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[EQUATION] as [MATH] , it is not necessarily true that [EQUATION] as [MATH] . However, the authors in totally ignored this issue and combined the transition equations for selection, mutation and crossover together (in Section III of |
) without any justification. In addition, there are several statements in the authors’ proofs in that are questionable. First, in the first paragraph of Appendix A (the proof for Theorem 1 in that paper), the authors considered a pair of parents [MATH] and [MATH] for the uniform crossover operator. [MATH] and [MATH] ar... |
is dubious at best. On the other hand, even if the authors’ modeling assumption is independence of individuals for the uniform crossover operator, this assumption is incompatible with the modeling assumption of exchangeability in |
for the operators of selection and mutation. Therefore, combining the transition equations for all these three operators is problematic, because the assumption of independence cannot hold beyond one iteration step. |
Another issue in is that the uniform crossover operator produces two dependent offspring at the same time. As a result, after uniform crossover, the intermediate population is not even exchangeable because it has pair-wise dependency between individuals. Then the same problem arises, that is the transition equation for... |
In summary, several issues arise from previous studies on IPMs for EAs on continuous optimization problems. Therefore, new frameworks and proof methods are needed for analyzing the convergence of IPMs and studying the issue of the stacking of operators and iterating the algorithm. |
III Proposed Framework In this section, we present our proposed analytical framework. In constructing the framework we strive to achieve the following three goals. |
1. The framework should be general enough to cover real world operators and to characterize the evolutionary process of real EA. |
2. The framework should be able to define the convergence of IPMs and serve as justifications of using them. The definition should match people’s intuitions and at the same time be mathematically rigorous. |
3. The framework should provide an infrastructure to study the issue of the stacking of operators and iterating the algorithm. The contents of this section roughly reflects the pursuit of the first two goals. The third goal is reflected in the analyses of the simple EA in Section IV . More specifically, in Section III-... |
To appreciate the significance of our framework, it is worth reviewing the methodology in studying the convergence of IPMs. Implicitly, the authors in |
used point-wise convergence of marginal p.d.f. as the criteria of defining the convergence of IPMs. Apart from the proofs being problematic and incomplete, this definition does not consider the joint distribution of individuals of the population. Thus, it loses information and cannot characterize the dynamics of the wh... |
III-A Notations and Preliminaries In the remainder of this paper we focus on the unconstrained continuous optimization problem [EQUATION] |
where [MATH] is some given objective function. Our framework is general enough such that it does not require other conditions on the objective function [MATH] . However, to prove the convergence of IPMs for mutation and recombination, conditions such as those in Theorem are sometimes needed. We will introduce them when... |
From now on we use [MATH] to denote the set of nonnegative integers and [MATH] the set of positive integers. For any two real numbers [MATH] and [MATH] , let [MATH] be the smaller one of them and [MATH] be the larger one of them. Let [MATH] be random elements of some measurable space [MATH] . We use [MATH] to represent... |
We use the notation [MATH] to represent the array [MATH] . When [MATH] [MATH] represents the infinite sequence [MATH] . We use [MATH] and [MATH] to represent the collections [MATH] and [MATH] , respectively. When the range is clear, we use [MATH] and [MATH] or [MATH] and [MATH] for short. |
Let [MATH] denote the solution space [MATH] . This simplifies our notation system when we discuss the spaces [MATH] and [MATH] . In the following, we define metrics and [MATH] -fields on [MATH] [MATH] and [MATH] and state properties of the corresponding measurable spaces. |
