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Experiments Since our analysis is asymptotic in nature we performed some elementary experiments in order to see whether besides the asymptotic runtime improvement (showing an improvement for an unspecified large problem size [MATH] ) we also see an improvement for realistic problem sizes. For this purpose we implemente... |
The first plot in Figure displays the average runtime over 10000 runs of the self-adjusting (1+ [MATH] ) EA on OneMax for [MATH] as given in Algorithm |
over [MATH] ; the second plot shows the corresponding interquartile ranges to support that the results are statistically significant. We set the initial mutation rate to [MATH] i. e., the minimum mutation rate the algorithm can attain. Moreover, the plot displays the average runtimes of the classic (1+ [MATH] ) EA usin... |
The average runtimes of both algorithms profit from higher offspring population sizes [MATH] leading to lower average runtimes as [MATH] increases. Interestingly, the two static settings of the classic (1+ [MATH] ) EA outperform the self-adjusting (1+ [MATH] ) EA for small values of [MATH] up to [MATH] For higher offsp... |
can in fact be relevant in practice. Furthermore, we implemented the self-adjusting (1+ [MATH] ) EA without the random steps, that is, when the rate is always adjusted according to how the best offspring are distributed over the two subpopulations. The experiments show that this variant of the self-adjusting (1+ [MATH]... |
[MATH] as presented in The experiments suggest that this scheme outperforms all other variants considered. Additionally we implemented another variant of the self-adjusting (1+ [MATH] ) EA using three equally-sized subpopulations i. e. the additional one is using (that is, exploiting) the current mutation rate. We comp... |
To gain some understanding on how the parameters influence the runtime, we implemented the self-adjusting (1+ [MATH] ) EA using different mutation rate update factors, that is, we consider the self-adjusting (1+ [MATH] ) EA as given in Algorithm |
where the mutation rate [MATH] is increased or decreased by some factor [MATH] (instead of the choice [MATH] made in Algorithm ). Note that we do not change the rule that we use the rates [MATH] and [MATH] |
to create the subpopulations. Furthermore, after initialization, the algorithm starts with rate [MATH] and the rate is capped below by [MATH] and above by [MATH] during the run, accordingly. |
The results are shown in Figure The plot displays the average runtime over 10000 runs of the self-adjusting (1+ [MATH] ) EA on OneMax for [MATH] |
over [MATH] using the update factors [MATH] The plot suggests that lower values of [MATH] yield a better performance. This result is not immediately obvious. Clearly, a large factor [MATH] implies that the rate changes a lot from generation to generation (namely by a factor of [MATH] ). These changes prevent the algori... |
Finally, to illustrate the nontrivial development of the rate during a run of the algorithm we plotted the rate of three single runs of the self-adjusting (1+ [MATH] ) EA using different factors [MATH] over the fitness in Figure Since the algorithm initialized with the rate [MATH] the rate increases after initializatio... |
[MATH] for [MATH] and [MATH] for [MATH] A similar, more pronounced behaviour can be seen for [MATH] we chose these particular values of [MATH] for illustrative purposes since for [MATH] the variance in the rate can be visually confusing for the reasons given above. |
While we would draw from this experiment the conclusion that a smaller choice of [MATH] is preferable in a practical application of our algorithm, the influence of the parameter on the runtime is not very large. So it might not be worth optimizing it and rather view Algorithm as a parameter-less algorithm. |
10 Conclusions We proposed and analyzed a new simple self-adjusting mutation scheme for the (1+ [MATH] ) EA. It consists of creating half the offspring with a slightly larger and the rest with a slightly smaller mutation rate. Based on the success of the subpopulations, the mutation rate is adjusted. This simple scheme... |
We proved rigorously that this self-adjusting (1+ [MATH] ) EA optimizes the OneMax test function in an expected number of [MATH] fitness evaluations. This matches the runtime shown in |
for a careful fitness-dependent choice of the mutation rate, which was also shown to be asymptotically optimal among all [MATH] -parallel black-box optimization algorithms. Hence our runtime result indicates that the self-adjusting mechanism developed in this work is able to find very good mutation rates. To the best o... |
