A Compass to Guide Genetic Algorithms - Semantic Scholar

A Compass to Guide Genetic Algorithms Jorge Maturana and Fr´ed´eric Saubion {maturana,saubion}@info.univ-angers.fr LERIA, Universit´e d’Angers 2, Bd Lavoisier 49045 Angers (France)

Abstract. Parameter control is a key issue to enhance performances of Genetic Algorithms (GA). Although many studies exist on this problem, it is rarely addressed in a general way. Consequently, in practice, parameters are often adjusted manually. Some generic approaches have been experimented by looking at the recent improvements provided by the operators. In this paper, we extend this approach by including operators’ effect over population diversity and computation time. Our controller, named Compass, provides an abstraction of GA’s parameters that allows the user to directly adjust the balance between exploration and exploitation of the search space. The approach is then experimented on the resolution of a classic combinatorial problem (SAT).

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Introduction

Genetic Algorithms (GA) are metaheuristics inspired by natural evolution, which manage a population of individuals that evolve thanks to operators’ applications. Since their introduction, GAs have been successfully applied to solve various complex optimization problems. From a general point of view, the performance of a GA is related to its ability to correctly explore and exploit the interesting areas of the search space. Several parameters are commonly used to adjust this exploration/exploitation balance (EEB), and the operator application rates are probably among the most influential ones. A suitable control of parameters is crucial to avoid two well-known problems: premature convergence, that occurs when the population gets trapped in a local optima, and the loss of computation time, due to the inability of the GA to detect the most promising areas of the search space. Most of the efforts on this subject are only applicable to specific algorithms, thus, in practice, parameter control is often achieved manually, supported by empirical observations. More recently, new methods have begun to rise up, proposing more generic control mechanisms. In this trend, our motivation is to design a new controller in which parameters could be handled by more general and abstract concepts, in order to be used by a wide range of GAs. Techniques for assigning values to parameters can be classified according to the taxonomy proposed by Eiben et al. [1]. A general class, named Parameter Setting [2], is divided in Parameter Tuning, where parameters are fixed before the run, and Parameter Control, where parameters are modified during the run. Parameter Control is further divided in Deterministic, where parameters are

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modified according to a fixed and predefined scheduling; Adaptive, where the current state of the search is used to modify parameters by means of rules; and Self-Adaptive [3], where parameters are encoded in the genotype and evolve together with the population. Within adaptive control, the central issue is to design rules able to guide the search and to make the suitable choices. A straightforward way consists in performing test runs to extract pertinent information in order to feed the system. However, this approach involves an extra computational time and does not really correspond to the idea of an “automatic self-driven” algorithm. A more sophisticated way to build a control system consists in adding a learning component, which is able to identify a correct control procedure. This reduces prior effort and increases adaptation abilities, according to the needs of different algorithms. In this context, two perspectives could be identified: The first approach consists in modeling the behavior of the GA using different parameters, typically during a learning phase. [4] presents two methods including a learning phase that tries different combinations of parameters and encodes the results in tables or rules. A similar approach is presented in [5], where population’s diversity and fitness evaluation are embedded in fuzzy logic controllers. Later this controllers are used to guide the search according to a high level strategy. [6] proposes an algorithm divided in periods of learning and control of parameters, by adjusting central and limit values of them. A second approach consists in providing a fast control, neglecting the modeling aspect. [7] presents a controller that adjusts operators’ rates according to recent performances. Similar ideas are presented in [8, 9]. In [10], this approach is extended by considering several statistics of individuals fitness and survival rate to evaluate operator quality. In [11], the population is resized, depending of several criteria based on the improvement of the best historical fitness. [12] presents an algorithm that oscillates between exploration and exploitation phases when diversity thresholds are crossed. [13] modifies parameters according to best fitness value. Some methods in this class require special features from the GA, such as [14], that maintains several populations with different parameter values, and moves the parameter’s values toward the value that produces the best results. In [15], a forking scheme is used: a parent population is in charge of exploration, while several child populations exploit particular areas of the search space. In [16], a parameterless GA gets rid of popsize parameter by comparing the performance of multiple populations of different size. In this paper, we investigate a combination of these two general approaches in order to benefit from their complementary strengths, providing an original abstract control of GAs’ operators. Our controller measures the variations of population’s diversity and mean fitness resulting from an operator application, as well as its execution time. A unique control parameter (Θ) allows us to adjust the desired level of EEB and determines the application rates assigned to each operator. We have tested our approach on the resolution of the famous boolean satisfaction problem (SAT) and compared it to other adaptive control methods.

