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; they continue dividing indefinitely (just as stem and germ cells too). An immortal living cell can only last as long as its error-correcting methods (such as conserving the length of its telomeres, up to the so-called Hayflick limit |
, and correcting DNA mutation mechanisms) preserve genome integrity and stability. In practice this is impossible, for purely chemical and physical reasons. For instance, DNA is believed to undergo more than 60 000 single mutations a day in the genomes of mammalian cells |
, and a fraction of DNA will never be repaired by specific DNA enzyme-repairing molecules, hence leading to genome instability and degenerative diseases such as cancer |
, and neuronal and neuromuscular diseases . Telomeres, for their part, cannot remain unchanged due to chemical degradation, so after about 40 to 60 divisions, they shorten and die |
. But all cells are potentially immortal. If they die at the end of a generation, it is because they are programmed to do so in order to serve as building blocks for multicellular organisms. The length of a cell’s telomeres is also believed to play a fundamental role in the fight against cancer: the length of the telom... |
. We can thus start seeing how information theory and effective programming are implicated in the most basic mechanisms of life and death. Hacking cells can be achieved simply by inhalation just as viruses do, and it is promising to be effective in a trial with Cystic fibrosis caused by a gene mutation located in the c... |
3.3 Immunity as computation and cancer as a software-engineering problem Cells are continually programmed and reprogrammed by the environment and by the cells themselves. Fig. (left) illustrates how this process happens. In the example, generic signal 2 first turns a stem cell into an immune system cell (e.g. CD4-posit... |
. Signals 2 and 4 may also prompt other cells to react in different ways. Signals are simply chemicals or chemical cocktails. For example, programming a T cell towards the regulatory T cell (Treg) subset requires proteins such as the transcription factor Foxp3 (forkhead box P3) |
and the cytokines TGF- [MATH] (transforming growth factor beta) and interleukin-2 (IL-2) that the body itself produces using the same or other cells, in effect producing its own signals for reprogramming itself. However, the same signals can also be artificially produced. A model based on cellular automata has shed lig... |
As a programming strategy, one needs for example to induce the production of Foxp3 and TGF- [MATH] in a naive CD4 T cell to produce a Treg-like cell (under ideal conditions and in the presence of IL-2). Importantly, immune system signals also contribute to the regulation of cancer development. In fact, it is believed t... |
What is known as morphological computation is based on the universal phenomenon in biology that form is function, that is, the shape of biological structures (epitomized by proteins) determine the structure’s biological function. While all forms of computation can be traced all the way back to classical computation and... |
Using a combination of the fundamentals of information theory and the principles of natural selection one may grasp how the immune system seems to have come into existence, it being a natural error-correcting mechanism continually streamlined and reprogrammed by natural selection. This latter is again constrained as to... |
, for if the replication process exceeds this limit, then its accuracy will be compromised and there will be a loss of information, eventually leading to failure to recode basic functions of living organisms such as genes. |
How does nature reprogram (or un-program , from the point of view of their original function) cells so that they become cancerous? There are various hypotheses, one of which posits the breaking of a “contract” that cells made when they went from being unicellular life forms, hence ”selfish”, to being components of mult... |
Methods for modifying the functioning of cells have already shown promise. Recently, for example, it has been shown that the immune system can be reprogrammed to fight cancer |
. As an example, in , it is shown that genetically modified T cells can be used in cancer therapy. Indeed, as is pointed out, tumour cells (tumours) follow many strategies to evade the immune system, including tampering with genes that would normally regulate a function related to the sensitivity of an immune cell to c... |
. However, tumour cells generate an immunosuppressive tumour environment that leads the immune system to neglect the tumour and therefore the great danger to the host multicellular organism (the host body) |
So how might we reprogram tumour cells themselves or immune cells to recognise and naturally fight tumour cells? For a cell to replicate it needs to become redundant in the face of environmental noise, hence more simple. We can target the genes that in the replication process contribute less to the information content ... |
The current common classification of cancer divides it into broad groups that are not related to the type of possible error leading to a cancer but rather to where a cancer originates and other physical properties. However, it has been found that different types of cancer under this classification can be deeply related... |
