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The possibility of exploiting self-organization for design purposes is especially relevant toward the development of living technologies , that is, technologies presenting features of living systems (Bedau et al.,, 2009 , such as robustness, adaptability, and self-organization, which can include self-reconfiguration, s... |
As a final discussion, and as food for thought, it is also worth considering when self-organization is not useful in the context of ALife. Tracing a clear line across the domain is of course impossible, but our reasoning above provides some suggestions. Indeed, self-organization does not account for every life-like pro... |
Needless to say, this mini-review we presented here is not meant to be a complete, comprehensive review of self-organization and ALife, given the limitation of space. We plan to expand the work substantially in the near future to develop a more thorough review for publication elsewhere. |
# Source: arxiv 1804.01314 # Title: When Hypermutations and Ageing Enable Artificial Immune Systems to Outperform Evolutionary Algorithms # Sections: all # Downloaded: 2026-03-03T01:57:58.878113+00:00 |
When Hypermutations and Ageing Enable Artificial Immune Systems to Outperform Evolutionary Algorithms Abstract We present a time complexity analysis of the Opt-IA artificial immune system (AIS). We first highlight the power and limitations of its distinguishing operators (i.e., hypermutations with mutation potential an... |
keywords: Artificial Immune Systems , Opt-IA, Runtime Analysis , Evolutionary Algorithms , Hypermutation , Ageing Introduction Artificial immune systems (AIS) are a class of bio-inspired computing techniques that take inspiration from the immune system of vertebrates |
Burnet’s clonal selection theory has inspired various AIS for function optimisation. The most popular ones are Clonalg , the B-Cell Algorithm |
and Opt-IA After numerous successful applications of AIS were reported, a growing body of theoretical work has gradually been built to shed light on the working principles of AIS. While initial work derived conditions that allowed to prove whether an AIS converges or not |
nowadays rigorous time complexity analyses of AIS are available. Initial runtime analyses focused on studying the performance of typical AIS operators in isolation to explain when and why they are effective. Such studies have been extensively performed for the contiguous somatic hypermutation operator employed by the B... |
the inversely proportional hypermutation operator of Clonalg and the ageing operator used by Opt-IA These studies formed a foundational basis which allowed the subsequent analysis of the complete B-Cell algorithm as used in practice for standard combinatorial optimisation |
Compared to the relatively well understood B-Cell algorithm, the theoretical understanding of other AIS for optimisation is particularly limited. In this paper we consider the complete Opt-IA algorithm |
. This algorithm has been shown experimentally to be successful at optimising instances of problems such as protein structure prediction |
, graph colouring and hitting set The main distinguishing features of Opt-IA compared to other AIS is their use of an ageing operator and of hypermutations with mutation potentials. In this work we will first analyse the characteristics of these operators respectively in isolation and afterwards consider a simple, but ... |
The idea behind the ageing operator is that old individuals should have a lower probability of surviving compared to younger ones. Ageing was originally introduced as a mechanism to maintain diversity. Theoretical analyses have strived to justify this initial motivation because the new random individuals (introduced to... |
and to escape local optima if most of the population dies except for one survivor that, at the same generation, moves out of the local optimum |
This effect was shown for a Random Local Search (RLS) algorithm equipped with ageing on the Balance dynamic optimisation benchmark function. An evolutionary algorithm (EA) using standard bit mutation (SBM) |
and ageing would not be able to escape the local optima due to their very large basin of attraction. Herein, we carefully analyse the ability of ageing to escape local optima on the more general Cliff benchmark function and show that using the operator with both RLS and EAs can make a difference between polynomial and ... |
Hypermutation operators are inspired by the high mutation rates occurring in the immune system. In Opt-IA the mutation potential is linear in the problem size and in different algorithmic variants may be static or either increase by a factor that is proportional to the fitness of the solution (i.e., the b-cell) undergo... |
, where inversely proportional hypermutations were considered, with and without the stop at first constructive mutation (FCM) strategy The analysis revealed that, without FCM, the operator requires exponential runtime to optimise the standard OneMax function, while by using FCM the algorithm is efficient. We consider a... |