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The paper is organized as follows. Sect. 2 exposes our approach, Sect. 3 describes the experimental framework we have used, and Sect. 4 discusses results. Finally, main conclusions and future directions are drawn in Sect. 5.

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Method Overview

We consider here a basic steady-state GA: at each step an operator is selected among several ones, according to a variable probability. Asexual operators are applied to the best of two randomly chosen individuals of the population, and the resulting individual replaces the worst one. Sexual operators work on two randomly chosen individuals, modifying them directly. The parameters considered here are therefore operators’ application rates. As mentioned in the introduction, adaptive control can be considered from two different points of view. In order to illustrate more precisely these differences, we may detail two recent and representative approaches by comparing the work of Thierens [7] and a method proposed by Wong et al. [6]. In [7], Adaptive Pursuit (AP) aims at adjusting the probabilities of associated operators, depending on their performances, measured typically by fitness improvement during previous applications. This method is able to quickly adapt these probabilities in order to award the most successful operators. AP does not care about understanding the behavior of the algorithm and focuses immediately on the best values, in order to increase the performance. At this point, we may remark that algorithms that are solely based on fitness improvement may experience premature convergence. In the APGAIN method [6], the search is divided in epochs, further divided in two periods. The first one is devoted to the measurement of operators’ performance by applying them randomly, and the second one applies operators according to a probability which is proportional to the observed performance. Three values (low, medium, high) are considered for each parameter, and adjusted by moving them towards the most successful value. Finally, a diversification mechanism is included in the fitness function. Roughly a quarter of the generations is dedicated to the first (learning) period, what could be harmful if there are disrupting operators. Here, we propose a new controller (Compass) based on the idea presented in [5], that considers both diversity and quality as pertinent criteria to evaluate algorithms’ performance. Parameters are abstracted, in order to guide the search by inducing a required level of EEB. The operators are evaluated after each application and, in addition to diversity and fitness variation, a third measure –operator’s execution time– is also considered. To get rid of previous drawbacks, we include some controllers’ features that adapt parameters’ rates during the search [7–9], namely the speed of response, to update the model. Operators are applied according to their application rate, which is updated at every generation. Since we are interested in a controller which could be used by any GA, it must be independent and placed at a different layer. We have then implemented Compass in a C++ class, included by the GA.

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2.1

Operator Evaluation and Applications Rates Updating

Given an operator i ∈ [1 . . . k] and a generation number t, let dit , qit , Tit be, respectively, the population’s mean diversity variation, mean quality (fitness) variation, and mean execution time of i over the last τ applications of this operator. At the beginning of the run, all operators can be applied with the same probability. We define a vector oit = (dit , qit ) to characterize the effects of the operator over the population in terms of variation of quality and diversity (axis ∆D and ∆Q of Fig. 1). Note that, since both quality and diversity improvements correspond to somewhat opposite goals, most vectors will lay on quadrants II (improvement of quality but a decrease in diversity) and IV (increase in diversity and a reduction of mean fitness), shown in Fig. 1a. Algorithms that just consider the fitness improvement to adjust the operator probabilities would only use the projection of oit over the y-axis (dotted lines in Fig. 1b). On the other hand, if diversity is solely taken in account, measures would be considered as the projection over the x-axis (Fig. 1c). Our goal is to control these two criteria together by choosing a search direction which will be expressed by a vector c (defined by its angle Θ ∈ [0, π2 ]) that characterizes also its orthogonal plane P (see Fig. 1d). Since measures of diversity and quality usually have different magnitudes, they are normalized as: dnit =

dit maxi {|dit |}

and

n qit =

qit maxi {|qit |}

n We thus have vectors onit = (dnit , qit ). Rewards are then based on the projection n of vectors oit over c, i.e., |oit |cos(αit ), αit being the angle between oit and c. A value of Θ close to 0 will encourage exploration, while a value close to π2 will favor exploitation. In this way, the management of application rates is abstracted by the angle Θ, that guides the direction of the search as the needle of a compass shows the north. Projections are turned into positive values by subtracting the smallest one and dividing them by execution time, in order to award faster operators (Fig. 1e).