. This may be because certain cancers may have a set of common causes related to a type of bogus program. In software terms, such a classification scheme amounts to classifying all operating system errors in the same bin simply because they involve operating systems, and not because, say, one has a software bug related... |
Some of these software bugs may lead to incremental error accumulation. We think these types of software errors may prompt the rethinking of current cancer classifications, leading to the grouping of cancers in terms of their information/computational type, i.e. the type of error that leads to cell reprogramming, and n... |
shows a Venn diagram depicting the proposed informational view of a software-engineering classification of diseases such as cancer, based on bug type rather than tissue origin. The diagram is in itself a simplification. For example, in biology a lack of signal X likely represents a lack of production of a protein or it... |
Some interesting initial software-engineering questions about cell biology, bugs and cancer can be found illustrated in Fig. (top). While the potential cell bug space can be very large (blue), there have to be natural selection mechanisms that prevent the bugs leading to fatal diseases (e.g. red set), keeping them smal... |
Diseases imply deviations of a cell towards pathological states that are encoded in the cell’s descriptions. Fig. (bottom) shows the intricate ways in which different descriptions of the cell interact with each other. All these causal interactions fully describing the cell constitute what is called the interactome . Th... |
. We have proposed ways to study and quantify the information content of biological networks based on the related concept of algorithmic probability, and we have found that it is possible to characterise and profile these networks in different ways and with considerable degrees of accuracy |
. This shows that the information approach may open a new pathway towards understanding key aspects of the inner workings of molecular biology causality-driven rather than correlation-driven. |
Important sources of information are epigenetic phenomena, an additional layer of complexity reversing the traditional molecular biology dogma that describes how information is transferred from the genome all the way to the upper levels. Epigenetics shows that information can flow bottom-down from all upper layers to t... |
Conclusions We have seen that uncomputability prescribes limits to what can be known about nature or models of nature, limits that are likely to apply to natural and biological systems or the models we build of them, and therefore we cannot help but develop an encompassing behavioural approach that can ultilise ideas a... |
An information computational approach to cancer and human diseases may be key to understanding molecular medicine from a new perspective. The most promising approach, as in software-engineering at the design stage, may involve prevention through permanent monitoring of the immune system (see Fig. ) based on systematic ... |
Acknowledgements We wish to thank the rest of the Unit of Computational Medicine team at Karolinska Institutet, and the support of AstraZeneca, the Strategic Area Neuroscience (StratNeuro), the Foundational Questions Institute (FQXi), the John Templeton Foundation and the Algorithmic Nature Group, LABORES. |
Hector Zenil (BSc Math, UNAM; Masters Logic, Paris 1 Sorbonne; PhD Computer Science, Lille 1) has held positions at the Behavioural and Evolutionary Lab, Department of Computer Science, University of Sheffield and at the Structural Biology Group at the Department of Computer Science, University of Oxford in the UK; and... |
Angelika Schmidt studied Biology at the Technical University of Darmstadt, Germany and was a visiting scholar at The Rockefeller University, New York, USA. She obtained her PhD in Immunology from the University of Heidelberg, Germany on a project about regulatory T cells in Peter Krammer’s group at the German Cancer Re... |
Jesper Tegnér is Chaired Strategic Professor in Computational Medicine at the Centre for Molecular Medicine and Sciences For Life Laboratory (SciLifeLab) at Karolinska Institutet and Karolinska University Hospital in Stockholm, Sweden. He heads a research group of 35 people, one-third working in the molecular biology l... |
# Source: arxiv 1509.07946 # Title: A Revisit of Infinite Population Models for Evolutionary Algorithms on Continuous Optimization Problems # Sections: all # Downloaded: 2026-03-03T01:56:55.870241+00:00 |
A Revisit of Infinite Population Models for Evolutionary Algorithms on Continuous Optimization Problems Abstract Infinite population models are important tools for studying population dynamics of evolutionary algorithms. They describe how the distributions of populations change between consecutive generations. In gener... |
Index Terms: Evolutionary algorithms, infinite population models, population dynamics, convergence in distribution, theoretical analysis |
Introduction Evolutionary algorithms (EAs) are general purpose optimization algorithms which saw great successes in real-world applications. They are inspired by the evolutionary process in nature. A certain number of candidate solutions to the problem at hand are modeled as individuals in a population, and through gen... |
