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A more designer-friendly way to work with dynamic parameters is to modify the parameters based on simple rules taking into account the recent performance. A number of recent results shows that such on the fly or self-adjusting parameter settings can give an equally good performance as the optimal fitness-dependent para... |
Such success-based self-adjusting parameter settings can work well when there is a simple monotonic relation between success and parameter value, e.g., when one speculates that increasing the size of the population in an EA helps when no progress was made. For parameters like the mutation rate, it is not clear what a s... |
, a relatively complicated learning mechanism was presented that tries to learn the right mutation strength by computing a time-discounted average of the past performance stemming from different parameter values. This learning mechanism needed a careful trade-off between exploiting the currently most profitably mutatio... |
1.1 A New Self-Adjusting Mechanism for Population-Based EAs In this work, we propose an alternative way to adjust the mutation rate on the fly for algorithms using larger offspring populations. It aims at overcoming some of the difficulties of the learning mechanism just described. The simple idea is to create half the... |
We add one modification to the very basic idea described in the first paragraph of this section. Instead of always modifying the mutation rate to the rate of the best offspring, we shall take this winner’s rate only with probability a half and else modify the mutation rate to a random one of the two possible values (tw... |
In this first work proposing this mechanism, we shall not spend much effort fine-tuning it, but rather show in a proof-of-concept manner that it can find very good mutation rates. In a real application, it is likely that better results are obtained by working with three subpopulations, namely an additional one using (t... |
1.2 Runtime Analysis for the Self-Adjusting (1+ [MATH] ) EA on OneMax To prove that the self-adjusting mechanism just presented can indeed find good dynamic mutation rates, we conduct a rigorous runtime analysis for our self-adjusting (1+ [MATH] ) EA on the classic test function |
[EQUATION] The runtime of the (1+ [MATH] ) EA with fixed mutation rates on OneMax is well understood In particular, Gießen and Witt show that the expected runtime (number of generations) is [MATH] |
when a mutation rate of [MATH] [MATH] a constant, is used. Thus for [MATH] not too large, the mutation rate determines the leading constant of the runtime, and a rate of [MATH] gives the asymptotically best runtime. |
As a consequence of their work on parallel black-box complexities, Badkobeh, Lehre, and Sudholt showed that the (1+ [MATH] ) EA with a suitable fitness-dependent mutation rate finds the optimum of OneMax in an asymptotically better runtime of [MATH] , where the improvement is by a factor of [MATH] This runtime is best-... |
Our main technical result is that the (1+ [MATH] ) EA adjusting the mutation rate according to the mechanism described above has the same (optimal) asymptotic runtime. Consequently, the self-adjusting mechanism is able to find on the fly a mutation rate that is sufficiently close to the one proposed in |
to achieve asymptotically the same expected runtime. Theorem 1 Let [MATH] and [MATH] Let [MATH] denote the number of generations of the (1+ [MATH] ) EA with self-adjusting mutation rate on OneMax Then, |
[EQUATION] This corresponds to an expected number of function evaluations of [MATH] To the best of our knowledge, this is the first time that a simple mutation-based EA achieves a super-constant speed-up via a self-adjusting choice of the mutation rate. |
As an interesting side remark, our proofs reveal that a quite non-standard but fixed mutation rate of [MATH] also achieves the [MATH] improvement as it implies the bound of [MATH] |
generations if [MATH] is not too small. Hence, the constant choice [MATH] as studied in does not yield the asymptotically optimal number of generations unless |
[MATH] is so small that the [MATH] -term dominates. Lemma 2 Let [MATH] and [MATH] Let [MATH] denote the number of generations of the (1+ [MATH] ) EA with fixed mutation rate [MATH] Then, |
[EQUATION] This corresponds to an expected number of function evaluations of [MATH] The paper is structured as follows: In Section we give an overview over previous analyses of the (1+ [MATH] ) EA and of self-adjusting parameter control mechanism in EAs from a theoretical perspective. In Section we give the algorithm a... |
gives a high-level overview of our main proof strategy. The following technical sections deal with the runtime analysis of the expected time spent by the (1+ [MATH] ) EA on OneMax in each of three regions of the fitness distance [MATH] We label these regions the far region middle region and near region , each of which ... |
