How to Escape Traps Using Clonal Selection Algorithms

This paper presents an experimental study on clonal selection algorithms (CSAs) to optimize simple and complex trap functions. Several settings of the proposed immune algorithms were tested in order to effectively face such a hard computational problem. The key feature to solve the trap functions, hence escape traps, is the usage of the hypermacromutation operator couple with a traditional perturbation immune operator. The experimental results show that the CSAs we designed are very competitive with the best algorithms in literature. 1 CLONAL SELECTION THEORY Artificial Immune Systems (AIS) are computational models, inspired by biological immune systems in a broader sense, that have often been shown to be effective for difficult combinatorial optimization (Cutello V., 2002), learning and solving problems (Timmis J., 2001) and many other areas appearing in various industrial, economical and academic domains (de Castro L. N., 2002b). In this paper, we use Clonal Selection Algorithms (CSAs) to solve two trap functions, i.e. for extracting the characteristic behavior when facing these computational problems. The trap functions are complex toy problem, that can help in understanding the efficiency of algorithms’ search ability. Toy problems (e.g., onescounting, Basin-with-a-barrier, Hurdle-problem) play a central role in understanding the dynamics of algorithms (Prugel-Bennett A., 2001). They allow algorithm designers to devise new tools for mathematical analysis and modelling. One can tackle toy problems to build-up a fruitful intuition about the algorithm workings. Moreover, toy problems can be used to show the main differences between different algorithms. In the present experimental research work, we will consider the main differences between clonal selection algorithms, and we will show which particular algorithms and implementation to use to solve effectively the trap functions, a paradigmatic example of toy problem. Clonal selection algorithms are special kind of Immune Algorithms (de Castro L. N., 2002b; Cutello V., 2004) which use the clonal expansion and the affinity maturation as the main forces of the evolutionary process. The theory of clonal selection (Burnet, 1959), suggests that among all possible cells with different receptors circulating in the host organism, only those who are actually able to recognize the antigen will start to proliferate by duplication (cloning). Hence, when a B cell is activated by binding an antigen, it produces many clones, in a process called clonal expansion. The resulting cells can undergo somatic hypermutation, creating offspring B cells with mutated receptors. Antigens compete for recognition with these new B cells, their parents and with other clones. The higher the affinity of a B cell to available antigens, the more likely it will clone. This results in a Darwinian process of variation and selection, called affinity maturation. Two key features of the clonal selection theory need to be taken into account: the hypermutation mechanism and the clonal expansion. Hypermutation can be seen as a local search procedure that leads to a fast ”maturation” during the learning phase. The clonal expansion phase triggers the growth of a new population of high-value B cells centered on a higher affinity value. We will describe, in what follows, the class of immune algorithms (IA) based on the theory of clonal selection. To this end, we will abstract a simplified model of the IS. We will consider only two entities: antigens 322 Cutello V., Narzisi G., Nicosia G., Pavone M. and Sorace G. (2004). HOW TO ESCAPE TRAPS USING CLONAL SELECTION ALGORITHMS. In Proceedings of the First International Conference on Informatics in Control, Automation and Robotics, pages 322-326 DOI: 10.5220/0001142503220326 Copyright c © SciTePress (Ag’s) and B cells. The input is the Ag (i.e.,the problem to tackle, the function to optimize); the output is basically the candidate solutions, the B cells, that have solved/recognized the Ag. All IAs based on the clonal selection theory are population based. Each individual of the population is a candidate solution belonging to the combinatorial fitness landscape of a given computational problem. Using the cloning operator, an immune algorithm produces individuals with higher affinities (higher fitness function values), introducing blind perturbation (by means of a hypermutation operator) and selecting their improved mature progenies. We will describe two different examples of Clonal Selection Algorithms. We start with the algorithm CLONALG (de Castro L. N., 2002a), which uses fitness values for proportional cloning, inversely proportional