δit =

|onit |cos(αit ) − mini {|onit |cos(αit )} Tit

Application rates are obtained proportionally to values of δit plus a constant ξt , that ensures that the smallest rate is equal to a minimal rate, Pmin , preventing the disappearance of the corresponding operator (Fig. 1f). δit + ξt pit = Pk i=1 δit + ξt 2.2

Operator Application

Operators’ application rates are updated at every generation. An interesting phenomenon, observed during previous experiments, is the displacement of points

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Fig. 1. (a) points (dit , qit ) and corresponding vectors oit , (b) quality-based ranking, (c) diversity-based ranking, (d) proposed approach, (e) values of δit , (f) final probabilities

n (dnit ,qit ) in the graphic during execution. Consider for instance that Θ is set to π4 , so an equal importance is given to ∆Q and ∆D. At the beginning of the search, just after the population was randomly created, the population diversity is high and mean fitness is low, thus it is easy for most operators to be situated in the quadrant II. After some generations, the population starts to converge to some optimum, so improvement becomes difficult, and points in II corresponding to exploitative operators move near x-axis. When improvements in this zone are exhausted, exploitation operators obtain worst rewards than exploration ones, causing a shift of the search to diversification, and escaping from that optimum. Such a visualization tool could be useful to understand the behavior of operators as well as for debugging purposes.

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Experimentation

For our experiments, we focus on the use of GAs for the resolution of combinatorial problems. Among the numerous possible classes, we have chosen the Boolean satisfiability problem (SAT) [17], which consists in assigning values to binary variables in order to satisfy a Boolean formula. The first reason is that this is probably the most known combinatorial problem, since it has been the first to be proved NP complete and therefore it has been used to encode and solve problems from many application areas. The second reason is that there exists an impressive library [18] of instances and their difficulty has been deeply studied with several interesting theoretical results (e.g.,

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phase transition), which allows us to select different instances with various search landscapes’ properties. More formally, an instance of the SAT problem is defined by a set of Boolean variables X = {x1 , ..., xn } and a Boolean formula F: {0, 1}n → {0, 1}. The formula is said to be satisfiable if there exists an assignment v: X → {0, 1}n satisfying F and unsatisfiable otherwise. Instances are classically formulated in conjunctive normal form (conjunctions of clauses) and therefore one has to satisfy all these clauses. To solve this problem, we consider a GA with a binary population that applies one operator at each generation. The fitness function evaluates the number of clauses satisfied by an individual and the associated problem is thus obviously a maximization one. The diversity is classically computed as the Hamming distance entropy (see [19]). In order to evaluate our control approach, we compare it with Adaptive Pursuit (AP) [7] and APGAIN [6]. As mentioned in Sect. 2, AP is representative of many controllers that consider fitness improvement as their guiding criterion while APGAIN is representative of methods that try to learn and model the behavior of the operators. Additionally, we also included a uniform choice (UC) among operators as the baseline of the comparison. In order to check the robustness of our method –but restricted by the lack of space–, we present 13 different instances from the SATLIB repository [18], mixing problems of different sizes and nature, including random-generated instances, graph coloring, logistics planning and blocks world problems. 3.1

Operators

The goal of this work is to create an abstraction of operators, regardless of their quality, and to compare controllers, and not to develop an efficient GA for SAT. The idea is also to use non standard operators, whose effect over diversity and quality is a priori unknown. Therefore, we propose six operators with different features, more or less specialized with regards to the SAT problem. One-point crossover chooses randomly two individuals and crosses them at a random position. In this operator exclusively, the best child replaces the worst parent. Contagion chooses randomly two individuals, and the variables in false clauses of the worst one are replaced with corresponding values of the best individual. Hill climbing checks all neighbors by swapping one variable, moves to the better one and repeats while improvement is possible. Tunneling swaps variables without decreasing the number of true clauses according to a tabu list of length equal to 14 of the number of variables. Badswap swaps all variables that appear in false clauses. Wave chooses the variable that appears in the highest number of false clauses and in the minimum number of clauses only supported by it, and swaps it. It repeats the same process at most 12 times the number of variables.