[MATH] is equipped with the ordinary metric [MATH] . Let [MATH] denote the Borel [MATH] -field on [MATH] [MATH] . Together [MATH] defines a measurable space. |
Similarly, [MATH] is equipped with the metric [MATH] , and the corresponding Borel [MATH] -field under [MATH] is denoted by [MATH] . Together [MATH] is the measurable space for [MATH] tuples. |
Next, consider the space of infinite sequences [MATH] It is equipped with the metric [EQUATION] The Borel [MATH] -field on [MATH] under [MATH] is denoted by [MATH] . Then [MATH] is the measurable space for infinite sequences. |
Since [MATH] is separable and complete, it can be proved that [MATH] and [MATH] are also separable and complete (Appendix M6 in ). In addition, because of separability, the Borel [MATH] -fields [MATH] and [MATH] are equal to [MATH] and [MATH] , respectively. In other words, the Borel [MATH] -fields [MATH] and [MATH] |
[MATH] -fields [MATH] ) and all measurable cylinder sets ( [MATH] ), respectively (Lemma 1.2 in ). Therefore, from now on we write [MATH] and [MATH] for the corresponding Borel [MATH] -fields. Finally, let [MATH] [MATH] and [MATH] denote the set of all random elements of [MATH] [MATH] and [MATH] , respectively. |
Let [MATH] be the natural projection: [MATH] . Since given [MATH] [MATH] defines a random element of [MATH] projected from [MATH] , we also use [MATH] to denote the mapping: [MATH] where [MATH] . By definition, [MATH] is the operator which truncates random sequences to random vectors. Given [MATH] , we use [MATH] to de... |
III-B Analytical Framework for EA and IPMs In this section, we present an analytical framework for the EA and IPMs. First, the modeling assumptions are stated. We only deal with operators which generate c.i.i.d. individuals. Then, we present an abstraction of the EA and IPMs. This abstraction serves as the basis for bu... |
III-B Modeling Assumptions We assume that the EA on the problem ( 24 ) is time homogeneous and Markovian, such that the next generation depends only on the current one, and the transition rule from the [MATH] th generation to the [MATH] th generation is invariant with respect to [MATH] . We further assume that individu... |
The main reason for introducing this assumption is to simplify the analysis. Conditional independence implies exchangeability, therefore individuals in the same generation [MATH] are always exchangeable. As a result, it is possible to exploit the symmetry in the population and study the transition equations of marginal... |
However, we admit that there are some exceptions to our assumption. A most notable one may be the mutation operator, though it does not pose significant difficulties. The mutation operator perturbs each individual in the current population independently, according to a common conditional p.d.f. If the current populatio... |
, where mutation is analyzed together with proportionate selection. On the other hand, an algorithm which only uses mutation is very simple. It can be readily modeled and analyzed without much difficulty. |
Perhaps more significant exceptions are operators such as selection without replacement, or the crossover operator which produces two dependent offspring at the same time. In fact, for these operators not satisfying the c.i.i.d. assumption, it is still possible to expand the random elements modeling finite-sized popula... |
III-B The Abstraction of EA and IPMs Given the modeling assumptions, we develop an abstraction to describe the population dynamics of the EA and IPMs. |
Let the EA with population size [MATH] be denoted by [MATH] , and the [MATH] th [MATH] generation it produces be modeled as a random element [MATH] , where [MATH] is a random element representing the [MATH] th individual in [MATH] . Without loss of generality, assume that the EA has two operators, [MATH] and [MATH] . I... |
Given these notations, the evolutionary process of [MATH] can be described by the sequence [MATH] , where the initial population [MATH] is known and the generation of [MATH] follows the recurrence equation |
[EQUATION] Then understanding the population dynamics of the EA can be achieved by studying the distributions and properties of [MATH] |
Let the IPM of the EA be denoted by [MATH] . The population dynamics it produces can be described by the sequence [MATH] , where [MATH] is known and the generation of [MATH] follows the recurrence equation |
