Towards a Runtime Comparison of Natural and Artificial Evolution
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Abstract
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 wellknown (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.
Keywords
Runtime analysis Evolutionary algorithms Natural evolution Population genetics Theory Strong selection weak mutation regime1 Introduction
Evolutionary algorithms are popular generalpurpose heuristics that have found countless applications in design and optimisation. For many largescale problems they provide good results in reasonable time where many exact techniques fail [6]. They are a method of choice in blackbox optimisation, where no information about the problem at hand is available, and the only way to obtain knowledge is to evaluate candidate solutions.
In the last 20 years evolutionary computation has developed a number of algorithmic techniques for the analysis of evolutionary and genetic algorithms. These methods typically focus on rigorously bounding the expected time required to reach a global optimum, or other wellspecified highfitness solutions. Problems studied include illustrative example functions from pseudoBoolean optimisation that isolate characteristics found in more complex problems as well as problems from combinatorial optimisation. Various studies showed that the simple evolutionary algorithm called (1+1) EA evolving a single search point (see Sect. 2) finds global optima across a range of combinatorial optimisation problems in polynomial expected time, including shortest paths [29], sorting (as maximising sortedness) [29], minimum spanning trees [18], matroid optimisation problems [26], and Eulerian cycles [16]. The (1+1) EA (with restarts) further constitutes a polynomialtime randomised approximation scheme for the NPhard Partition problem [35].
The runtime analysis of evolutionary algorithms has become one of the dominant concepts in evolutionary computation, leading to a plethora of results for evolutionary algorithms [1, 10, 20] as well as novel optimisation paradigms such as swarm intelligence [19, 20, 31] and artificial immune systems [11].
Interestingly, although evolutionary algorithms are heavily inspired by natural evolution, these methods have seldom been applied to natural evolution as studied in mathematical population genetics. This is a missed opportunity: the time it takes for a natural population to reach a fitness peak is an important question for the study of natural evolution. The kinds of results obtained from runtime analysis, namely how the runtime scales with genome size and mutation rate, are of general interest to population genetics. Moreover, recently there has been a renewed interest in applying computer science methods to problems in evolutionary biology with contributions from unlikely fields such as game theory [2], machine learning [33] and Markov chain theory [3]. Here, we present a first attempt at applying runtime analysis to the socalled Strong Selection Weak Mutation regime of natural populations.
This choice of acceptance function introduces two main differences to the (1+1) EA. First, while the (1+1) EA never decreases its current fitness (a property called elitism), SSWM accepts solutions of lower fitness (worsenings) with some positive probability. This is reminiscent of the Metropolis algorithm (Simulated Annealing with constant temperature) which can also accept worsenings (see e. g. [12]). Second, and in contrast to the Metropolis algorithm, solutions of higher fitness can be rejected, since they are accepted with a probability that is roughly proportional to the relative advantage they have over the current solution.
We cast this model of natural evolution as a stochastic process, referred to as SSWM, using common mutation operators from evolutionary algorithms. We then present first runtime analyses of this process using techniques from the analysis of randomised and evolutionary algorithms.

to explore the performance of natural evolution in the context of runtime, comparing it against simple evolutionary algorithms like the (1+1) EA,

to investigate the nonelitistic selection mechanism implicit to SSWM and its usefulness in the context of evolutionary algorithms, and

to show that techniques for the analysis of evolutionary algorithms can be applied to simple models of natural evolution, aiming to open up a new research field at the intersection of evolutionary computation and population genetics (see [24] for recent work on unifying these two fields).
We then illustrate the particular features of the selection rule in more depth. In Sect. 4, we consider a function \({\textsc {Cliff}}_{d} \) where a fitness valley of Hamming distance d needs to be crossed in order to reach the global optimum. For \(d = \omega (\log n)\) the (1+1) EA needs time \(\varTheta (n^d)\), but SSWM is faster by a factor of \(e^{\varOmega (d)}\) because of its ability to accept worse solutions. Finally, in Sect. 5 we illustrate on the function Balance [27] that SSWM can drastically outperform the (1+1) EA because the fitnessdependent selection drives it to follow the steepest gradient. While the (1+1) EA needs exponential time in expectation with \(\varOmega (1)\) probability, SSWM with overwhelming probability finds an optimum in polynomial time.