The main technical challenge in this work is to analyze the quality of the best offspring. In contrast to most previous runtime analyses, where only the asymptotic order of the fitness gain was relevant, we needed a much higher degree of precision as we needed to make statements about in which sub-population the best o... |
As a side-result of our analyses, we have observed that using a fixed rate of [MATH] gives the bound [MATH] , which is also asymptotically optimal unless [MATH] is small. However, this setting is far off the usual constant choice of [MATH] . It is the first time that a significantly larger mutation rate was shown to be... |
From this work, a number of open problems arise. A technical challenge is to prove that our algorithm also without the random rate adjustments performs well. This requires an even more precise analysis of the qualities of the offspring in the two sub-populations, for which we currently do not have the methods. From the... |
Acknowledgments This work was supported by a public grant as part of the Investissement d’avenir project, reference ANR-11-LABX-0056-LMH, LabEx LMH, and by a grant by the Danish Council for Independent Research (DFF-FNU 4002–00542). |
# Source: arxiv 1704.08774 # Title: Genealogical Distance as a Diversity Estimate in Evolutionary Algorithms # Sections: all # Downloaded: 2026-03-03T01:57:17.196933+00:00 |
Genealogical Distance as a Diversity Estimate in Evolutionary Algorithms (2017) Abstract. The evolutionary edit distance between two individuals in a population, i.e., the amount of applications of any genetic operator it would take the evolutionary process to generate one individual starting from the other, seems like... |
1. Introduction Diversity has been a central point of research in the area of evolutionary algorithms. It is a well-known fact that maintaining a certain level of diversity aids the evolutionary process in preventing premature convergence , i.e., the phenomenon that the population focuses too quickly on a local optimum... |
We encountered this problem from an industry point of view when designing learning components for a system that needs to guarantee certain levels of quality despite being subjected to the probabilistic nature of its physical environment and probabilistic behavior of its machine learning parts (Belzner et al 2016 . Of c... |
Among the copious amount of different techniques to introduce diversity-awareness to evolutionary algorithms, many do not immediately make the job of adjusting a given evolutionary algorithm easier but instead require additional engineering effort: For example, one may need to define a distance metric specifically for ... |
We discuss related work in the following Section . We then explain the target metric called “evolutionary edit distance” in Section . Section continues by introducing the notion of “genealogical diversity” as means to approximate that concept. We improve this approach in Section by using a much simpler and computationa... |
2. Related Work The importance of diversity for evolutionary algorithms is discussed throughout the body of literature on evolutionary computing ranging from entry level (Eiben and Smith, 2003 ; Kruse et al 2016 to specialized papers (Ursem, 2002 ; Segura et al 2016 . In many cases, authors refer to diversity as a meas... |
An extensive overview of current approaches to increase diversity in evolutionary algorithms is provided in (Squillero and Tonda, 2016 , which also defines a helpful taxonomy of said approaches. Whenever a diversity objective can be quantified, it can be used to build a classic multi-objective optimization problem and ... |
The authors of (Wineberg and Oppacher, 2003 address the very important issue of how to efficiently compute diversity estimates requiring to compare every individual of a population to every other. They develop an approach to reduce the complexity of said computation to linear time. However, it might still be interestin... |
3. Evolutionary Edit Distance As described in the previous Section, there exists a vast amount of approaches to compute a population’s diversity (and an individual’s diversity with respect to that population). We found, among other things, that from an engineering point of view, many (if not most) of these approaches r... |
The concept this line of thought is based on could be called evolutionary edit distance : Given two individuals [MATH] and [MATH] we want to estimate how many applications of a genetic operator it would take to turn one of these individuals into the other. First, we start off by defining a lower bound on the number of ... |
We can assume that a given evolutionary process provides the genetic operator [MATH] where [MATH] is the problem domain in which our individuals live and [MATH] is a list of arbitrary many elements of [MATH] . Most evolutionary algorithms define exactly two instances of genetic operators called mutation [MATH] and reco... |