[EQUATION] in which [MATH] are operators in [MATH] modeled after [MATH] and [MATH] . Then, the convergence of [MATH] basically requires that [MATH] converges to [MATH] for every generation [MATH] |
III-B The Proposed Framework As stated before, for each generation [MATH] , the elements of the sequence [MATH] and the limit [MATH] are all random elements of different metric spaces. Therefore, the core of developing our model is to expand [MATH] to random sequences, while ensuring that this expansion will not affect... |
The expansion of [MATH] and the relationships between [MATH] [MATH] and [MATH] are the core of our framework. In the following, we present them rigorously. |
III-B The Expansion of [MATH] We start by decomposing each of [MATH] and [MATH] to two operators. One operator is from [MATH] to [MATH] . It corresponds to how to convert random sequences to random vectors. A natural choice is the projection operator [MATH] |
To model the evolutionary process, we also have to define how to expand random vectors to random sequences. In other words, we have to define the expansions of [MATH] and [MATH] , which are functions from [MATH] to [MATH] |
Definition 2 (The expansion of operator) For an operator [MATH] satisfying the condition that for any [MATH] , the elements of [MATH] are c.i.i.d. given [MATH] , the expansion of [MATH] is the operator [MATH] , satisfying that for any [MATH] |
1. [MATH] 2. The elements of [MATH] are c.i.i.d. given [MATH] In Definition , the operator [MATH] is the expansion of [MATH] . Condition ) ensures that [MATH] can be safely replaced by [MATH] . Condition ) ensures that the paddings for the sequence are generated according to the same conditional probability distributio... |
By Definition , the operators in [MATH] can be decomposed as [MATH] and [MATH] , respectively. Then, the evolutionary process of [MATH] can be described by the sequence of random sequences [MATH] , satisfying the recurrence equation |
[EQUATION] where [MATH] follows the recurrence equation ( 25 ), and [MATH] . It can also be proved that [EQUATION] Essentially, ( 27 ) and ( 28 ) describe how the algorithm progresses in the order [MATH] . It fully characterizes the population dynamics [MATH] , and it is clear that the extra step of generating [MATH] d... |
For [MATH] , because [MATH] , there is no need for expansion. For convenience we simply let [EQUATION] for [MATH] In summary, the relationships between [MATH] [MATH] and [MATH] are better illustrated in Fig. . This is the core of our framework for modeling the EA and IPMs. For clarity, we also show the intermediate pop... |
In Fig. , a solid arrow with an operator on it means that the item at the arrow head equals the result of applying the operator on the item at the arrow tail. For example, from the figure it can be read that [MATH] . Dashed arrow with a question mark on it signals the place to check whether convergence in distribution ... |
Finally, one distinction needs special notice. For [MATH] and [MATH] [MATH] ), consider the operators to generate [MATH] and [MATH] . It is clear that [MATH] and [MATH] are two different operators because their domains and ranges are all different. The distinction still exists when we consider [MATH] , though it is mor... |
III-C Convergence of IPMs Given the framework modeling the EA and IPMs, first, we define convergence in distribution for random elements of [MATH] . This is standard material. Then, the convergence of IPMs is defined by requiring that the sequence [MATH] converges to [MATH] for every [MATH] |
III-C Convergence in Distribution As [MATH] are random elements of [MATH] , in the following we define convergence in distribution for sequences of [MATH] -valued random elements. Convergence in distribution is equivalent to weak convergence of induced probability measures of the random elements. We use the former theo... |
. The definition of Prokhorov metric is collected from Section 11.3 in Let [MATH] be random elements defined on a hidden probability space [MATH] taking values in some separable metric space [MATH] [MATH] is coupled with the Borel [MATH] -field [MATH] . Let [MATH] be a separable measurable space other than [MATH] |
Definition 3 (Convergence in distribution) If the sequence [MATH] satisfies the condition that [MATH] for every bounded, continuous function [MATH] , we say [MATH] converges in distribution to [MATH] , and write [MATH] |