and show an upper bound on its runtime that is at most by a linear factor larger than the upper bound obtained for any RLS algorithm using AFL. To achieve this, we present a theorem that allows to derive an upper bound on the runtime of sophisticated hypermutation operators by analysing much simpler RLS algorithms with... |
We then concentrate on the analysis of the complete Opt-IA algorithm. The standard Opt-IA uses both hypermutations and hypermacromutation (both with FCM) mainly because preliminary experimental studies for trap functions indicated that this setting led to the best results |
Our analysis reveals that it is unnecessary to use both operators for Opt-IA to be efficient on trap functions. To this end, we will consider the simple version using only static hypermutations as in |
. We will first consider the algorithm with the simplification that we allow genotypic duplicates in the population, to simplify the analysis and enhance the probabilities of ageing to create copies and escape from local optima. Afterwards we extend the analysis to the standard version using a genotype diversity mechan... |
Compared to its conference version , this paper has been improved in several ways. Firstly, we have extended our analyses of the ageing operator and Opt-IA to include the genotype diversity mechanism as in the algorithm proposed in the literature |
. Another addition is the introduction of a class of functions where Opt-IA fails to find the optimum efficiently, allowing us to complete the picture by highlighting problem characteristics where Opt-IA succeeds and where it does not. Finally, this paper includes some proofs which were omitted from the conference vers... |
The rest of the paper is structured as follows. In Section , we introduce and define Opt-IA and its operators. In Section , we present the results of our analyses of the static hypermutation operator in a simple framework to shed light on its power and limitations in isolation. In Section , we present our analyses of t... |
. Finally, we conclude the paper with a discussion of the results and directions for future work. Preliminaries 1: [MATH] 2: initialise [MATH] , a population of |
[MATH] b-cells uniformly at random and set [MATH] for [MATH] 3: while termination condition is not reached do 4: [MATH] = Cloning ( [MATH] ), |
5: [MATH] = Hypermutation ( [MATH] 6: [MATH] = Hypermacromutation ( [MATH] 7: Ageing [MATH] 8: [MATH] 9: [MATH] 10: end while Algorithm 1 |
Opt-IA Subroutines are described in Algorithms and In this section we first present the standard Opt-IA as applied in (called Opt-IA from now on) for the maximisation of [MATH] and then a slightly different version which we will analyse. |
The Opt-IA pseudo-code is given in Algorithm It is initialised with a population of [MATH] b-cells, representing candidate solutions, generated uniformly at random with [MATH] In each generation, the algorithm creates a new parent population consisting of dup copies of each b-cell (i.e., Cloning) which will be the subj... |
1: [MATH] 2: for all [MATH] do 3: copy [MATH] [MATH] times, 4: add the copies to [MATH] 5: end for Algorithm 2 Cloning [MATH] The variation stage in Opt-IA uses a hypermutation operator with mutation potential sometimes followed by hypermacromutation |
, sometimes not . The Hypermacromutation operator is essentially the same as the well-studied contiguous somatic mutation operator of the B-Cell algorithm; it chooses two integers [MATH] and [MATH] at random such that [MATH] , then mutates at most [MATH] values in the range of [MATH] |
If both operators are applied, they act on the clone population (i.e., not in sequence) such that they generate [MATH] mutants each. The number of bits [MATH] that are flipped by the hypermutation operator is determined by a function called mutation potential . Three different potentials have been considered in the lit... |
static , where the number of bits that are flipped is linear in the problem size and does not depend on the fitness function fitness proportional (i.e., a linear number of bits are always flipped but increasing proportionally with the fitness of the mutated b-cell) and inversely fitness proportional . The latter potent... |
What is unclear from the literature is whether the [MATH] bits to be flipped should be distinct or not and, when using FCM, whether a constructive mutation is a strictly improving move or whether a solution of equal fitness suffices. In this paper we will consider the static hypermutation operator with pseudo-code give... |
, where individuals have a probability [MATH] of dying only once they reach an age of [MATH] In it was shown that the hybrid version allows to escape local optima, hence we employ this version in this paper and give its pseudo-code in Algorithm |