Related Work Since this is a theoretically oriented work on how a dynamic parameter choice speeds up the runtime of the (1+ [MATH] ) EA on the test function OneMax , let us briefly review what is known about the theory of this EA and dynamic parameter choices in general. |
2.1 The (1+ [MATH] ) EA The first to conduct a rigorous runtime analysis of the (1+ [MATH] ) EA were Jansen, De Jong, and Wegener |
. They proved, among other results, that when optimizing OneMax a linear speed-up exists up to a population size of [MATH] , that is, for [MATH] , finding the optimal solution takes an expected number of [MATH] generations, whereas for larger [MATH] at least [MATH] generations are necessary. This picture was completed ... |
with a proof that the expected number of generations taken to find the optimum is [MATH] . The implicit constants were determined in |
giving the bound of [MATH] , for any constant [MATH] , as mentioned in the introduction. Aside from the optimization behavior on OneMax , not too much is known for the (1+ [MATH] ) EA, or is at least not made explicit (it is easy to see that waiting times for an improvement which are larger than [MATH] reduce by a fact... |
, the worst-case [MATH] expected runtime on linear functions , and the [MATH] runtime estimate for minimum spanning trees valid for [MATH] |
, where [MATH] denotes the number of vertices of the input graph, [MATH] the number of edges, and [MATH] the maximum of the integral edge weights. |
2.2 Dynamic Parameter Choices While it is clear the EAs with parameters changing during the run of the algorithm (dynamic parameter settings) can be more powerful than those only using static parameter settings, only recently considerable advantages of dynamic choices could be demonstrated by mathematical means (for di... |
and its extension from 2.2.1 Deterministic Parameter Control In this language, deterministic parameter control means that the dynamic choice of a parameter does not depend on the fitness landscape. The first to rigorously analyze a deterministic parameter control scheme are Jansen and Wegener |
. They regard the performance of the (1+1) EA which uses in iteration [MATH] the mutation rate [MATH] , where [MATH] is chosen such that [MATH] . In other words, they cyclically use the mutation rates [MATH] , where [MATH] is the largest power of two that is less than [MATH] . Jansen and Wegener demonstrate that there ... |
In , a rank-based mutation rate was analyzed for the ( [MATH] +1) EA. A previous experimental study suggested that this is a profitable approach, but the mathematical runtime analysis in |
rather indicates the opposite. While there are artificial examples where a huge runtime gain could be shown and also the worst-case runtime of the ( [MATH] +1) EA reduces from essentially [MATH] to [MATH] , a rigorous analysis on the OneMax function rather suggests that the high rate of offspring generated with a mutat... |
For two non-standard settings in evolutionary computation, deterministic parameter control mechanisms also gave interesting results. For problems where the solution length is not known |
, more precisely, where the number or the set of bits relevant for the solution quality is unknown, again random mutation rates gave good results |
. Here however, not a power-law scheme was used, but rather one based on very slowly decreasing summable sequences. For problems where the discrete variables take many values, e.g., the search space is [MATH] for some large [MATH] , the question is how to change the value of an individual variable. The results in |
suggest that a harmonic mutation strength, that is, changing a variable value by [MATH] with [MATH] chosen randomly with probability proportional to [MATH] , can be beneficial. This distribution was analyzed earlier in |
for the one-dimensional case, where it was also shown to give the asymptotically best performance on a OneMax type problem. For randomized search heuristics outside evolutionary computation, Wegener |
showed that simulated annealing (using a time-dependent temperature) can beat the Metropolis algorithm (using a static temperature). |
2.2.2 Adaptive Parameter Control A parameter control scheme is called adaptive if it used some kind of feedback from the optimization process. This can be functionally dependent (e.g., the mutation rate depends on the fitness of the parent) or success-based (e.g., a 1/5th rule). |
The first to conduct a runtime analysis for an adaptive parameter control mechanism (and show a small advantage over static choices) were Böttcher, Doerr, and Neumann |
. They proposed to use the fitness-dependent mutation rate of [MATH] for the optimization of the LeadingOnes test function. They proved that with this choice, the runtime of the (1+1) EA improves to roughly [MATH] compared to a time of [MATH] stemming from the classic mutation rate [MATH] or a runtime of [MATH] stemmin... |