For [MATH] , let [MATH] Then it is well known that convergence in distribution on separable metric spaces can be metricized by the Prokhorov metric. |
Definition 4 (Prokhorov metric) For two random elements [MATH] and [MATH] , the Prokhorov metric is defined as [EQUATION] Call a set [MATH] in [MATH] an [MATH] -continuity set if [MATH] , where [MATH] is the boundary set of [MATH] |
Theorem 2 (The Portmanteau theorem) The following statements are equivalent. 1. [MATH] 2. [MATH] for all closed set [MATH] 3. [MATH] for all open [MATH] |
4. [MATH] for all [MATH] -continuity set [MATH] Theorem 3 (The mapping theorem) Suppose [MATH] is a measurable function. Denote by [MATH] the set of discontinuities of [MATH] . If [MATH] and [MATH] , then [MATH] |
Let [MATH] be random elements of [MATH] [MATH] be random elements of [MATH] , then [MATH] and [MATH] are random elements of [MATH] . Note that [MATH] is separable. |
Theorem 4 (Convergence in distribution for product spaces) If [MATH] is independent of [MATH] and [MATH] is independent of [MATH] for all [MATH] , then [MATH] if and only if [MATH] and [MATH] |
Theorem is adapted from Theorem 2.8 (ii) in Let [MATH] be random elements of [MATH] Theorem 5 (Finite-dimensional convergence) [MATH] if and only if [MATH] for any [MATH] |
Theorem basically asserts that convergence in distribution for countably infinite dimensional random elements can be studied through their finite-dimensional projections. It is adapted from Example 1.2 and Example 2.4 in |
. In , the metric space under consideration is [MATH] . However, as both [MATH] and [MATH] are separable, it is not difficult to adapt the proofs for [MATH] to a proof for Theorem . Note that [MATH] are random elements defined on [MATH] taking values in [MATH] , and [MATH] for every [MATH] . The same is true for [MATH] |
III-C Convergence of IPM As convergence in distribution is properly defined, we can use the theory to define convergence of IPMs. The idea is that IPM is convergent (thus justifiable) if and only if it can predict the limit distribution of the population dynamics of [MATH] for every generation [MATH] as the population ... |
Definition 5 (Convergence of IPMs) An infinite population model [MATH] is convergent if and only if for every [MATH] [MATH] as [MATH] , where [MATH] [MATH] and the underling [MATH] [MATH] are generated according to ( 27 ), ( 29 ), ( 25 ) and ( 26 ). |
Definition is essentially the core of our proposed framework. It defines the convergence of IPM and is rigorous and clear. III-D |
Summary In this section, we built a framework to analyze the convergence of IPMs. The most significant feature of the framework is that we model the populations as random sequences, thereby unifying the ranges of the random elements in a common metric space. Then, we gave a rigorous definition for the convergence of IP... |
Our framework is general. It only requires that operators produce c.i.i.d. individuals. In fact, any EA and IPM satisfying this assumption can be put into the framework. However, to obtain meaningful results, the convergence of IPMs has to be proved. This may require extra analyses on IPM and the inner mechanisms of th... |
Finally, there is one thing worth discussing. In our framework, the expansion of operator is carried out by padding the finite population with c.i.i.d. individuals following the same marginal distribution. Then a question naturally arises: why not pad the finite population with some other random elements, or just with ... |
[MATH] s. This restricts our option in proving the convergence of IPMs. IV Analysis of the Simple EA In this section, we analyze the simple EA using our framework. In Section IV-A , we give sufficient conditions for the convergence of IPMs. To appreciate the necessity, consider the framework in Fig. . To prove the conv... |
In view of this, a “smarter” way to prove the convergence of IPM may be the following method. First, the convergence of IPM for one iteration step for each operator is proved. Then, the results are combined and extended to cover the whole population dynamics. The idea is that if the convergence holds for one generation... |
[EQUATION] In other words, [MATH] can model [MATH] for one iteration step. Then, after obtaining similar results for [MATH] and [MATH] , we combine the results together and the convergence of the overall IPM is proved. |