The main technical difficulties are that in contrast to the simple (1+1) EA, SSWM is a nonelitist algorithm, hence fitnesslevel arguments based on elitism are not applicable. Popular techniques such as levelbased theorems for nonelitist populations [4] are not applicable either because they require population sizes larger than 1. Moreover, while for the (1+1) EA transition probabilities to better solutions are solely determined by probabilities for flipping bits during mutation, for SSWM these additionally depend on the fitness difference. The analysis of SSWM is more challenging than the analysis of the (1+1) EA, and requires tailored proof techniques. We hope that these techniques will be helpful for analysing other evolutionary algorithms with fitnessbased selection schemes.
2 Preliminaries
We define the optimisation time of SSWM as the number of the first iteration at which the optimum is accepted as a new individual.
Since the acceptance function in this algorithm depends on the difference in fitness between genotypes, we include a parameter \(\beta > 0\) that effectively scales the fitness function and that in population genetics models the strength of selection on a phenotype. By incorporating \(\beta \) as a parameter of this function (and hence of the algorithm) we avoid having to explicitly rescale the fitness functions we analyse, while allowing us to explore the performance of this algorithm on a family of functions.
Next, we derive upper and lower bounds for \(p_{\mathrm {fix}}(\Delta f) \) that will be useful throughout the manuscript. The bounds for \(\Delta f>0\) show that \(p_{\mathrm {fix}}\) is roughly proportional to the fitness difference between solutions \(\beta \Delta f\).
Lemma 1
Proof
In the following we frequently use \(1 + x \le e^x\) and \(1e^{x}\le 1\) for all \(x \in {\mathbb {R}}\) as well as \(e^x \le \frac{1}{1x}\) for \(x < 1\).
The next lemma shows that the probability of accepting an improvement of \(\Delta f\) is exponentially larger (in \(N\beta \Delta f\)) than accepting its symmetric fitness variation \(\Delta f\).
Lemma 2
Proof
3 SSWM on OneMax
The function \({\textsc {OneMax}} (x) := \sum _{i=1}^n x_i\) has been studied extensively in natural computation because of its simplicity. It represents an easy hill climbing task, and it is the easiest function with a unique optimum for all evolutionary algorithms that only use standard bit mutation for variation [30]. Showing that SSWM can optimise OneMax efficiently serves as proof of concept that SSWM is a reasonable optimiser. It further sheds light on how to set algorithmic parameters such as the selection strength \(\beta \) and the population size N. To this end, we first show a polynomial upper bound for the runtime of SSWM on OneMax for a selection strength of \(2( N  1) \beta \ge \ln ( cn )\). We then show that SSWM exhibits a phase transition on its runtime as a function of \(2N\beta \); decreasing this parameter by a constant factor below \(\ln n\) leads to exponential runtimes on OneMax.
Another reason why studying OneMax for SSWM makes sense is because not all evolutionary algorithms that use a fitnessdependent selection perform well on OneMax. Neumann et al. [17] as well as Oliveto and Witt [23] showed that evolutionary algorithms using fitnessproportional selection, including the Simple Genetic Algorithm, fail badly on OneMax even within exponential time, with very high probability.
3.1 Upper Bound for SSWM on OneMax
We first show the following simple lemma, which gives an upper bound on the probability of increasing or decreasing the number of ones in a search point by k in one mutation.
Lemma 3
The proof can be found in the appendix; it uses arguments from the proof of Lemma 2 in [30]. The second inequality follows immediately from the first one due to the symmetry \({{\mathrm{mut}}}(i,ik)={{\mathrm{mut}}}(ni,ni+k)\).
