When non-elitism outperforms elitism for crossing fitness valleys

Crossing fitness valleys is one of the major obstacles to function optimization. In this paper we investigate how the structure of the fitness valley, namely its depth d and length ℓ, influence the runtime of different strategies for crossing these valleys. We present a runtime comparison between the (1+1) EA and two non-elitist nature-inspired algorithms, Strong Selection Weak Mutation (SSWM) and the Metropolis algorithm. While the (1+1) EA has to jump across the valley to a point of higher fitness because it does not accept decreasing moves, the non-elitist algorithms may cross the valley by accepting worsening moves. We show that while the runtime of the (1+1) EA algorithm depends critically on the length of the valley, the runtimes of the non-elitist algorithms depend crucially only on the depth of the valley. In particular, the expected runtime of both SSWM and Metropolis is polynomial in ℓ and exponential in d while the (1+1) EA is efficient only for valleys of small length. Moreover, we show that both SSWM and Metropolis can also efficiently optimize a rugged function consisting of consecutive valleys

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

A Computational View on Natural Evolution: On the Rigorous Analysis of the Speed of Adaptation

Inspired by Darwin’s ideas, Turing (1948) proposed an evolutionary search as an automated problem solving approach. Mimicking natural evolution, evolutionary algorithms evolve a set of solutions through the repeated application of the evolutionary operators (mutation, recombination and selection). Evolutionary algorithms belong to the family of black box algorithms which are general purpose optimisation tools. They are typically used when no good specific algorithm is known for the problem at hand and they have been reported to be surprisingly effective (Eiben and Smith, 2015; Sarker et al., 2002). Interestingly, although evolutionary algorithms are heavily inspired by natural evolution, their study has deviated from the study of evolution by the population genetics community. We believe that this is a missed opportunity and that both fields can benefit from an interdisciplinary collaboration. The question of how long it takes for a natural population to evolve complex adaptations has fascinated researchers for decades. We will argue that this is an equivalent research question to the runtime analysis of algorithms. By making use of the methods and techniques used in both fields, we will derive plenty of meaningful results for both communities, proving that this interdisciplinary approach is effective and relevant. We will apply the tools used in the theoretical analysis of evolutionary algorithms to quantify the complexity of adaptive walks on many landscapes, illustrating how the structure of the fitness landscape and the parameter conditions can impose limits to adaptation. Furthermore, as geneticists use diffusion theory to track the change in the allele frequencies of a population, we will develop a brand new model to analyse the dynamics of evolutionary algorithms. Our model, based on stochastic differential equations, will allow to describe not only the expected behaviour, but also to measure how much the process might deviate from that expectation

On the Analysis of Trajectory-Based Search Algorithms: When is it Beneficial to Reject Improvements?

We investigate popular trajectory-based algorithms inspired by biology and physics to answer a question of general significance: when is it beneficial to reject improvements? A distinguishing factor of SSWM (strong selection weak mutation), a popular model from population genetics, compared to the Metropolis algorithm (MA), is that the former can reject improvements, while the latter always accepts them. We investigate when one strategy outperforms the other. Since we prove that both algorithms converge to the same stationary distribution, we concentrate on identifying a class of functions inducing large mixing times, where the algorithms will outperform each other over a long period of time. The outcome of the analysis is the definition of a function where SSWM is efficient, while Metropolis requires at least exponential time. The identified function favours algorithms that prefer high quality improvements over smaller ones, revealing similarities in the optimisation strategies of SSWM and Metropolis respectively with best-improvement (BILS) and first-improvement (FILS) local search. We conclude the paper with a comparison of the performance of these algorithms and a (1, λ ) RLS on the identified function. The algorithm favours the steepest gradient with a probability that increases with the size of its offspring population. The results confirm that BILS excels and that the (1, λ ) RLS is efficient only for large enough population sizes

First Steps 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 their runtime 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 occurrence of new mutations is much longer than the time it takes for a new beneficial mutation to take over the population. In this situation, the population only contains copies of one genotype and evolution can be modelled as a (1+1)-type process where the probability of accepting a new genotype (improvements or worsenings) depends on the change in fitness. We present an initial runtime analysis of SSWM, quantifying its performance for various parameters and investigating differences to the (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

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

How Well Does the Metropolis Algorithm Cope With Local Optima?

The Metropolis algorithm (MA) is a classic stochastic local search heuristic. It avoids getting stuck in local optima by occasionally accepting inferior solutions. To better and in a rigorous manner understand this ability, we conduct a mathematical runtime analysis of the MA on the CLIFF benchmark. Apart from one local optimum, cliff functions are monotonically increasing towards the global optimum. Consequently, to optimize a cliff function, the MA only once needs to accept an inferior solution. Despite seemingly being an ideal benchmark for the MA to profit from its main working principle, our mathematical runtime analysis shows that this hope does not come true. Even with the optimal temperature (the only parameter of the MA), the MA optimizes most cliff functions less efficiently than simple elitist evolutionary algorithms (EAs), which can only leave the local optimum by generating a superior solution possibly far away. This result suggests that our understanding of why the MA is often very successful in practice is not yet complete. Our work also suggests to equip the MA with global mutation operators, an idea supported by our preliminary experiments.Comment: To appear in the proceedings of GECCO 2023. With appendix containing all proofs. 28 page

Evolving developmental, recurrent and convolutional neural networks for deliberate motion planning in sparse reward tasks

Motion planning algorithms have seen a diverse set of approaches in a variety of disciplines. In the domain of artificial evolutionary systems, motion planning has been included in models to achieve sophisticated deliberate behaviours. These algorithms rely on fixed rules or little evolutionary influence which compels behaviours to conform within those specific policies, rather than allowing the model to establish its own specialised behaviour. In order to further these models, the constraints imposed by planning algorithms must be removed to grant greater evolutionary control over behaviours. That is the focus of this thesis. An examination of prevailing neuroevolution methods led to the use of two distinct approaches, NEAT and HyperNEAT. Both were used to gain an understanding of the components necessary to create neuroevolution planning. The findings accumulated in the formation of a novel convolutional neural network architecture with a recurrent convolution process. The architecture’s goal was to iteratively disperse local activations to greater regions of the feature space. Experimentation showed significantly improved robustness over contemporary neuroevolution techniques as well as an efficiency increase over a static rule set. Greater evolutionary responsibility is given to the model with multiple network combinations; all of which continually demonstrated the necessary behaviours. In comparison, these behaviours were shown to be difficult to achieve in a state-of-the-art deep convolutional network. Finally, the unique use of recurrent convolution is relocated to a larger convolutional architecture on an established benchmarking platform. Performance improvements are seen on a number of domains which illustrates that this recurrent mechanism can be exploited in alternative areas outside of planning. By presenting a viable neuroevolution method for motion planning a potential emerges for further systems to adopt and examine the capability of this work in prospective domains, as well as further avenues of experimentation in convolutional architectures