[EQUATION] Note that as long as we assume the genetic operator [MATH] to be symmetric (which they usually are), mdist is symmetric as well. |
The minimal edit distance is not an accurate estimate of the actual effort it would take the evolutionary process to turn [MATH] into [MATH] since the required indexed instances [MATH] of the genetic operator [MATH] may be arbitrarily unlikely to occur in the process. Instead, we want to estimate the expected amount of... |
However, if we want to use mdist to compute the diversity of individuals for a given evolutionary process, we never want to compare arbitrary solution candidates [MATH] but will only ever compare individuals within the current population [MATH] or at most individuals from the set [MATH] with [MATH] , which is the set o... |
We can thus define the factual evolutionary edit distance [MATH] as the total amount of operations it actually took to turn [MATH] into [MATH] |
[EQUATION] [EQUATION] Note that edist can only be defined this way when we assume that its parameters [MATH] and [MATH] have actually been generated through the application of genetic operators from a single base individual only. This is an unrealistic assumption: Completely unrelated individuals can be generated durin... |
4. Paths in the Genealogical Tree In the context of evolutionary processes it seems natural to think of individuals as forming genealogical relationships between each other. These relations correspond to the genetic operators applied to an individual [MATH] to create the individual [MATH] . Connecting all individuals (... |
Formally, we write [MATH] for the genealogical tree of [MATH] consisting of vertices [MATH] and edges [MATH] . For an evolutionary process producing (over all generations) the set of individuals [MATH] , it holds for all [MATH] that [MATH] iff [MATH] is the result of a variation of [MATH] . If we consider an evolutiona... |
However, most evolutionary algorithms also feature a mutation operator that works independently from recombination. For the genealogical tree, we treat it like a one-parent recombination in that we consider a mutated individual an ancestor of the original one. This approach does not reflect the fact that a single mutat... |
Given these graphs, we can then trivially define the ancestral distance from an individual [MATH] to another individual [MATH] as follows: |
[EQUATION] Note that adist as defined here is still not symmetric, i.e., it returns the amount of variation steps it took to get from [MATH] to [MATH] , which is a finite number iff [MATH] is an ancestor of [MATH] . This also usually means that if [MATH] is finite, [MATH] |
Given two individuals [MATH] and [MATH] , we can use these definitions to compute their latest common ancestor [MATH] , i.e., the individual with the closest relationship to either [MATH] or [MATH] that appears in the respective other individual’s genealogical tree. Formally, if a (latest) common ancestor exists it is ... |
[EQUATION] Note that [MATH] is symmetric, so [MATH] . For our definition of genealogical distance we consider the ancestral distance from the latest common ancestor to the given individuals. However, we also want to normalize the distance values with respect to the maximally achievable distance for a certain individual... |
[EQUATION] Note that for all [MATH] it holds that [MATH] is finite. We can now use the ancestral distance to an individual’s earliest ancestor to normalize the distance to the latest common ancestor with respect to the age of the evolutionary process. Note that if [MATH] and [MATH] share no common ancestor, we set [MAT... |
[EQUATION] This genealogical distance function gdist then describes for two individuals [MATH] how close their latest common ancestor is in comparison to their combined “evolutionary age”, i.e., the total amount of variation operations they went through. |
Following up from the previous Section, we claim that this genealogical distance correlates to the factual evolutionary edit distance between two individuals. It is not an exact depiction, though, because for cousins, e.g., we choose the minimum distance to their common ancestor instead of adding both paths through whi... |
In effect, the metric of gdist still appears to be needlessly exact for the application purpose inside the highly stochastic nature of an evolutionary algorithm. And while a lot of algorithmic optimizations and caching of ancestry values can help to cut down the computation time of the employed metric, comparing two in... |