Now we introduce the concept of drift and find some bounds for its forward and backward expression.
Definition 1
Lemma 4
Proof
Finally, the case for local mutations is straightforward since the probability of a local mutation increasing the number of ones is \(\frac{ni}{n}\) and that of decreasing it is at most 1. \(\square \)
The following theorem shows that SSWM is efficient on OneMax whenever \({2(N1) \beta } \ge \ln (cn)\) for some constant \(c>1.2\), since then \(p_{\mathrm {fix}}(1)\) starts being greater than \({n\cdot p_{\mathrm {fix}}(1)}\), allowing for a positive drift even on the hardest fitness level (\(n1\) ones).
Theorem 5
The factor \(\frac{1}{p_{\mathrm {fix}}(1)} \le 1+\frac{1}{2\beta }\) (by Lemma ) on the runtime bound represents the extra time paid due to the probability of rejecting a better search point. For small selection strength \(\beta \ll 1\) the upper bound essentially increases with \(1/(2\beta )\). This makes sense as for small \(\beta \) (and \(N\beta \gg 0\)) we have \(p_{\mathrm {fix}}(1) \approx 2\beta \) (cf. Lemma ). In this regime absolute fitness differences are small and improvements are only accepted with a small probability.
Proof of Theorem 5
We only give a proof for global mutations here. The analysis for local mutations follows the same way, with simpler calculations, and without the factor “e” in the running time bound.
3.2 A Critical Threshold for SSWM on OneMax
The upper bound from Theorem 5 required \(2(N1) \beta \ge \ln (cn)\), or equivalently, \(2N\beta \ge \ln (n) + \ln (c) + 2\beta \). This condition is vital since if \(N \beta \) is chosen too small, the runtime of SSWM on OneMax is exponential with very high probability, as we show next.
If \(2N \beta \) is smaller than \(\ln (n)\) by a factor of \(1\varepsilon \), for some constant \(\varepsilon > 0\), the optimisation time is exponential in n, with overwhelming probability. SSWM therefore exhibits a phase transition behaviour: changing \(N \beta \) by a constant factor makes a difference between polynomial and exponential expected optimisation times on OneMax.
Theorem 6
If \(2 \le 2N\beta \le (1\varepsilon ) \ln n\) for some \({0< \varepsilon < 1}\), then the optimisation time of SSWM with local or global mutations on OneMax is at least \(2^{c n^{\varepsilon /2}}\) with probability \({12^{\varOmega (n^{\varepsilon /2})}}\), for some constant \(c > 0\).
The condition \(N\beta \ge 1\) is used to ease the presentation; it is not essential and we believe it can be dropped when using more detailed calculations. The idea behind the proof of Theorem 6 is to show that for all search points with at least \(n  n^{\varepsilon /2}\) ones, there is a negative drift for the number of ones. This is because for small \(N \beta \) the selection pressure is too weak, and worsenings in fitness are more likely than steps where mutation leads the algorithm closer to the optimum.
We then use the negative drift theorem with selfloops presented in Rowe and Sudholt [28] (an extension of the negative drift theorem [22] to stochastic processes with large selfloop probabilities). It is stated in the following for the sake of completeness. The theorem uses “\(p_{k, k \pm d} \le x\)” as a shorthand for “\(p_{k, k+d} \le x\) and \(p_{k, kd} \le x\)”.
Theorem 7
Proof of Theorem 6
We only give a proof for global mutations; the same analysis goes through for local mutations with similar, but simpler calculations.
The drift theorem, Theorem 7, will be applied to the number of zeros at the current point in time as distance function to the optimum, when the number of zeros is in the interval \([0, n^{\varepsilon /2}]\). By Chernoff bounds, SSWM starts with a fitness of at most \(nn^{\varepsilon /2}\) with probability \(12^{\varOmega (n)}\). We assume in the following that this happens; the claimed probability bound then follows from a union bound of the failure probability \(2^{\varOmega (n)}\) and a failure probability \(2^{\varOmega (n^{\varepsilon /2})}\) which will result from an application of the drift theorem.