Though conceptually simple, the underlying evolutionary processes and the behaviors of EAs remain to be fully understood. The difficulties lie in the fact that EAs are customizable population-based iterative stochastic algorithms, and the objective function also has great influence on their behaviors. A successful mode... |
Although dynamical system approach brings many insights about EAs, the state spaces of the models tend to grow rapidly as the population size increases. This is because in order to characterize the population dynamics accurately, the state space in the model has to be large enough to describe all the interdependencies ... |
In this paper, we follow this line of research and study IPMs of EAs on continuous space. More specifically, we aim at rigorously proving the convergence of IPMs. Notice that in this study by convergence we usually mean a certain property of IPMs. That an IPM converges loosely means that as the population size goes to ... |
To our knowledge, there are very few research efforts which directly studied the convergence of IPMs. Among them, the studies of Qi et al. in |
are the classic and most relevant ones. Qi et al. studied the population dynamics of simple EA on continuous space. In the first part of their research |
, the authors built an IPM to analyze the population dynamics of simple EA with proportionate selection and mutation. Traditionally, a transition equation is constructed to describe how the probability density functions (p.d.f.s) of the joint distributions of individuals change between consecutive generations. The nove... |
[EQUATION] where [MATH] is the solution space, [MATH] is the predicted marginal p.d.f. of the [MATH] th generation, [MATH] is the objective function to be maximized and [MATH] is the conditional p.d.f. decided by the mutation operator. Though the transition equation of marginal distributions loses information of interd... |
, it is accurate in the limiting case when the population size goes to infinity. Furthermore, in the second part of the research |
, the authors analyzed the crossover operator and modified the transition equation to include all three operators in the simple EA. Overall, the studies of Qi et al. are inspiring, especially the idea of combining the modeling assumption that individuals are exchangeable with the mathematical analysis of point-wise con... |
However, as will be shown in Section II , the convergence proof for ( ) in is problematic . We provide a counterexample to show that in the authors’ proof a key assertion about the law of large numbers (LLN) for exchangeable random vectors is generally not true. Therefore the whole proof is unsound. Furthermore, we sho... |
In addition to the aforementioned problems, we also show that the authors’ proofs in both and are incomplete . The authors did not address the convergence of the stacking of operators and of recursively iterating the algorithm. In essence, the authors only attempted to prove the convergence of the IPM for one iteration... |
Besides , we found no other studies which attempted to prove the convergence of IPMs for EAs on continuous space. Therefore, to fill the research gap, in Section III we propose a general analytical framework. The novelty of our framework is that from the very start of the analysis, we model generations of the populatio... |
To understand the issues and appreciate our framework, consider an EA operating in [MATH] on a fixed continuous optimization problem with different population sizes. When the population size is [MATH] , denote the algorithm by [MATH] . The [MATH] th generation produced by [MATH] can be described by the joint distributi... |
However, it is not obvious how one can rigorously define the convergence for the sequence [MATH] . This is because [MATH] and the limit [MATH] are all random elements taking values in different metric spaces. The range of [MATH] is the Cartesian product of [MATH] copies of [MATH] , whereas the range of [MATH] is the in... |
In this research, we took a different approach. We extended [MATH] , unified the ranges of random elements in a common metric space and gave a mathematically rigorous definition of sequence convergence. We assume for each generation [MATH] [MATH] first generates an intermediate infinite sequence of individuals [MATH] b... |
To illustrate the effectiveness of our framework, we perform convergence analysis of IPM of the simple EA. As our analyses show, the modeling assumption of exchangeability cannot yield the transition equation. Therefore, to obtain meaningful results, we adopt a “stronger” modeling assumption that individuals of the sam... |
To be complete, regarding , there is a comment from Yong with reply. However, the comment was mainly about the latter part of , where the authors analyzed the properties of EAs based on the IPM. It did not discuss the proof for the model itself. For IPMs of EAs on discrete optimization problems, extensive research were... |
. The problems under consideration were discrete optimization problems with finite solution space. The staring point of the authors’ analysis was to model each generation of the population as an “incidence vector”, which describes for each point in the solution space the proportion of the population it occupies. Based ... |