The generation ends with a selection phase for which the pseudo-code is given in Algorithm If the total number of b-cells that have survived the ageing operator is larger than [MATH] then a standard [MATH] selection scheme is used with the exception that genotype duplicates are not allowed. If the population size is le... |
In this paper we give evidence that disallowing genotypic duplicates may be detrimental because, as we will show, genotypic copies may help the ageing operator to escape local optima more efficiently. Considering the results of the investigations made in Sections and the Opt-IA we will analyse does not apply hypermacro... |
1: [MATH] 2: [MATH] 3: for all [MATH] do 4: if FCM is not used then 5: create [MATH] by flipping [MATH] distinct bits selected uniformly at random. |
6: else 7: create [MATH] by flipping at most [MATH] distinct bits selected uniformly at random one after another until a constructive mutation happens. |
8: end if 9: if [MATH] then 10: [MATH] 11: else 12: [MATH] 13: end if 14: add [MATH] to [MATH] 15: end for Algorithm 3 Static hypermutation ( [MATH] |
1: for all [MATH] do 2: [MATH] 3: if [MATH] then 4: remove [MATH] with probability [MATH] 5: end if 6: end for Algorithm 4 Hybrid ageing [MATH] |
1: [MATH] 2: if [MATH] =1 then 3: remove any offspring with the same genotype as individuals in [MATH] 4: end if 5: if [MATH] then |
6: remove the [MATH] individuals with the lowest fitness breaking ties uniformly at random. 7: end if 8: if [MATH] then 9: add [MATH] individuals initialised uniformly at random. |
10: end if Algorithm 5 Selection [MATH] 1: [MATH] 2: initialise [MATH] , a population of [MATH] b-cells uniformly at random and set [MATH] for [MATH] |
3: while the optimum is not found do 4: [MATH] = Cloning ( [MATH] ), 5: [MATH] = Static hypermutation ( [MATH] 6: Hybrid ageing [MATH] |
7: Selection [MATH] 8: [MATH] 9: end while Algorithm 6 Opt-IA Static Hypermutation The aim of this section is to highlight the power and limitations of the static hypermutation operator in isolation. For this purpose we embed the operator into a minimal AIS framework that uses a population of only one b-cell and create... |
If no improvement is found in the first step , then the operator will perform at most [MATH] useless fitness function evaluations before one hypermutation process is concluded. |
We formalise this result in Theorem for two cases: when FCM only accepts strict improvements as constructive solutions (we formally call such algorithm (1+1) IA [MATH] ) and for the case when FCM also accepts points of equal fitness as constructive solutions (we name such algorithm (1+1) IA [MATH] ). |
We prove in Theorem that the (1+1) IA [MATH] cannot be too slow compared to the standard RLS (i.e., flipping one bit per iteration). We show that the presented results are tight for some standard benchmark functions by proving that the (1+1) IA [MATH] has expected runtimes of [MATH] for OneMax and [MATH] for LeadingOne... |
Nevertheless, we conclude the section by showing for the standard benchmark functions Jump and Cliff that the (1+1) IA [MATH] can be particularly efficient on functions with local optima that are generally difficult to escape from. |
We start by highlighting the limitations of static hypermutation when FCM is not used. Since [MATH] distinct bits have to be flipped at once, the outcome of the hypermutation operator is characterised by a uniform distribution over the set of all solutions which have Hamming distance [MATH] to the parent. Since [MATH] ... |
1: [MATH] 2: initialise a solution uniformly at random and assign it to [MATH] 3: while an optimum is not found do 4: Cloning ( [MATH] ), |
5: Static hypermutation ( [MATH] 6: if [MATH] then 7: [MATH] 8: else 9: [MATH] 10: end if 11: [MATH] 12: end while Algorithm 7 (1+1) IA hyp |
Theorem 1 For any function with a polynomial number of optima, the (1+1) IA [MATH] without FCM needs expected exponential time to find any of the optima. |
Proof. We will first consider the probability that the initial solution is optimal and then the probability of finding an optimal solution in a single step given that the current solution is suboptimal. Note that if [MATH] , sampling the complementary bit string of an optimal solution allows the hypermutation to flip a... |
We now analyse the expected time of the last step before an optimal solution is found given that the current solution is neither an optimal solution nor the complementary bit string of an optimal solution. When [MATH] the probability of finding the optima is zero since the hyper mutation deterministically samples the c... |
The theorem explains why poor results were achieved in previous experimental work both on benchmark functions and real world applications such as the hard protein folding problem |
The authors indeed state that “With this policy, however, and for the problems which are faced in this paper, the implemented IA did not provide good results” |