Let \(p_{k, j}\) be the probability that SSWM will make a transition from a search point with k ones to one with j ones. Note that, although the drift theorem applies to the number of zeros, our notation of transition probabilities \(p_{k, j}\) refers to numbers of ones for simplicity and consistency with other parts of the paper. Throughout the remainder of the proof we assume \(k \ge n  n^{\varepsilon /2}\).
We remark that for many evolutionary algorithms, such as the (1+1) EA and the (1+\(\lambda \)) EA, a lower bound on OneMax transfers to all functions with a unique global optimum. The reason is that in these cases OneMax is an easiest function amongst those with a single optimum [30]. This generalisation does not apply to SSWM. In fact, OneMax is probably not the easiest function with a single global optimum, even when the fitness range is normalised to [0, n]. The reason here is that acceptance is determined by absolute fitness differences, and a convex fitness curve (beautifully illustrated in [32, Fig. 1]) might amplify fitness differences close to the optimum, in order to compensate for small probabilities of mutation creating fitness improvements. We leave the discovery of an easiest function for SSWM (with a unique optimum) as an open topic for future work.
4 On Traversing Fitness Valleys
We have shown that with the right parameters, SSWM is an efficient hill climber. On the other hand, in contrast to the (1+1) EA, SSWM can accept worse solutions with a probability that depends on the magnitude of the fitness decrease. This is reminiscent of the Metropolis algorithm—although the latter accepts every improvement with probability 1, whereas SSWM may reject improvements.
Jansen and Wegener [12] compared the ability of the (1+1) EA and a Metropolis algorithm in crossing fitness valleys and found that both showed similar performance on smooth integer functions: functions where two Hamming neighbours have a fitness difference of at most 1 [12, Sect. 6].
Definition 2
The (1+1) EA typically optimises \({\textsc {Cliff}}_{d} \) through a direct jump from the top of the cliff to the optimum, which takes expected time \(\varTheta (n^d)\).
Theorem 8
The expected optimisation time of the (1+1) EA on \({\textsc {Cliff}}_{d} \), for \(2 \le d \le n/2\), is \(\varTheta (n^d)\).
In order to prove Theorem 8, the following lemma will be useful for showing that the top of the cliff is reached with good probability. More generally, it shows that the conditional probability of increasing the number of ones in a search point to j, given it is increased to some value of j or higher, is at least 1 / 2.
Lemma 9
The proof of this lemma is presented in the appendix.
Proof of Theorem 8
From any search point with \(i < nd\) ones, the probability of reaching a search point with higher fitness is at least \(\frac{ni}{en}\). The expected time for accepting a search point with at least \(nd\) ones is at most \(\sum _{i=0}^{nd1} \frac{en}{ni} = O(n \log n)\). Note that this is \(O(n^d)\) since \(d \ge 2\).
We claim that with probability \(\varOmega (1)\), the first such search point has \(nd\) ones: with probability at least 1 / 2 the initial search point will have at most \(nd\) ones. Invoking Lemma 9 with \(j := nd\), with probability at least 1 / 2 the top of the cliff is reached before any other search point with at least \(nd\) ones.
Once on the top of the cliff the algorithm has to jump directly to the optimum to overcome it. The probability of such a jump is \(\frac{1}{n^d} \left( 1\frac{1}{n}\right) ^{nd}\) and therefore the expected time to make this jump is \(\varTheta (n^d)\). \(\square \)
SSWM with global mutations also has an opportunity to make a direct jump to the optimum. However, compared to the (1+1) EA its performance slightly improves when considering shorter jumps and accepting a search point of inferior fitness. The following theorem shows that for large enough cliffs, \(d = \omega (\log n)\), the expected optimisation time is by a factor of \(e^{\varOmega (d)} = n^{\omega (1)}\) smaller than that of the (1+1) EA. Although both algorithms need a long time for large d, the speedup of SSWM is significant for large d.