II Discussion of the Works of Qi et al. In this section we analyze the results of Qi et al. in . We begin by introducing some preliminaries for the analysis. Then, in Section II-B , following the notations and derivations in the authors’ papers, we provide a counterexample to show that the convergence proof for the tra... |
is problematic. We further show that the modeling assumption of exchangeability cannot yield the transition equation in general. In Section II-C , we show that the analyses in both |
and are incomplete. The authors did not prove the convergence of IPMs in the cases where operators are stacked together and the algorithm is iterated for multiple generations. |
II-A Preliminaries In the authors’ paper , the problem to be optimized is [EQUATION] where [MATH] is the solution space and [MATH] is some given objective function. The analysis intends to be general; therefore no explicit form of [MATH] is assumed. The algorithm to be analyzed is the simple EA with proportionate selec... |
[EQUATION] After selection, each individual in [MATH] is mutated to generate individuals in [MATH] . The mutation is conducted following the conditional p.d.f. |
[EQUATION] Overall the algorithm is illustrated in Fig. After presenting the optimization problem and the algorithm, the authors proved the convergence of the IPM. It is the main result in |
. It can be reiterated as follows. Theorem 1 (Theorem 1 in Qi et al. Assume that the fitness function [MATH] in ( ) and the mutation operator of simple EA described by ( ) satisfy the following conditions: |
1. [MATH] 2. [MATH] Then as [MATH] , the time history of the simple EA can be described by a sequence of random vectors [MATH] with densities |
[EQUATION] In Theorem [MATH] is the marginal p.d.f. of the [MATH] th generation predicted by the IPM. As the proof for Theorem in |
and the analyses in this paper use the concept of exchangeability in probability theory, we list its definition and some basic facts. |
Definition 1 (Exchangeable random variables, Definition 1.1.1 in A finite set of random variables [MATH] is said to be exchangeable if the joint distribution of [MATH] is invariant with respect to permutations of the indices [MATH] . A collection of random variables [MATH] is said to be exchangeable if every finite sub... |
Definition can also be extended to cover exchangeable random vectors or exchangeable random elements by replacing the term “random variables” in the definition with the respective term. One property of exchangeability is that if [MATH] are [MATH] exchangeable random elements, then the joint distributions of any [MATH] ... |
). When [MATH] this property indicates that [MATH] have the same marginal distribution. Another property is that a collection of random elements are exchangeable if and only if they are c.i.i.d. given some [MATH] -field [MATH] (Theorem 1.2.2 in |
). Conversely, a collection of c.i.i.d. random elements are always exchangeable. Finally, an obvious fact is that i.i.d. random elements are exchangeable, but the converse is not necessarily true. |
It can be seen that the simple EA generates c.i.i.d. individuals given the current population. Therefore, the individuals within the same generation are exchangeable, and they have the same marginal distribution. This leads to the transition equation of marginal p.d.f.s in Theorem . To analyze its proof and construct o... |
II-B Convergence Proof of the Transition Equation In this section we analyze the proof of Theorem and show that it is incorrect. The proof by Qi et al. is in Appendix A of |
. In the proof the authors assumed that individuals in the same generation are exchangeable, therefore they have the same marginal distribution. After a series of derivation steps, the authors managed to obtain a transition equation between the density functions of [MATH] and [MATH] |
[EQUATION] where in ( ), [EQUATION] ) and ( ) are exact. They accurately describe how the marginal p.d.f. for any individual in the next generation can be calculated from the joint p.d.f. of individuals in the current generation. Noticing that [MATH] is the average of the exchangeable random variables [MATH] , by the L... |
[EQUATION] The authors further asserted that [MATH] is itself a random variable, satisfying [EQUATION] 10 ) and ( 11 ) correspond to (A13) and (A14) in Appendix A of |
, respectively. The authors’ proof is correct until this step. However, the authors then asserted that [MATH] is independent of [MATH] for any finite N. In particular, [MATH] is independent of [MATH] for all [MATH] |
\tagform@ 12 Based on this assertion the authors then proved that for all [MATH] and [MATH] [EQUATION] Therefore, the p.d.f. in ( ) converges point-wisely to [MATH] . Noticing that the expression of [MATH] is equal to the right hand side of ( ), the authors claimed that Theorem is proved. |
In the following, we provide a counterexample to show that assertion ( II-B ) is not true. Then, we carry out further analysis to show that under the modeling assumption of exchangeability, the conclusion in ( 13 ), or equivalently Theorem , cannot be true in general. |
II-B On Assertion ( II-B We first reformulate the assertion. Since [MATH] are exchangeable, [MATH] are exchangeable (Property 1.1.2 in |
