√logn) both algorithms efficiently optimize OneMax in expected time Θ(nlogn). Our analyses provide new insights into the stochastic behavior of PMBGAs and propose new guidelines for setting the update strength in global optimization

Lower Bounds for Non-Elitist Evolutionary Algorithms via Negative Multiplicative Drift

International audienceA decent number of lower bounds for non-elitist population-based evolutionary algorithms has been shown by now. Most of them are technically demanding due to the (hard to avoid) use of negative drift theorems -- general results which translate an expected movement away from the target into a high hitting time. We propose a simple negative drift theorem for multiplicative drift scenarios and show that it can simplify existing analyses. We discuss in more detail Lehre's (PPSN 2010) \emph{negative drift in populations} method, one of the most general tools to prove lower bounds on the runtime of non-elitist mutation-based evolutionary algorithms for discrete search spaces. Together with other arguments, we obtain an alternative and simpler proof of this result, which also strengthens and simplifies this method. In particular, now only three of the five technical conditions of the previous result have to be verified. The lower bounds we obtain are explicit instead of only asymptotic. This allows to compute concrete lower bounds for concrete algorithms, but also enables us to show that super-polynomial runtimes appear already when the reproduction rate is only a $(1 - \omega(n^{-1/2}))$ factor below the threshold. For the special case of algorithms using standard bit mutation with a random mutation rate (called uniform mixing in the language of hyper-heuristics), we prove the result stated by Dang and Lehre (PPSN 2016) and extend it to mutation rates other than $\Theta(1/n)$, which includes the heavy-tailed mutation operator proposed by Doerr, Le, Makhmara, and Nguyen (GECCO 2017). We finally use our method and a novel domination argument to show an exponential lower bound for the runtime of the mutation-only simple genetic algorithm on \onemax for arbitrary population size

Not an Ordinary Killer- Just an Ordinary Guy: When men murder an intimate woman partner

The Murder in Britain Study was designed to examine in detail different types of murder. Using a subset of case files from this study, men who murder other men (MM;n = 424) are compared with men who murder an intimate partner (IP;n = 106) to reflect on the relative conventionality of each group. In terms of many of the characteristics of childhood and adulthood, the IP murder group differs from theMMgroup and appears to be more “ordinary” or “conventional.” However, the IP group is less conventional in that they are more likely to have intimate relationships that had broken down, to have used violence against a previous woman partner as well as against the victim they killed, and to “ specialize” in violence against women