Theorem 10
The expected optimisation time of SSWM with global mutations and \(\beta =1, N = \frac{1}{2}\ln (9n)\) on \({\textsc {Cliff}}_{d} \) with \(d = \omega (\log n)\) is at most \(n^{d}/e^{\varOmega (d)}\).
Proof
We define R as the expected time for reaching a search point with either \(nd\) or n ones, when starting with a worst possible nonoptimal search point. Let \(T_{{\mathrm {cliff}}}\) be the random optimisation time when starting with any search point of \(nd\) ones, hereinafter called the top of the cliff or a local peak. Then the expected optimisation time from any initial point is at most \(R + \text {E}\left( T_{{\mathrm {cliff}}}\right) \).
The remainder of the proof now shows a lower bound on \(p_\mathrm {success}\), the probability of a trial being successful. A sufficient condition for a successful trial is that the following events occur: the next mutation creates a search point with \(nd+k\) ones, for some integer \(1 \le k \le d\) chosen later, this point is accepted, and from there the global optimum is reached before returning to the top of the cliff.
We estimate the probabilities for these events separately in order to get an overall lower bound on the probability of a trial being successful.
We first show that the drift is typically equal to that on OneMax. For every search point with more than a ones, in order to reach \(S_1\), at least k / 2 bits have to flip. Until this happens, SSWM behaves like on OneMax and hence reaches either a global optimum or a point in \(S_1\) in expected time \(O(n \log n)\). The probability for a mutation flipping at least k / 2 bits is at most \(1/(k/2)! = (\log n)^{\varOmega (\log n)} = n^{\varOmega (\log \log n)}\), so the probability that this happens in expected time \(O(n \log n)\) is still \(n^{\varOmega (\log \log n)}\).
This implies that following a lengthk jump, a trial is successful with probability \(1n^{\omega (1)}\). This establishes \(p_\mathrm {success}:= \mathord {\varOmega }\mathord {\left( n^{d+1/2} \cdot \left( \frac{5}{4}\right) ^{d}\right) }\). Plugging this into (10), adding time R for the time to reach the top of the cliff initially, and using that \(O(n^{1/2}\log n) \cdot (4/5)^d = e^{\varOmega (d)}\) for \(d = \omega (\log n)\) yields the claimed bound. \(\square \)
5 SSWM Outperforms (1+1) EA on Balance
Finally, we investigate a feature that distinguishes SSWM from the (1+1) EA as well as the Metropolis algorithm: the fact that larger improvements are more likely to be accepted than smaller improvements.
To this end, we consider the function Balance, originally introduced by Rohlfshagen et al. [27] as an example where rapid dynamic changes in dynamic optimisation can be beneficial. The function has also been studied in the context of stochastic ageing by Oliveto and Sudholt [21] and it goes back to an earlier idea by Witt [36].
In its static (nondynamic) form, Balance can be illustrated by a twodimensional plane, whose coordinates are determined by the number of leading ones (LO) in the first half of the bit string, and the number of ones in the second half, respectively. The former has a steeper gradient than the latter, as the leading ones part is weighted by a factor of n in the fitness (see Fig. 4).
Definition 3
The function is constructed in such a way that all points with a maximum number of leading ones are global optima, whereas increasing the number of ones in the second half beyond a threshold of 7n / 16 (or decreasing it below a symmetric threshold of n / 16) leads to a trap, a region of local optima that is hard to escape from.
Rohlfshagen et al. [27, Theorem 3] showed the following lower bound for the (1+1) EA. The statement is specialised to nondynamic optimisation and slightly strengthened by using a statement from their proof.
Theorem 11
([27]) With probability \(\varOmega (1)\) the (1+1) EA on Balance reaches a trap, and then needs at least \(n^{\sqrt{n}}\) further generations in expectation to find an optimum from there. The expected optimisation time of the (1+1) EA is thus \(\varOmega (n^{\sqrt{n}})\).