However, this approach still seems difficult because we have to prove this pass-on relation ( 30 ) holds for every [MATH] . In essence, this corresponds to whether the operators in IPM can be stacked together and iterated for any number of steps. This is the issue of the stacking of operators and iterating the algorith... |
To model real EAs, IPM has to be constructed reasonably. As shown in Section II , exchangeability cannot yield the transition equation for the simple EA. This creates the research problem of finding a suitable modeling assumption to derive IPM. Therefore, in Section IV-B , we discuss the issue and propose to use i.i.d.... |
Then, we use the sufficient conditions to prove the convergence of IPMs for various operators. The operators of mutation and [MATH] -ary recombination are readily analyzed in Section IV-C and Section IV-D , respectively. In Section IV-E , we summarize this section and discuss our results. |
IV-A Sufficient Conditions for Convergence of IPMs To derive sufficient conditions for the convergence of the overall IPM, the core step is to derive conditions under which the operators in the IPM can be stacked and iterated. |
As before, let [MATH] and [MATH] denote the EA with population size [MATH] and the IPM under analysis, respectively. Let [MATH] be an operator in the EA, and [MATH] and [MATH] be its corresponding expanded operators in [MATH] and [MATH] , respectively. Note that [MATH] and [MATH] generate random elements of [MATH] . To... |
We define a property under which [MATH] can be stacked with some other operator [MATH] satisfying the same property without affecting the convergence of the overall IPM. In other words, for an EA using [MATH] and [MATH] as its operators, we can prove the convergence of IPM by studying [MATH] and [MATH] separately. We c... |
Let [MATH] be random elements in [MATH] for [MATH] . We have the following results. Definition 6 (The stacking property) Given [MATH] , if for any converging sequence [MATH] [MATH] as [MATH] always holds, then we say that [MATH] has the stacking property on [MATH] |
Theorem 6 If [MATH] and [MATH] have the stacking property on [MATH] , then [MATH] has the stacking property on [MATH] Proof. For any converging sequence [MATH] , because [MATH] has the stacking property on [MATH] , we have [MATH] . Then, [MATH] is also a converging sequence. Since [MATH] has the stacking property on [M... |
By Theorem , any composition of [MATH] and [MATH] has the stacking property on [MATH] . In particular, [MATH] has the stacking property on [MATH] . The stacking property essentially guarantees that the convergence on [MATH] can be passed on to subsequent generations. |
Theorem 7 (Sufficient condition 1) For an EA consisting of a single operator [MATH] , let [MATH] be modeled by [MATH] in the IPM [MATH] and [MATH] have the stacking property on some space [MATH] . If the initial populations of both EA and [MATH] follow the same distribution [MATH] for some [MATH] , then [MATH] converge... |
Proof. Note that for [MATH] and [MATH] , the [MATH] th populations they generate are [MATH] and [MATH] , respectively. By Theorem [MATH] has the stacking property on [MATH] . Because the sequence [MATH] converges to [MATH] , by Definition [MATH] as [MATH] . Since this holds for any [MATH] , by Definition [MATH] converg... |
By Theorem and Theorem , we can prove the convergence of the overall IPM by proving that the operators in the IPM have the stacking property. Comparing with ( 30 ), it is clear that the stacking property is a sufficient condition. This is because the stacking property requires that [MATH] converges to a point in [MATH]... |
Another point worth discussing is the introduction of [MATH] in Definition . Of course, if we omit [MATH] (or equivalently let [MATH] ), the stacking property will become “stronger” because if it holds, the convergence of the IPM is proved for the EA starting from any initial population. However, in that case the condi... |
In Definition , it is required that [MATH] as [MATH] . The sequence under investigation is [MATH] , which is a sequence of changing operators [MATH] on a sequence of changing inputs [MATH] . As both the operators and the inputs change, the convergence of [MATH] may still be difficult to prove. Therefore, in the followi... |