For the [MATH] GA, a fitness-dependent offspring population size of order [MATH] was suggested in , where [MATH] is the fitness-distance of the parent individual to the optimum. This choice improves the optimization time (number of fitness evaluations until the optimum is found) on OneMax from [MATH] stemming from the ... |
to [MATH] . Since in this adaptive algorithm the mutation rate [MATH] is functionally dependent on the offspring population size, namely via [MATH] , the dynamic choice of [MATH] is equivalent to a fitness-dependent mutation rate of [MATH] |
In the aforementioned work by Badkobeh et al. , a fitness-dependent mutation rate of [MATH] was shown to improve the classic runtime of [MATH] to [MATH] . In |
, the (1+1) EA using a [MATH] -bit flip mutation operator together with a fitness-dependent choice of [MATH] was shown to give a performance on OneMax that is very close to the theoretical optimum (among all unary unbiased black-box algorithms), however, this differs only by lower order terms from the performance of th... |
proved that fitness-dependent mutation rates can be beneficial in artificial immune systems. 2.2.3 Self-adjusting and Self-adaptive Parameter Control |
While all these results show an advantage of adaptive parameter settings, it remains questionable if an algorithm user would be able to find such a functional dependence of the parameter on the fitness. This difficulty can be overcome via self-adjusting parameter choices, where the parameter is modified according to a ... |
, however, is promising and shows examples where self-adaptation can lead to significant speed-ups for non-elitist evolutionary algorithms. |
In contrast to this, the last years have produced a profound understanding of self-adjusting parameter choices. The first to perform a mathematical analysis were Lässig and Sudholt , who considered the (1+ [MATH] ) EA and a simple parallel island model together with two self-adjusting mechanisms for population size or ... |
In it was shown that the fitness-dependent choice of [MATH] for the [MATH] GA described above can also be found in a self-adjusting way. To this aim, another success-based mechanism was proposed, which imitates the [MATH] -th rule from evolution strategies. With some modifications, this mechanism also works on random s... |
. For the problem of optimizing an [MATH] -valued OneMax function, a self-adjustment of the step size inspired by the [MATH] -th rule was found to find the asymptotically best possible runtime in |
These results indicate that success-based dynamics work well for adjusting parameters when a monotonic relation like “if progress is difficult, then increase the population size” holds. For adjusting a parameter like the mutation rate, it is less obvious how to do this. For example, in the search space [MATH] both a to... |
, in a learning mechanism was proposed that from the medium-term past estimates the efficiency of different parameter values. As shown there, this does find the optimal mutation strength sufficiently well to obtain essentially the runtime stemming from the fitness-dependent mutation strength exhibited before. |
In the light of these works, our result from the methodological perspective shows that some of the difficulties of the learning mechanism of |
, e.g., the whole book-keeping being part of it and also the setting of the parameters regulating how to discount information over time, can be overcome by the mechanism proposed in this work. In a sense, the use of larger populations enables us to adjust the mutation rate solely on information learned in the current i... |
to intentionally use parameter settings which appear to be slightly off the current optimum to gain additional insight. 2.2.4 Selection Hyper-Heuristics |
We note that so-called selection hyper-heuristics may lead to processes resembling dynamic parameter choices. Selection hyper-heuristics are methods that select, during the run of the algorithm, which one out of several pre-specified simpler algorithmic building blocks to use. When the different pre-specified choices a... |
The first to conduct a rigorous runtime analysis for selection hyper-heuristics were Lehre and Öczan . They show that the (1+1) EA using the mixed strategy of choosing the mutation operator randomly between the 1-bit flip operator (with probability [MATH] ) and the [MATH] -bit flip operator (with probability [MATH] ) o... |
[MATH] It appears to us that, most likely, this result is not absolutely correct, since, e.g., in the case [MATH] the expected optimization time is clearly infinite (if the random initial search point has an odd Hamming distance from the optimum, then the optimum cannot be reached only via [MATH] -bit flips). So most l... |
Lehre and Öczan further construct an example function GapPath , which has the property that it can only be optimized when both the [MATH] -bit and the [MATH] -bit flip mutation operator occur with positive probability and use this to discuss various static and dynamic ways to randomly decide between the two operators. ... |