We believe that the probability bound \(\varOmega (1)\) can be strengthened to \(1e^{\varOmega (n^{1/2})}\) with a more detailed analysis, which would show that the (1+1) EA gets trapped with an overwhelming probability.
We next show that SSWM with high probability finds an optimum in polynomial time. For appropriately small \(\beta \) we have sufficiently many successes on the LOpart such that we find an optimum before the OneMaxpart reaches the region of local optima. This is because for small \(\beta \) the probability of accepting small improvements is small. The fact that SSWM for \(\beta <1\) is slower than the (1+1) EA on OneMax by a factor of \(O(1/\beta )\) turns into an advantage over the (1+1) EA on Balance.
The following lemma shows that SSWM effectively uses elitist selection on the LOpart of the function in a sense that every decrease is rejected with overwhelming probability.
Lemma 12
For every \(x = ab\) with \(n/16< b_1 < 7n/16\) and \(\beta = n^{3/2}\) and \(N \beta = \ln n\), the probability of SSWM with local or global mutations accepting a mutant \(x'=a'b'\) with \({\textsc {LO}}(a') < {\textsc {LO}}(a) \) and \(n/16< b'_1 < 7n/16\) is \(O(n^{n})\).
Proof
The following lemma establishes the optimisation time of the SSWM algorithm on either the OneMax or the LOpart of Balance.
For global mutations we restrict our considerations to relevant steps, defined as steps where no leading ones in the first half of the bit string is flipped. The probability of a relevant step is always at least \((11/n)^{n/2} \approx e^{1/2}\). When using local mutations, all steps are defined as relevant.
Lemma 13
Proof
We use the method of typical runs [34]: we consider the typical behaviour of the algorithm, and show that events where the algorithm deviates from a typical run are very unlikely. A union bound over all such failure events proves the claimed probability bound.
Consider a relevant step, implying that global mutations will leave all leading ones intact. With probability 1 / n a local or global mutation will flip the first 0bit. This increases the fitness by \(k \cdot n  \Delta _{\mathrm {OM}}\), where \(\Delta _{\mathrm {OM}}\) is the difference in the OneMaxvalue of b caused by this mutation and k is the number of consecutive 1bits following the first 0bit, after mutation. The latter bits are called free riders and it is well known (see [15, Lemma 1 and proof of Theorem 2]) that the number of free riders follows a geometric distribution with parameter 1 / 2, only capped by the number of bits to the end of the bit string a.
The probability of flipping at least \(\sqrt{n}\) bits in one global mutation is at most \(1/(\sqrt{n})! = e^{\varOmega (\sqrt{n})}\) and the probability that this happens at least once in T relevant steps is still of the same order (using that \(T = \text {poly}\left( n\right) \) as \(p_{\mathrm {fix}}(n\sqrt{n}) \ge 1/N \ge 1/\text {poly}\left( n\right) \)). We assume in the following that this does not happen, which allows us to assume \(\Delta _{\mathrm {OM}} \le \sqrt{n}\). We also assume that the number of leading ones is never decreased during nonrelevant steps as the probability of accepting such a fitness decrease is \(O(n^{n})\) by Lemma 12 and the expected number of nonrelevant steps before T relevant steps have occurred is O(T).
Now, a LOvalue of n / 2 is reached if (event A) in T relevant steps at least \(n/4 + n^{3/4}/8\) improvements happen and if (event B) the first \(n/4 + n^{3/4}/8\) improvements lead to a total of at least \(n/4  n^{3/4}/8\) free riders (unless the number of leading ones hits n / 2). Note that these two events are independent, as improvements are due to the current mutation and the number of free riders is due to the uniform random distribution of bits following the first 0bit [15, Lemma 1].
By Chernoff bounds [5] , the probability that the typical event A does not occur, that is, less than \(n/4 + n^{3/4}/8\) improvements happen, is \(e^{\varOmega (n^{1/2})}\). For the same reason also the probability of event B not occurring is \(e^{\varOmega (n^{1/2})}\). Taking the union bound over all rare failure probabilities proves the claim. \(\square \)
We now show that the OneMax part is not optimised before the LO part.