. Theorem shows that this is the case for any optimisation function of practical interest. In it had already been shown that inversely proportional hypermutations cannot optimise OneMax in less than exponential time (both in expectation and w.o.p.). Although static hypermutations are the focus of this paper, we point o... |
and to the proportional hypermutations from the literature From now on we will only consider hypermutations coupled with FCM. We start by showing that hypermutations cannot be too slow compared to local search operators. We first state and prove the following helper lemma. |
Lemma 1 The probability that the static hypermutation applied to [MATH] either evaluates a specific [MATH] with Hamming distance [MATH] to [MATH] (i.e., event [MATH] ), or that it stops earlier on a constructive mutation (i.e., event |
[MATH] is lower bounded by [MATH] (i.e., [MATH] . Moreover, if there are no constructive mutations with Hamming distance smaller than [MATH] , then [MATH] |
Proof. Since the bits to be flipped are picked without replacement, each successive bit-flip increases the Hamming distance between the current solution and the original solution by one. The lower bound is based on the fact that the first [MATH] bit positions to be flipped have [MATH] different and equally probable out... |
We are ready to show that static hypermutations cannot be too slow compared to any upper bound obtained by applying the AFL method on the expected runtime of the (1+1) RLS which flips exactly [MATH] bits to produce a new solution and applies non-strict elitist selection. AFL require a partition of the search space [MAT... |
Theorem 2 Let [MATH] be any upper bound on the expected runtime of algorithm A established via the artificial fitness levels method. Then, [MATH] . Additionally, for the special case of [MATH] [MATH] |
Proof. Let the function [MATH] for solution [MATH] return the number of solutions which are at Hamming distance [MATH] away from [MATH] and belong to set |
[MATH] for some [MATH] The upper bound on the expected runtime of the (1+1) RLS to solve any function obtained by applying the AFL method is |
[MATH] where [MATH] Since the hypermutation operator wastes at most [MATH] bit mutations when it fails to improve, to prove the first claim it is sufficient to show that for any current solution [MATH] , the probability that the (1+1) IA [MATH] finds an improvement is at least |
[MATH] . This follows from Lemma and the definition of a constructive mutation for the (1+1) IA [MATH] , since for each one of the [MATH] search points, the probability of either finding it or finding a constructive mutation is lower bounded by [MATH] |
Note that, for the (1+1) IA [MATH] , the probability of improving the current solution [MATH] can be smaller than [MATH] since it is also necessary that the first [MATH] sampled solutions have strictly worse fitness than [MATH] to allow the hypermutation operator to flip at least [MATH] bits before stopping. However, f... |
[MATH] produces the same solution. Considering the fact that static hypermutation wastes at most [MATH] fitness evaluations in every failure to obtain an improvement in the first step, the second claim follows. |
In the following we show that the upper bounds of the previous theorem are tight for well-known benchmark functions. Theorem 3 The expected runtime of the (1+1) IA [MATH] and of the (1+1) IA [MATH] to optimise |
OneMax [MATH] is [MATH] Proof. The upper bounds for the [MATH] and [MATH] FCM selection versions follow from Theorem since it is easy to derive an upper bound of [MATH] for RLS using AFL |
For the lower bound of the [MATH] FCM selection version, we follow the analysis in Theorem 3 of for inversely proportional hypermutation (IPH) with FCM to optimise ZeroMin The proof there relies on IPH wasting [MATH] function evaluations every time it fails to find an improvement. This is obviously also true for static... |
We now turn to the LeadingOnes benchmark function, which simply returns the number of consecutive [MATH] -bits before the first 0-bit. |
Theorem 4 The expected runtime of the (1+1) IA [MATH] on LeadingOnes [MATH] is [MATH] Proof. The upper bound is implied by Theorem because AFL gives an [MATH] runtime of RLS for LeadingOnes |
Let [MATH] be the expected number of fitness function evaluations until an improvement is made, considering that the initial solution has [MATH] leading 1-bits. The initial solutions consist of [MATH] leading 1-bits, followed by a 0-bit and [MATH] |
bits which each can be either one or zero with equal probability. Let events [MATH] [MATH] and [MATH] be that the first mutation step flips one of the leading ones (with probability [MATH] ), the first 0-bit (with probability [MATH] or any of the remaining bits (with probability [MATH] ), respectively. If [MATH] |
occurs, then the following mutation steps cannot reach any solution with fitness value [MATH] or higher and all [MATH] mutation steps are executed. Since no improvements have been achieved, the remaining expected number of evaluations will be the same as the initial expectation (i.e., |