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In order to observe the effect of population size over the performance of controllers, we performed experiments with populations of 3, 5, 10 and 20 individuals. 10.000 generations were processed, in order to observe the long-term behavior of controllers. 3.2

Control strategy

Previous experiments have shown that values of Θ around 0.25π produced good results. To observe the sensitivity of this value, we ran experiments with values of 0.20π, 0.25π and 0.30π. Note that, even when the value of Θ remains fixed along the run, it does not mean that Compass falls in the category of parameter tuning. It is necessary to distinguish the parameters of the GA (operator’s application rates) from the parameter(s) of the control strategy (θ in this case). Controller parameters provide an abstraction of GA’s parameters. It is pertinent to wonder whether it is worth replacing GA parameters by controller parameters. We think that this substitution is beneficial in two cases: – When the effect of controller parameters is less sensitive than GA’s parameters. Consider, for instance, the case of mutation rate: small changes in this parameter have a drastic effect over GA performances; so it is interesting to use a controller which is able to wrap these parameters, providing a more stable operation, even by including additional control parameters. – When the controller provides a more comprehensible abstraction of GA parameters. This is the case in our approach: it is easier for a human to think in terms of raising and lowering EEB instead of modifying multiple operators’ parameters, specially when their behavior is ill-known. 1 (see Sect. 2.1). Each run, The parameter τ is set to 100, and Pmin to 3k consisting of a specific problem instance, population size, controller and Θ (just for Compass), was replicated 30 times for significant statistical comparisons. AP and APGAIN parameters were set to published values, or tuned to obtain good performance. According to the notations used in [7, 6], for AP: α = 0.8, β = 0.8, 1 Pmin = 2k . For APGAIN: vL = 0, vU = 1, δ = 0.05, σ = 700, ρ = σ4 , ξ = 10, φ = 0.045 (about 10% of re-evaluations).

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Results and discussion

The average number of false clauses obtained over 30 runs is shown in table 1. Comparisons were done using a student-t test with a significance level of 5%. Values are boldfaced when Compass outperforms UC, and italicized when UC is better than Compass. No font modification means that results are statistically indistinguishable. Cells are grey when Compass outperforms AP, and black when AP outperforms Compass. White cells means indistinguishability. Finally, Compass outperformed APGAIN in all cases, except in those indicated with underlined values, where results are indistinguishable. Average execution times of AP, APGAIN and Compass, relative to those of UC, are shown at the rightest

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column of the table. Total number of clauses of each problem appear in the bottom of the table. From now on we will refer as C.2, C.25, C.3 to Compass with Θ values of 0.20π, 0.25π and 0.30π, respectively.

sw100-1

16.7 8.9 14.0 8.1 8.1 8.1

7.7 3.5 5.4 2.0 2.0 3.0

124.2 23.2 71.3 16.2 109.2 20.6 41.6 13.7 38.4 13.4 47.2 15.5

9.4 3.3 6.0 1.3 1.8 2.2

UC AP APGAIN 5 C.2 C.25 C.3

13.8 8.9 11.2 6.2 6.3 7.8

3.3 61.4 7.6 34.6 3.0 47.4 4.8 23.3 4.9 60.1 6.6 31.1 2.2 27.5 3.0 15.5 1.9 27.2 2.7 16.2 2.5 36.5 4.2 20.0

16.5 11.3 14.0 9.1 8.8 9.0

7.9 4.9 6.4 2.6 2.6 3.5

126.2 24.6 88.2 18.4 118.7 20.1 45.0 15.0 43.4 14.8 66.7 16.8

UC AP APGAIN 10 C.2 C.25 C.3

13.8 9.9 11.7 8.3 8.0 9.1

UC AP APGAIN 20 C.2 C.25 C.3

13.8 9.1 11.3 9.1 9.2 9.4

8.3 4.5 6.0 2.8 2.9 3.4

11.2 14.0 8.3 10.2 9.8 13.6 4.6 6.7 4.4 7.1 5.9 10.3

1.00 0.88 1.09 0.89 0.91 0.80

3.3 54.2 6.3 28.8 15.5 7.0 110.0 19.4 3.9 55.2 5.0 26.3 12.9 5.7 98.6 18.1 4.8 66.0 5.8 30.4 16.3 6.2 120.4 18.8 3.5 44.9 3.9 21.9 11.3 4.5 72.5 17.9 2.9 42.7 4.4 23.1 11.2 4.2 72.6 17.8 3.7 49.3 5.7 24.7 11.5 5.1 96.8 19.1

6.1 5.9 6.8 4.3 4.6 5.9

10.1 12.2 1.00 9.3 12.2 1.00 11.3 13.9 0.62 7.5 10.4 0.92 7.2 9.5 0.83 9.1 12.6 0.78

2.8 42.0 2.4 54.1 4.3 68.0 3.5 55.2 3.3 53.2 3.9 58.6

3.4 6.4 10.3 1.00 5.7 8.4 11.3 1.28 6.3 11.1 13.7 2.38 6.2 8.1 12.5 0.92 6.2 8.7 13.2 0.93 6.1 10.4 12.4 0.88