Lemma 14
Let \(\beta = n^{3/2}\), \(N \beta = \ln n\), and T be as in Lemma 13. The probability that SSWM starting with \(a_0b_0\) such that \(n/4 \le b_0_1 \le n/4 + n^{3/4}\) creates a search point ab with \(b_1 \le n/16\) or \(b_1 \ge 7n/16\) in T relevant steps is \(e^{\varOmega (n^{1/2})}\).
It will become obvious that in T relevant steps SSWM typically makes a progress of O(n) on the OneMax part. The proof of Lemma 14 requires a careful and delicate analysis to show that the constant factors are small enough such that the stated thresholds for \(b_1\) are not surpassed.
Proof of Lemma 14
We only prove that a search point with \(b_1 \ge 7n/16\) is unlikely to be reached with the claimed probability. The probability for reaching a search point with \(b_1 \le n/16\) is clearly no larger, and a union bound for these two events leads to a factor of 2 absorbed in the asymptotic notation.
The proof is divided into two parts: we first estimate the increase of \(b_1\) in steps where the number of leading ones in a does not change. We refer to these as regular steps. Steps where mutation increases the number of leading ones in a are called special steps; during these steps every mutation of b is accepted as the fitness gain through additional leading ones guarantees that any change in b will be accepted. In the following, we first show that the progress in \(b_1\) in regular steps is close to 1.14n / 9 and then we show that the progress in special steps is bounded by \(O(n^{3/4})\) with high probability.
Now, by Chernoff bounds, the probability of having more than \(S := (1+n^{1/4}) \cdot p^+ \cdot T\) improving steps in T relevant steps is \(e^{\varOmega (n^{1/2})}\). Using a Chernoff bound for geometric random variables [5, Theorem 1.14], the probability of S improving steps yielding a total progress of at least \({(1+n^{1/4}) \cdot 4/3 \cdot S}\) is \(e^{\varOmega (n^{1/2})}\).
We grant the algorithm an advantage if we assume that, after initialising with \(b_1 \ge n/4\), no search point with \(b_1 < n/4\) is ever reached.^{1} Under this assumption we always have at least as many 1bits as 0bits in b, and mutation in expectation flips at least as many 1bits to 0 as 0bits to 1.
Then the progress in \(b_1\) in one special step increasing the number of leading ones by \(d\ge 1\) can be described as follows. Imagine a matching (pairing) between all bits in b such that each pair contains at least one 1bit. Let \(X_i\) denote the random change in \(b_1\) by the ith pair. If the pair has two 1bits, \(X_i \le 0\) with probability 1. Otherwise, we have \(X_i = 1\) if the 0bit in the pair is flipped, the 1bit in the pair is not flipped, and the mutant is accepted (which depends on the overall \(b_1\)value in the mutant). The potential fitness increase is at most \(dn + n/2\) as the range of \(b_1\)values is n / 2. Likewise, we have \(X_i = 1\) if the 0bit is not flipped, the 1bit is flipped, and the mutant is accepted (which again depends on the overall \(b_1\)value in the mutant). The fitness increase is at least \(dn  n/2\). With the remaining probability we have \(X_i = 0\). Hence for global mutations (for local mutations simply drop the \(11/n\) term) the total progress in a special step increasing \({\textsc {LO}}(a) \) by \(d\) is stochastically dominated by a sum of independent variables \(Y_1, \dots , Y_{n/4}\) where \(\mathrm {Pr}\left( Y_i = \pm 1\right) = 1/n \cdot (11/n) \cdot p_{\mathrm {fix}}(dn \pm n/2)\) and \(Y_i = 0\) with the remaining probability.