5. Estimating Genealogical Distance on the Genome At first, it seems impossible or at east overly difficult to estimate the genealogical distance (or for that matter, the evolutionary edit distance) of two individuals without knowing about their ancestry inside the evolutionary process. However, life sciences are facin... |
To most artificial evolutionary processes, this approach is not directly applicable for a few reasons: (i) Most evolutionary algorithms use genomes that are much smaller than that of living beings. Thus, it is much harder to derive statistical similarity estimates and the analysis is much more prone to be influenced by... |
(ii) In many cases, the genomes used are not homogeneous but include various fields of different data types. Comparing similarity between different types of data requires a rather complex combined similarity metric. |
(iii) The way genomes are usually structured in evolutionary algorithms means that most to all parts of the genome are subject to selection pressure reducing the variety found between different genomes. |
The last point may seem odd because, obviously, genomes found in nature are subject to selection pressure as well. However, biology has found that, in fact, most parts of the human genome are not expressed at all when building the phenotype (i.e., a human body) (Mills et al 2007 and are thus not directly subjected to s... |
We can, however, mitigate these problems making a rather simple addition to an arbitrary evolutionary algorithm: Add more genes . As these additional genes do not carry any meaning for the solution candidate encoded by the genome, they are not subjected to selection pressure (iii). We can choose any data type we want f... |
For our experiments thus far, we have chosen a simple bit vector of a fixed length [MATH] to encode the added trash genes. Choosing [MATH] too small ( [MATH] where [MATH] is the population size) can obviously be detrimental to the distance estimate, but choosing very large [MATH] [MATH] ) has not shown any negative eff... |
Formally, to any individual [MATH] we assign a bit vector [MATH] with [MATH] for all [MATH] , which is initialized at random when the individual [MATH] is created. Every time a mutation operation is performed on [MATH] , we perform a random single bit flip on [MATH] For each recombination of [MATH] and [MATH] , we gene... |
We can then compute a trash bit distance tdist between two individuals [MATH] and [MATH] simply by returning the Hamming distance between their respective trash genes: |
[EQUATION] This metric clearly is symmetric. Again, we normalize the output by dividing it by [MATH] . Furthermore, trash bit vectors allow for a more detailed distinction between the impact of various genetic operators: The expected distance between two randomly generated individuals [MATH] and [MATH] is [MATH] . Howe... |
These examples should illustrate that the computed trash bit diversity is able to express genealogical relations between individuals. It stresses recombination over mutation but in doing so reflects the impact the respective operators have on the individual’s actual genome. We thus propose trash bit vectors as a much s... |
As is clear from the usage of the “expected value” [MATH] in these computations, the actual distance between parents and offspring is now always subject to random effects. However, so is their similarity on the non-trash genes as well. This kind of probabilistic behavior is an intrinsic part of evolutionary algorithms.... |
Finally, the computational effort to compute the trash bit distance is at most times negligible. Computing the distance between two individuals is an operation that can be performed in [MATH] and while we expect there to be a lose connection between population size and the optimal [MATH] , for a given evolution process... |
6. Experiment To verify the practical applicability of the concept of genealogical diversity and its realizations presented in the previous Sections and , respectively, we constructed a simple experimental setup: We define a simple routing task in which a robot has to choose a sequence of [MATH] continuous actions [MAT... |
We solved this scenario with four different evolutionary algorithms. All of these use a population size of [MATH] individuals and have been executed for [MATH] generations. For this kind of continuous optimization problem, that is not enough time for them to fully converge. We constructed a standard setup of an evoluti... |
Within this setup, we define a standard genetic algorithm using a fitness function that simply returns the aforementioned bonus for each individual. It performs well but seems to suffer from premature convergence in this setup (see Figure for all plots). This is the baseline approach all diversity-enabled versions of t... |