Note that for [MATH] , the randomized selection heuristic is the classic randomized search heuristic using [MATH] -bit and [MATH] -bit flips, which was regarded, among others, in |
In , Alanazi and Lehre extent the previous work to several classic selection hyper-heuristics like simple random (take a random low-level heuristic in each iteration), random gradient (take a random low-level heuristic and repeat using it as long as a true fitness improvement is obtained), greedy (in each iteration, us... |
Given that the probabilities to find a true improvement are very low in this discrete optimization problem, one would expect that all these four heuristics have the same runtime (apart from lower order terms), and this is indeed the first set of results by Lissovoi, Oliveto, and Warwicker |
, who show that the expected runtime in all cases is [MATH] . They build on this strong result by proposing to use a slower adaptation. For the random gradient method, they propose to use a random low-level heuristic for up to [MATH] iterations. If an improvement is found, immediately another phase with this operator i... |
For this generalized random gradient heuristic using a phase length of [MATH] for a constant [MATH] , they show (still for the LeadingOnes problem and the [MATH] -bit and [MATH] -bit mutation operator) an expected runtime of [MATH] , where [MATH] is a constant depending on [MATH] only that tends to [MATH] when [MATH] i... |
[MATH] The generalized random gradient heuristic was further extended in . There an operator was defined as successful (which leads to another phase using this operator) if it leads to [MATH] improvements in a phase of at most [MATH] iterations. Hence in this language, the previous generalized random gradient heuristic... |
to determine the phase length [MATH] in a self-adjusting manner. While the previous work does not state this explicitly, the choice of [MATH] is crucial. A [MATH] -value of smaller asymptotic order than [MATH] leads to typically no improvement within a phase and thus reverts the algorithms to the simple random heuristi... |
Since the choice of [MATH] is that critical, a mechanism successfully adjusting it to the right value is desirable. In it is shown that by choosing [MATH] —note that this is a quite wide range—the value of [MATH] can be easily adjusted on the fly via a multiplicative update rule resembling vaguely the [MATH] -th succes... |
Preliminaries We shall now formally define the algorithm analyzed and present some fundamental tools for the analysis. 3.1 Algorithm |
We consider the (1+ [MATH] ) EA with self-adjusting mutation rate for the minimization of pseudo-boolean functions [MATH] defined as Algorithm |
The general idea of the mutation scheme is to adjust the mutation strength according to its success in the population. We perform mutation by applying standard bit mutation with two different mutation probabilities [MATH] and [MATH] |
and we call [MATH] the mutation rate More precisely, for an even number [MATH] the algorithm creates [MATH] offspring with mutation rate [MATH] and with [MATH] each. |
The mutation rate is adjusted after each selection. With probability a half, the new rate is taken as the mutation rate that the best individual (i. e. the one with the lowest fitness, ties broken uniformly at random) was created with ( success-based adjustment ). With the other 50% probability, the mutation rate is ad... |
If an adjustment of the rate results in a new rate [MATH] outside the interval [MATH] , we replace this rate with the corresponding boundary value. Note that in the case of [MATH] , a subpopulation with rate less than [MATH] would be generated, which means flipping less than one bit in expectation. At a rate [MATH] , a... |
We formulate the algorithm to start with an initial mutation rate [MATH] The only assumption on [MATH] is to be greater than or equal to [MATH] The (1+ [MATH] ) EA with this self-adjusting choice of the mutation rate is given as pseudocode in Algorithm |
Algorithm 1 (1+ [MATH] ) EA with two-rate standard bit mutation Select [MATH] uniformly at random from [MATH] and set [MATH] for |
[MATH] do for [MATH] do Create [MATH] by flipping each bit in a copy of [MATH] independently with probability [MATH] if [MATH] and with probability [MATH] otherwise. |
[MATH] (breaking ties randomly). if [MATH] then [MATH] Perform one of the following two actions with prob. [MATH] Replace [MATH] with the mutation rate that [MATH] has been created with. |
Replace [MATH] with either [MATH] or [MATH] , each with probability [MATH] Replace [MATH] with [MATH] Let us explain the motivation for the random adjustments of the rate. Without such random adjustments, the rate can only be changed into some direction if a winning offspring is generated with this rate. For simple fun... |