The total progress in all m special steps is hence stochastically dominated by a sequence of \(m \cdot n/4\) random variables \(Y_i\) as defined above, with \(d:= 1\). Invoking Lemma 17 (basically a Hoeffding bound on the nonzero outcomes of the variables), stated in the appendix, with \(\delta := n^{3/4}\), the total progress in all special steps is at most \(\delta + m \cdot n/4 \cdot \text {E}\left( Y_i\right) = \delta + O(n^{1/2}) = O(n^{3/4})\) with probability \(1e^{\varOmega (n^{1/2})}\).
Hence the net gain in the number of ones in all special steps is at most \(n^{3/4} + O(mn/4 \cdot n^{3/2}) = O(n^{3/4})\) with probability \({1e^{\varOmega (n^{1/2})}}\).
Together with all regular steps, the progress on the OneMax part is at most \(1.14n/9 + O(n^{3/4})\), which for large enough n is less than the distance \(7n/16  (n/4+n^{3/4})\) to reach a point with \(b_1 \ge 7n/16\) from initialisation. This proves the claim. \(\square \)
Finally, we put the previous lemmas together into our main theorem that establishes that SSWM can optimise Balance in polynomial time.
Theorem 15
With probability \(1e^{\varOmega (n^{1/2})}\) SSWM with \(\beta = n^{3/2}\) and \(N \beta = \ln n\) optimises Balance in time \(O(n/\beta ) = O(n^{5/2})\).
Proof
By Chernoff bounds, the probability that for the initial solution \(x_0 = a_0 b_0\) we have \(n/4  n^{3/4} \le b_0_1 \le n/4 +n^{3/4}\) is \(1e^{\varOmega (n^{1/2})}\). We assume pessimistically that \(n/4 \le b_0_1 \le n/4 +n^{3/4}\). Then Lemma 14 is in force, and with probability \(1e^{\varOmega (n^{1/2})}\) within T relevant steps, T as defined in Lemma 13, SSWM does not reach a trap or a search point with fitness 0. Lemma 13 then implies that with probability \(1e^{\varOmega (n^{1/2})}\) an optimal solution with n / 2 leading ones is found.
The time bound follows from the fact that \(T = O(n/\beta )\) and that, again by Chernoff bounds, we have at least T relevant steps in 3T iterations of SSWM, with probability \(1e^{\varOmega (n^{1/2})}\). \(\square \)
6 Conclusions
The field of evolutionary computation has matured to the point where techniques can be applied to models of natural evolution. Our analyses have demonstrated that runtime analysis of evolutionary algorithms can be used to analyse a simple model of natural evolution, opening new opportunities for interdisciplinary research with population geneticists and biologists.
Our conclusions are highly relevant for biology, and open the door to the analysis of more complex fitness landscapes in this field and to quantifying the efficiency of evolutionary processes in more realistic scenarios of evolution. One interesting aspect of our results is that they impose conditions on population size (N) and strength of selection (\(\beta \)) which represent fundamental limits to what is possible by natural selection. We hope that these results may inspire further research on the similarities and differences between natural and artificial evolution.
From a computational perspective, we have shown that SSWM can overcome obstacles such as posed by \({\textsc {Cliff}}_{d} \) and \({\textsc {Balance}} \) in different ways to the (1+1) EA, due to its nonelitistic selection mechanism. We have seen how the probability of accepting a mutant can be tuned to enable hill climbing, where fitnessproportional selection fails, as well as tunnelling through fitness valleys, where elitist selection fails. For Balance we showed that SSWM can take advantage of information about the steepest gradient. The selection rule in SSWM hence seems to be a versatile and useful mechanism. Future work could investigate its usefulness in the context of populationbased evolutionary algorithms.
Footnotes
 1.
Otherwise, we restart our considerations from the first point in time where \(b_1 \ge n/4\) again, replacing T with the number of remaining steps. With overwhelming probability we will then again have \(b_1 \le n/4 + n^{3/4}\).
Notes
Acknowledgments
The research leading to these results has received funding from the European Union Seventh Framework Programme (FP7/20072013) under Grant Agreement No. 618091 (SAGE). The authors thank the anonymous reviewers for their many constructive comments.
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