To introduce the diversity of the solutions to the genetic algorithm, we choose the approach to explicitly include the distance of the individual [MATH] to other individuals of [MATH] in [MATH] ’s fitness. But we do not construct a multi-objective optimization problem (as in (Laumanns et al 2002 ; Segura et al 2016 , e... |
[EQUATION] It is important to note that while we use [MATH] for the purpose of selection inside the evolutionary algorithm, all external analysis (plotting, e.g.) is performed on the value of [MATH] only in order to keep the results comparable. Also note that we reduce the computational effort to calculate any distance... |
Furthermore, we determined the optimal [MATH] for each algorithm using grid search on this hyperparameter. In a scenario like this, where higher diversity yields better results overall, it appears reasonable to think that [MATH] could be determined adaptively during the evolutionary process. This is still up to further... |
For evaluation purposes, we provided a domain-specific distance function. In this simple scenario, this can be defined quickly as well and we chose to use the sum of all differences between actions at the same position in the sequence. Figure shows that this approach takes a bit longer to learn but can then evade local... |
Finally, we implemented both genealogical distance metrics presented in this paper. We can see in Figure that both approaches in fact perform comparably, even though trash bit vectors require much less computational effort. For this experiment, we used [MATH] |
7. Conclusion In this paper, we have introduced the expected evolutionary edit distance as a promising target for diversity-aware optimization within evolutionary algorithms. Having found that it cannot be reasonably computed within another evolutionary process, we developed approaches to estimate that distance more ef... |
The experimental results show the initial viability of the approach used here and allow for many future applications. Some of these have been realized in (Gabor, 2017 . Other promising directions for future work have been mentioned throughout and include plans to omit the hyperparameter [MATH] by using genealogical div... |
# Source: arxiv 1705.00094 # Title: The Impact of Coevolution and Abstention on the Emergence of Cooperation # Sections: all # Downloaded: 2026-03-03T02:01:15.550850+00:00 |
The Impact of Coevolution and Abstention on the Emergence of Cooperation Abstract This paper explores the Coevolutionary Optional Prisoner’s Dilemma (COPD) game, which is a simple model to coevolve game strategy and link weights of agents playing the Optional Prisoner’s Dilemma game. We consider a population of agents ... |
Keywords: Coevolution; Optional Prisoner’s Dilemma Game; Evolutionary Game Theory; Cooperation. Introduction Evolutionary game theory in spatial environments has attracted much interest from researchers who seek to understand cooperative behaviour among rational individuals in complex environments. Many models have con... |
, small-world graphs , scale-free graphs and, bipartite graphs . It has been shown that the spatial organisation of strategies on these topologies affects the evolution of cooperation |
The Prisoner’s Dilemma (PD) game remains one of the most studied games in evolutionary game theory as it provides a simple and powerful framework to illustrate the conflicts inherent in the formation of cooperation. In addition, some extensions of the PD game, such as the Optional Prisoner’s Dilemma (OPD) game, have be... |
The vast majority of the spatial models in previous work have used static and unweighted networks. However, in many social scenarios that we wish to model, such as social networks and real biological networks, the number of individuals, their connections and environment are often dynamic. Thus, recent studies have also... |
where it has been shown that the coevolution of both networks and game strategies can play a key role in resolving social dilemmas in a more realistic scenario. |
In this paper we define and explore the Coevolutionary Optional Prisoner’s Dilemma (COPD) game, which is a simple coevolutionary spatial model where both the game strategies and the link weights between agents evolve over time. In this model, the interaction between agents is described by an OPD game. Previous research... |
Thus, given the Coevolutionary Optional Prisoner’s Dilemma game (i.e., an OPD game in a spatial environment, where links between agents can be evolved), the aims of the work are to understand the effect of varying the parameters [MATH] |
(temptation to defect), [MATH] (loner’s payoff), [MATH] and [MATH] for both unbiased and biased environments. By investigating the effect of these parameters, we aim to: |