). Let [MATH] . Then the premises of Theorem are equivalent to [EQUATION] Let [MATH] . According to ( ), ( 10 ) and ( 11 ), [MATH] has the properties that |
[EQUATION] Since [MATH] is a general function, there is no other restrictions for [MATH] and [MATH] . Therefore, ( II-B ) is equivalent to the following assertion: |
For any [MATH] and [MATH] satisfying ( 14 ), ( 15 ) and ( 16 ), [MATH] and [MATH] are independent for any finite [MATH] . In particular, [MATH] is independent of [MATH] for any [MATH] |
\tagform@ 17 However, we use the following counterexample (modified from Example 1.1.1 and related discussions on pages 11-12 in |
) to show that assertion ( II-B ) is not true. Therefore ( II-B ) is not true. II-B Counterexample Let [MATH] be a sequence of i.i.d. random variables satisfying |
[EQUATION] for all [MATH] . Let [MATH] be a random variable independent of [MATH] satisfying [EQUATION] and [EQUATION] Finally, let [MATH] for all [MATH] |
It can easily be verified that [MATH] and [MATH] satisfy ( 14 ) and ( 16 ). Since [MATH] is bounded, [MATH] for any [MATH] . By the strong law of large numbers (SLLN) for i.i.d. random variables, |
[EQUATION] Therefore ( 15 ) is also satisfied, i.e. [MATH] is the limit of [MATH] as [MATH] . However, because [MATH] and [MATH] is independent of [MATH] , it can be seen that [MATH] is not independent of [MATH] except for some degenerate cases (for example when [MATH] equals to a constant). In particular, in general [... |
II-B Further Analysis In the authors intended to prove Theorem , or equivalently ( 13 ). As shown by the counterexample, assertion ( II-B ) is not true. This renders the authors’ proof for ( 13 ) invalid. |
In the following, we carry out further analysis to show that ( 13 ) cannot be true even considering other methods of proof and adding new sufficient conditions. Therefore, in general, Theorem cannot be true. |
To begin with, consider the random variable [MATH] . We prove the following lemma. Lemma 1 [MATH] as [MATH] Proof. According to ( 10 ), [MATH] . Since [MATH] [MATH] almost surely. |
Since [MATH] is continuous on [MATH] , we have [EQUATION] Then we have [EQUATION] Finally, by the conditions in Theorem [MATH] . By the Lebesgue’s Dominated Convergence Theorem (Proposition 11.30 in |
), we have [MATH] as [MATH] . ∎ Now by Lemma , ( 13 ) is equivalent to [EQUATION] Now it is clear that if the only assumption is exchangeability, ( [MATH] ) is not true even considering other methods of proof. Of course, if ( II-B ) is true, [MATH] and [MATH] are independent, then ( II-B ) is true. However, as already ... |
A natural question then arises: is it possible to introduce some reasonable sufficient conditions such that ( [MATH] ) can be proved? One of such conditions frequently used is that [MATH] , i.e. [MATH] converges to its expectation, a constant which equals [MATH] for any [MATH] . However, the following analysis shows th... |
For exchangeable random variables [MATH] , we have [EQUATION] where [MATH] is the variance of [MATH] and [MATH] is the covariance of [MATH] and [MATH] . ( 18 ) is by the boundedness of [MATH] and the Lebesgue’s Dominated Convergence Theorem, ( 19 ) is by the exchangeability of [MATH] , and ( 20 ) is by the boundedness ... |
II-B Summary As the analyses in this section show, the transition equation ( ) does not hold under the modeling assumption of exchangeability. However, it does not preclude the possibility of enhancing the modeling assumption so that it can yield analytical results similar to the transition equation ( ). We deal with t... |
and are incomplete II-C The Issue of the Stacking of Operators and Iterating the Algorithm In the following, we discuss IPMs from another perspective. Consider an EA with only one operator. Let the operator be denoted by [MATH] . When the population size is [MATH] , denote this EA by [MATH] and the operator it actually... |
in the following way. Let [MATH] represent the combined operator of proportionate selection and mutation. Though the authors originally developed the transition equation from the [MATH] th to the [MATH] th generation, without loss of generality we can consider only the populations from the initial generation to the onw... |
[EQUATION] where m.p.w. stands for point-wise convergence of marginal p.d.f.s. However, apart from the fact that this proof is problematic, the authors’ proof covers only one iteration step, corresponding to the column-wise convergence of the [MATH] column in Table . The problem is that even if ( 21 ) is true, it does ... |
To give an example, consider the column of [MATH] in Table . To prove column-wise convergence, the authors need to prove that given ( 21 ), |
[EQUATION] as [MATH] . Comparing ( 21 ) with ( 22 ) and ( 23 ), ( 22 ) has the same sequence of operators but with a sequence of converging inputs, ( 23 ) has the same input but with a sequence of different operators. Therefore, they are not necessarily true even if ( 21 ) is proved. In fact, different techniques may h... |
ignored this issue, we believe their proofs are incomplete The issue of the stacking of operators is similar. Given some operator [MATH] satisfying ( 21 ) and some operator [MATH] satisfying |
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