A different way to implement a mechanism allowing larger changes of the rate to cross unfavorable regions would have been to not only generate offspring with rates [MATH] and [MATH] , but to allow larger deviations from the current rate with some small probability. One idea could be choosing for each offspring independ... |
The runtime , also called the optimization time , of the (1+ [MATH] ) EA is the smallest [MATH] such that an individual of minimum [MATH] -value has been found. Note that [MATH] corresponds to a number of iterations (also called generations), where each generation creates [MATH] offspring. Since each of these offspring... |
Throughout the paper, all asymptotic notation will be with respect to the problem size [MATH] 3.2 Drift Theorems Our results are obtained by drift analysis, which is also used in previous analyses of the (1+ [MATH] ) EA without self-adaptation on OneMax and other linear functions |
The first theorems stating upper bounds on the hitting time using variable drift go back to We take a formulation from but simplify it to Markov processes for notational convenience. |
Theorem 3 (Variable Drift, Upper Bound) Let [MATH] , be random variables describing a Markov process over a finite state space [MATH] , where [MATH] Let [MATH] be the random variable that denotes the earliest point in time [MATH] |
such that [MATH] If there exists a monotone increasing function [MATH] , where [MATH] is integrable on [MATH] , such that for all [MATH] with |
[MATH] we have [EQUATION] then for all [MATH] with [MATH] [EQUATION] The variable drift theorem is often applied in the special case of additive drift in discrete spaces: assuming [MATH] for some constant [MATH] one obtains [MATH] |
Since we will make frequent use of it in the following sections as well, we will also give the version of the Multiplicative Drift Theorem for upper bounds, due to |
. Again, this is implied by the previous variable drift theorem. Theorem 4 (Multiplicative Drift Let [MATH] be random variables describing a Markov process over a finite state space [MATH] |
and let [MATH] Let [MATH] be the random variable that denotes the earliest point in time [MATH] such that [MATH] If there exist [MATH] such that for all [MATH] with [MATH] we have |
[EQUATION] then for all [MATH] with [MATH] [EQUATION] 3.3 Chernoff Bounds For reasons of self-containedness, we state two well-known multiplicative Chernoff bounds and a lesser known additive Chernoff bound that is also known in the literature as Bernstein’s inequality. These bounds can be found, e.g., in |
Theorem 5 (Bernstein’s inequality, Chernoff bounds) Let [MATH] be independent random variables and [MATH] (a) Let [MATH] be such that [MATH] for all [MATH] . Let [MATH] . Then for all [MATH] |
[EQUATION] (b) Assume that for all [MATH] , the random variable [MATH] takes values in [MATH] only. Then [MATH] for all [MATH] [MATH] for all [MATH] |
3.4 Occupation Probabilities As mentioned above, we will be analyzing two depending stochastic processes: the random decrease of fitness and the random change of the mutation rate. Often, we will prove by drift analysis that the rate is drifting towards values that yield an almost-optimal fitness decrease. However, onc... |
Note that in the paper a slightly more general version including a self-loop probability is stated, which we do not need here. Theorem 6 |
(Theorem 7 in Let a Markov process [MATH] on [MATH] , where [MATH] with additive drift of at least [MATH] towards [MATH] be given (i. e., [MATH] ), starting at [MATH] (i.e. [MATH] ). Then we have, for all [MATH] and [MATH] |
[EQUATION] We can readily apply this theorem in the following lemma that will be used throughout the paper to bound the rate [MATH] |
Lemma 7 If there is a point [MATH] such that [MATH] for some constant [MATH] , then for all [MATH] and all [MATH] it holds [MATH] |
Proof. Apply Theorem to the process [MATH] Note that this process is on [MATH] , moves by an absolute value of at most [MATH] and has drift [MATH] We use [MATH] and [MATH] in the theorem and estimate [MATH] |
3.5 Useful Inequalities The following well-known estimates will be used regularly in this work. Lemma 8 (a) For all [MATH] , we have [MATH] |
(b) For all [MATH] , we have [MATH] (c) For all [MATH] , we have [MATH] Proof. Since [MATH] , then [MATH] is the maximum point of [MATH] . Thus part (a) holds. |
We notice that [MATH] , then [MATH] is the maximal point of [MATH] . Therefore part (b) holds. For the proof of part (c) , the second inequality can be easily obtained, since [MATH] monotonically increases and [MATH] when [MATH] . To prove the first inequality, let [MATH] for [MATH] . Then [MATH] and [MATH] . We notice... |
Proof Overview Since the following runtime analysis of our self-adjusting (1+ [MATH] ) EA on the OneMax function is slightly technical, let us outline the main proof ideas here in an informal manner. Let always [MATH] denote the parent individual of the current iterations and [MATH] its fitness distance from the optimu... |
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