hypermutation and a birth operator to introduce diversity in the current population along with a mutation rate to flip a bit of a B cell memory. Extended algorithms use also threshold values to clone the best cells in the present population. We will, then, describe an immune algorithm that uses a static cloning operator, hypermutation and hypermacromutation operators, without memory cells and an aging phase, a deterministic elimination process; we will refer to the algorithm using the acronym opt-IA. CLONALG. CLONALG (de Castro L. N., 2002a) is characterized by two populations: a population of antigens Ag and a population of antibodies Ab (denoted with P (t) ). The individual antibody, Ab, and antigen, Ag, are represented by string attributes m = mL, . . . ,m1, that is, a point in an L−dimensional real-valued shape space S,m ∈ S ⊆ . The Ab population is the set of current candidate solutions, and the Ag is the environment to be recognized. After a random initialization of the first population P , the algorithm loops for a predefined maximum number of generations (Ngen ). In the first step, it determines the fitness function values of all Abs in relation to the Ag. Next, it selects n Abs that will be cloned independently and proportionally to their antigenic affinities, generating the clone population P . Hence, the higher the affinity-fitness, the higher the number of clones generated for each of the n Abs with respect to the following function: Nc = ∑ i=1...n (β ∗ n)/i where β is a multiplying factor to be experimentally determined. Each term of the sum corresponds to the clone size of each Ab. The hypermutation operator performs an affinity maturation process inversely proportional to the fitness values generating the matured clone population P. After computing the antigenic affinity (i.e., the fitness function) of the population P, CLONALG creates randomly d new antibodies that will replace the d lowest fit Abs in the current population (for the pseudo-code of CLONALG see (de Castro L. N., 2002a)). opt-IA. The opt-IA algorithm uses only two entities: antigens (Ag) and B cells like CLONALG. At each time step t, we have a population P (t) of size d. The initial population of candidate solutions, time t = 0, is generated randomly. The function Evaluate(P) computes the affinity (fitness) function value of each B cell x ∈ P. The designed IA , like all immune algorithms based on the clonal selection principle, is characterized by clonal expansion, the cloning of B cells with higher antigenic affinity. The implemented IA uses three immune operators, cloning, hypermutation and aging. The cloning operator, simply, clones each B cell dup times producing an intermediate population P clo of size d× dup. The hypermutation operator acts on the the B cell receptor of P . The number of mutations M is determined by a mutation potential. It is possible define various mutation potential. We tested our IA using static, and inversely proportional hypermutation operators, hypermacromutation operator, and combination of hypermutation operators and hypermacromutation. The two hypermutation operators and the Hypermacromutation perturbs the receptors using different mutation potentials, depending upon a parameter c In particular, it is worthwhile to note here, that all the implemented operators try to mutate each B cell receptor M times without using probability mutation. The mutation potentials used in this research work are the following: Static Hypermutation (H1): the number of mutations is independent from the fitness function f , so each B cell receptor at each time step will undergo at most Ms( x) = c mutations. Inversely Proportional Hypermutation (H2): the number of mutations is inversely proportional to the fitness value, that is it decrease as the affinity function of the current B cell increases. So at each time step t, the operator will perform at most Mi(f( x)) = ((1 − E∗ f( x) ) × (c × )) + (c × )) mutations. In this case, Mi(f( x)) has the shape of an hyperbola branch. Hypermacromutation (M): the number of mutations is independent from the fitness function f and the parameter c. In this case, we choose at random two integers, i and j such that (i + 1) ≤ j ≤ the operator mutates at most Mm( x) = j − i + 1 directions, in the range [i, j]. The aging operator eliminates old B cells, in the populations P , P (hyp) and/or P , to avoid premature convergence. To increase the population diversity, new B cells are added by the Elitist Merge function. The parameter τB sets the maximum number of generations allowed to B cells to remain in the population. When a B cell is τB + 1 old it is erased HOW TO ESCAPE TRAPS USING CLONAL SELECTION ALGORITHMS