First, let [MATH] , where [MATH] . Then, we have the following sufficient conditions for the stacking property. Theorem 8 (Sufficient condition 2) |
For a space [MATH] and all converging sequences [MATH] , if the following two conditions 1. [MATH] , such that for all [MATH] [MATH] uniformly as [MATH] , i.e. [MATH] as [MATH] |
2. [MATH] as [MATH] are both met, then [MATH] has the stacking property on [MATH] Theorem 9 (Sufficient condition 3) For a space [MATH] and all converging sequences [MATH] , if the following two conditions |
1. [MATH] , such that for all [MATH] [MATH] uniformly as [MATH] , i.e. [MATH] as [MATH] 2. [MATH] as [MATH] are both met, then [MATH] has the stacking property on [MATH] |
In the following, we prove Theorem . Since Theorem and Theorem are symmetric in [MATH] and [MATH] , proving one of them leads to the other. Recall that [MATH] is the Prokhorov metric (Definition ) and [MATH] gets the maximal in the expression. |
Proof. [MATH] , by condition in Theorem [MATH] s.t. [MATH] for all [MATH] . By condition in Theorem [MATH] s.t. [MATH] for all [MATH] . Now for all [MATH] |
[EQUATION] Therefore, [MATH] as [MATH] To understand these two theorems, consider the relationships between [MATH] and [MATH] illustrated by Fig. . In the figure, the solid arrow represents the premise in Definition , i.e. [MATH] as [MATH] . The double line arrow represents the direction to be proved for the stacking p... |
Now it is clear that Theorem and Theorem bring benefits. For example, for Theorem , instead of proving the convergence for a sequence [MATH] ), this sufficient condition considers the convergence of sequences same operator on changing inputs ( [MATH] ) and of the sequence same input ( [MATH] ). |
The reason we introduce [MATH] and [MATH] in Theorem and Theorem respectively is to exclude some of the starting columns and rows in Fig. , if necessary. This is useful in proving the convergence of the IPM of the [MATH] -ary recombination operator. |
IV-B The I.I.D. Assumption In this section, we address the issue of how to construct IPM. This issue also corresponds to how to choose the space [MATH] for the stacking property. |
Before introducing the i.i.d. assumption, let us give an example. Consider the space [MATH] . If the initial population follows some distribution from [MATH] , then the population consists of all identical individuals. If an EA with proportionate selection and crossover operates on this initial population, then all sub... |
On the other hand, if [MATH] [MATH] may be too big to derive meaningful results. This can be seen from our analysis in Section II which shows that under exchangeability it is not possible to derive transition equations of marginal distributions for the simple EA. |
Therefore, choosing [MATH] should strike a balance between the capacity and the complexity of the IPM. In the following analysis, we choose [MATH] to be [MATH] . IPMs of EAs are constructed using the i.i.d. assumption, and we prove the convergence of the overall IPM by proving that the operators in the IPM have the sta... |
We choose [MATH] for the following reasons. First, in the real world, many EAs generate i.i.d. initial populations. Therefore this assumption is realistic. Secondly, i.i.d. random elements have the same marginal distributions. Therefore IPM can be described by transition equations of marginal distributions. Finally, th... |
In the following, we show how to construct IPM under the i.i.d. assumption. This process also relates to condition in Theorem . It essentially describes how the IPM generates new populations. |
Let the operator in the EA be [MATH] , and the corresponding operator in [MATH] be [MATH] . Recall that in our framework we only study EAs consisting of c.i.i.d. operators, therefore [MATH] generates c.i.i.d. outputs by using the first [MATH] elements of its input. The process that [MATH] generates each output can be d... |
[EQUATION] for every [MATH] To derive the IPM [MATH] for [MATH] , consider the case when [MATH] and [MATH] in ( IV-B ). Noting that in this case [MATH] , we have |
[EQUATION] Now taking [MATH] , ( 32 ) in the limit becomes the transition equation describing how [MATH] generates each new individual. Let the transition equation be |
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