4.5 23.1 4.9 26.0 5.5 30.0 4.8 26.0 4.8 25.2 4.9 27.5

13.8 14.6 17.8 13.3 13.3 13.4

par16

1.00 0.86 0.97 0.88 0.89 0.73

CBS

11.7 8.3 10.8 5.5 5.9 6.4

f1000

8.3 5.6 8.8 3.3 3.6 4.3

aim

time

sw100-0

3.2 52.6 5.4 38.4 2.1 37.7 3.4 19.6 3.3 51.6 5.0 27.5 2.1 25.5 2.1 13.2 1.6 26.7 2.3 11.7 1.6 26.8 2.8 15.9

uuf250

medium

12.9 7.5 11.8 5.8 6.4 6.1

uf250

logistics

UC AP APGAIN 3 C.2 C.25 C.3

popsize control

flat200

4-blocks

Table 1. Average false clauses and comparative execution times

5.2 88.5 16.5 5.3 102.6 17.2 6.2 120.1 18.7 5.2 90.4 18.9 4.8 90.6 17.8 6.2 99.2 18.7

Total clauses 47820 320 4250 449 2237 6718 953 3310 3100 3100 1065 1065

The mean number of generations required to reach the best values varies between 1000 and 7000, therefore, the 10000 allowed generations seem sufficient for all controllers to insure a fair comparison. Of course, the results are not competitive with specific SAT evolutionary solvers, since we do not use the best dedicated operators and neither try to optimize ours. Our purpose here is rather to highlight the differences between controllers. Better results for SAT using GAs were obtained by a hierarchical memetic algorithm [19]. However, given the early stage of this research, we preferred a simpler GA that applies one operator each time, in order to facilitate understanding. Further research will consider more complex operator architectures. The predominance of Compass, and specially C.25 over UC, AP and APGAIN is noticeable, particularly for small populations. Something similar happens with C.2, and, to some extent, with C.3. As mentioned previously, a value Θ = 0.25π works well with all kind of problems.

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Small populations lose diversity easily, so controlling diversity is a critical issue. APGAIN does it by penalizing common individuals. However, when all individuals are the same, this penalization is not effective. It seems that in practice, diversity is mostly induced by the first period in APGAIN (operator evaluation). 1 . This AP controls diversity by defining a minimum application rate equal to 2k value could be excessive if operators are mostly exploitative. A smaller value of 1 3k , used in Compass, grants the controller a greater range to balance EEB. Small populations provide better results than larger ones. This is probably due to the operators, that were inspired by local search heuristics: they are applied more repetitively over the same individual in smaller populations than in large ones, thus producing better results. Surprisingly, UC is quite competitive as population size increases. It seems that applying operators of both low dit and qit produce “bad” individuals that are able, however, to escape from local optima. Nevertheless, this practice is beneficial only if the population is big enough to keep their good elements at the same time. Execution times of AP and APGAIN are shorter than those of UC for the smallest populations. Compass has stable short execution times for different population sizes. This is interesting because it means that the effort spent in performing control induces savings in total execution time. From an implementation point of view, we found that Compass and AP were more independent from the logic of the GA than APGAIN, which introduces its diversity control mechanism in the GA fitness function. Both AP and Compass provide a separate layer of control. Parameterization of Compass is quite intuitive. We have already discussed the effect of Θ and Pmin . The last parameter, τ , is quite stable, we have replicated experiments with several values for this parameter without detecting a considerable influence over the performances.

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Conclusions

In this paper we have presented Compass, a GA controller that provides an abstraction of parameters and simplifies control by adjusting the level of exploration/exploitation along the search. This controller measures operators’ effects over population’s mean fitness, diversity and execution time. Compass is independent from the GA, in order to provide an additional control layer that could be used by other of population-based algorithms. Experiments were performed using a 6-operators GA to solve instances of the SAT problem. Results were favorably compared against a basic uniform choice and state-of-the-art controllers. The twofold evaluation of operators (quality and diversity variation) is coherent with the guiding principles of population-based search algorithms, i.e. maximizing quality of solutions while avoiding the concentration of the population, in order to benefit from their parallel nature. By considering both measures, we observed a natural mechanism to escape from local optima. The search direction is easily apprehensible by observing a dynamic vectorial representation, thus Compass could also be used as a tool for understanding the role of operators.

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The management of nonstandard unknown operators also opens the perspective of using Compass to evaluate operators generated automatically, for example by means of Genetic Programming.

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