Compare the outcomes of the COPD game with other games. Explore the impact of the link update rules and its properties. Investigate the evolution of cooperation when abstainers are present in the population. |
Investigate how many abstainers would be necessary to guarantee robust cooperation. The results show that cooperation emerges even in extremely adverse scenarios where the temptation to defect is almost at its maximum. It can be observed that the presence of the abstainers are fundamental in protecting cooperators from... |
The paper outline is as follows: Section presents a brief overview of the previous work in both spatial evolutionary game theory with dynamic networks and in the Optional Prisoner’s Dilemma game. Section gives an overview of the methodology employed, outlining the Optional Prisoner’s Dilemma payoff matrix, the coevolut... |
Related Work The use of coevolutionary rules constitute a new trend in evolutionary game theory. These rules were first introduced by Zimmermann et al. |
, who proposed a model in which agents can adapt their neighbourhood during a dynamical evolution of game strategy and graph topology. Their model uses computer simulations to implement two rules: firstly, agents playing the Prisoner’s Dilemma game update their strategy (cooperate or defect) by imitating the strategy o... |
In fact, as stated by Perc and Szolnoki , the spatial coevolutionary game is a natural upgrade of the traditional spatial evolutionary game initially proposed by Nowak and May |
, who considered static and unweighted networks in which each individual can interact only with its immediate neighbours. In general, it has been shown that coevolving the spatial structure can promote the emergence of cooperation in many scenarios |
, but the understanding of cooperative behaviour is still one of the central issues in evolutionary game theory. Szolnoki and Perc |
proposed a study of the impact of coevolutionary rules on the spatial version of three different games, i.e., the Prisoner’s Dilemma, the Snow Drift and the Stag Hunt game. They introduce the concept of a teaching activity, which quantifies the ability of each agent to enforce its strategy on the opponent. It means tha... |
, they also consider coevolution affecting either only the defectors or only the cooperators. They discuss that, in both cases and irrespective of the applied game, their coevolutionary model is much more beneficial to the cooperators than that of the traditional model. |
Huang et al. present a new model for the coevolution of game strategy and link weight. They consider a population of [MATH] agents arranged on a regular lattice network which is evolved through a Monte Carlo simulation. An agent’s interaction is described by the classical Prisoner’s Dilemma with a normalized payoff mat... |
[MATH] . They found that some values of [MATH] can provide the best environment for the evolution of cooperation. They also found that their coevolutionary model can promote cooperation efficiently even when the temptation of defection is high. |
In addition to investigations of the classical Prisoner’s Dilemma on spatial environments, some extensions of this game have also been explored as a means to favour the emergence of cooperative behaviour. For instance, the Optional Prisoner’s Dilemma game, which introduces the concept of abstention, has been studied si... |
. In their work, they proposed the opt-out or “loner’s” strategy in which agents could choose to abstain from playing the game, as a third option, in order to avoid cooperating with known defectors. There have been a number of recent studies exploring this type of game |
Cardinot et al. discuss that, with the introduction of abstainers, it is possible to observe new phenomena and, in a larger range of scenarios, cooperators can be robust to invasion by defectors and can dominate. |
Although recent work has discussed the inclusion of optional games with coevolutionary rules , this still needs to be investigated in a wider range of scenarios. Therefore, our work aims to combine both of these trends in evolutionary game theory in order to identify favourable configurations for the emergence of coope... |
Methodology The goal of the experiments outlined in this section is to investigate the environmental settings when coevolution of both strategy and link weights of the Optional Prisoner’s Dilemma on a weighted network takes place. |
This section includes a complete description of the Optional Prisoner’s Dilemma (PD) game, the spatial environment and the coevolutionary rules for both the strategy and link weights. Finally, we also outline the experimental set-up. |
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