[MATH] ). If [MATH] occurs, then a new solution with higher fitness value is acquired and the mutation process stops (i.e., [MATH] ). However if [MATH] occurs, since the number of leading 1-bits in the new solution is [MATH] , the hypermutation operator stops without any improvement (i.e., [MATH] ). According to the la... |
[MATH] . When, this equation is solved for [MATH] , we obtain, [MATH] Since the expected number of consecutive 1-bits that follow the leftmost 0-bit is less than two |
the probability of not skipping a level [MATH] is [MATH] . The initial solution on the other hand will have more than [MATH] leading ones with probability at most [MATH] . Thus, we obtain a lower bound on the expectation, |
[MATH] by summing over fitness levels starting from level [MATH] We now focus on establishing that hypermutations may produce considerable speed-ups if local optima need to be overcome. The Jump function, introduced in |
, consists of a OneMax slope with a gap of length [MATH] bits that needs to be overcome for the optimum to be found. The function is formally defined as: |
[EQUATION] for [MATH] and [MATH] Figure illustrates this function Mutation-based EAs require [MATH] expected function evaluations to optimise Jump and recently a faster upper bound by a linear factor has been proved for standard crossover-based steady-state GAs |
Hence, EAs require increasing runtimes as the length of the gap increases, from superpolynomial to exponential as soon as [MATH] The following theorem shows that hypermutations allow speed-ups by an exponential factor of [MATH] , when the jump is hard to perform. A similar result has been shown for the recently introdu... |
Theorem 5 Let [MATH] . Then the expected runtime of the (1+1) IA [MATH] to optimise Jump is at most [MATH] Proof. The (1+1) IA [MATH] reaches the fitness level [MATH] (i.e., local optima in [MATH] steps according to Theorem All local optima have Hamming distance [MATH] to the optimum and the probability that static hyp... |
Obviously, hypermutations can jump over large fitness valleys also on functions with other characteristics. For instance the Cliff function was originally introduced to show when non-elitist EAs may outperform elitist ones |
This function is formally defined as follows: [EQUATION] Figure shows an illustration of Cliff . Similarly to Jump , this function has a OneMax slope with a gap of length [MATH] bits that needs to be overcome for the optimum to be found. Differently to Jump though, the local optimum is followed by another OneMax slope ... |
Corollary 1 Let [MATH] . Then the expected runtime of the (1+1) IA [MATH] to optimise Cliff is [MATH] The analysis can also be extended to show an equivalent speed-up compared to the (1+1) EA for crossing the fitness valleys of arbitrary length and depth recently introduced in |
. In the next section we will prove that the ageing operator can lead to surprising speed-ups for Cliff and other functions with similar characteristics. |
Ageing It is well-understood that the ageing operator can allow algorithms to escape from local optima. This effect was shown on the Balance function from dynamic optimisation for an RLS algorithm embedding a hybrid ageing operator |
However, for that specific function, an SBM operator would fail to escape, due to the large basin of attraction of the local optima. In this section we highlight the capabilities of ageing in a more general setting (i.e., the standard Cliff benchmark function) and show that ageing may also be efficient when coupled wit... |
Ageing allows to escape from a local optimum if one not locally optimal b-cell is created and survives while all the other b-cells die. For this to happen, it is necessary that all the b-cells are old and have similar age. This is achieved on a local optimum by creating copies of the locally optimal b-cell (the b-cells... |
1: [MATH] 2: initialise [MATH] , a population of [MATH] individuals uniformly at random and set [MATH] for [MATH] 3: while the optimum is not found do |
4: select [MATH] uniformly at random, 5: with probability [MATH] create [MATH] by flipping one bit of [MATH] , otherwise [MATH] 6: Hybrid ageing [MATH] |
7: Selection [MATH] 8: [MATH] 9: end while Algorithm 8 [MATH] +1) RLS [MATH] The Cliff benchmark function is generally used to highlight circumstances when non-elitist EAs outperform elitist ones. Algorithms that accept inferior solutions can be efficient for the function by jumping to the bottom of the cliff and then ... |
is neither too large nor too small . This makes the difference between polynomial and exponential expected runtimes compared to elitist EAs (i.e., [MATH] if the cliff is located far away from the optimum. A smaller, but still exponential, speed-up was recently shown for the population-genetics-inspired SSWM (Strong Sel... |
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