K. D. Jong, W. Spears and D. Gordon. Using Markov Chains to Analyze GAFOs. In D. Whitley and M. Vose, editors, FOGA-3. Morgan Kaufmann, 1995.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....of using the infinite population model is that one cannot take into account the sampling bias introduced by using finite populations. The finite population Markov model on the other hand, is too expensive to actually execute except for extremely small problems and extremely small populations (e.g. [10]) One way in which we can introduce finite effects into the infinite population model is by initializing the infinite population model using a distribution taken from a finite population. In this section, we present a very simple application of the infinite population model to look at the ....

Ken De Jong, William Spears, and Diana Gordon. Using Markov Chains to Analyze GAFOs. In D. Whitley and M. Vose, editors, FOGA - 3. Morgan Kaufmann, 1995.

.... has moved away from schemata to land onto Markov chains (Nix and Vose 1992, Davis and Principe 1993, Rudolph 1997c) These are very accurate models and have been very useful to obtain theoretical results on the convergence properties of GAs (for example, on the expected time to hit a solution (De Jong et al. 1995, Rudolph 1997a) or on GA asymptotic convergence (Nix and Vose 1992, Rudolph 1994, Rudolph 1997b) However, although results based on Markov chains are very important in principle, very few useful recipes for practical GA users have been drawn from these fine grain stochastic models. Somewhere ....

De Jong, Kenneth A., William M. Spears and Diana F. Gordon (1995). Using Markov chains to analyze GAFOs. In: Proceedings of the Third Workshop on Foundations of Genetic Algorithms (L. Darrell Whitley and Michael D. Vose, Eds.). Morgan Kaufmann. San Francisco. pp. 115--138.

....Investigation of GA Performance Results for Different Cardinality Alphabets Jackie Rees and Gary J. Koehler Decision and Information Sciences 351 BUS University of Florida Gainesville, FL 32611 352 392 9600 352 392 5438 (FAX) jrees grove.ufl.edu, koehler nervm.nerdc.ufl.edu February 6, 1997 ABSTRACT Theoretical and empirical results give mixed advice for choosing the cardinality for GA representation. Using GA models that capture the exact expected behavior of both the binary and higher cardinality cases, the determination of which representation is best for a given GA can be ....

....and cover the general higher cardinality GA models. In Section 3 we discuss our goals and various issues that impact the study of crosscardinality comparisons of GA performance. In Section 4 we describe our experimental approach. Results are given in Section 5. We discuss these results in Section 6 and give conclusions and future directions in Section 7. 3 2. Notation and Models A. Simple GA Algorithm There are many variants of the simple GA algorithm. For this paper we use the algorithm shown in Table 1. The models given in this paper apply to this algorithm. Algorithm 1: Genetic ....

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De Jong, K. A., W. M. Spears and D. F. Gordon, "Using Markov Chains to Analyze GAFOs," in Foundations of Genetic Algorithms 3, 1995, Morgan Kaufmann, San Francisco, pp. 115-137.

....difficulties can be identified in: 1) the suggestions that evolutionary algorithms incorporating crossover operators will generate more efficient searches than those that do not (e.g. 2, p. 17 18] 6, p. 106] 7] which have been contradicted with empirical evidence in [3] 4] 8] [10]; and many others) 2) the idea that the schema theorem suggests that genetic algorithms offer a near optimal procedure for searching among alternative solutions (e.g. 6, p. 38] which was discounted in [11] also see [12] where it was shown that the schema theorem holds even when the ....

.... answers regarding questions about the asymptotic behavior of various algorithms (e.g. typical instances of evolution strategies and evolutionary programming exhibit asymptotic global convergence [13] whereas the canonical genetic algorithm [1] is not convergent at all) Further, De Jong et al. [10] have used Markov chains and brute force computation to analyze the exact transient behavior of genetic algorithms under small populations (e.g. size five) and small chromosomes (e.g. two or three bits) mainly with respect to the expected waiting time until the global optimum is found for the ....

K.A. De Jong, W.M. Spears, and D.F. Gordon, "Using Markov Chains to Analyze GAFOs," Foundations of Genetic Algorithms 3, L.D Whitley and M.D. Vose (eds.), Morgan Kaufmann, San Mateo, CA, pp. 115-137, 1995.

....X(T ) Then we define Q(X(T ) as the maximum fitness of any individual x i in X(T ) We will denote as X any state which includes an optimal individual x . Note that if U (T ) is fixed at U (0) T 0, then the system described by f is a finite Markov chain, as described, for example, in [9][10] etc. However, for the systems to be described here, with time varying operators and parameters, the notation of state models is simpler. Rather than discussing attainment of a global optimum solution x , we shall use function G(U; X i ; X f ; g; ffl) to denote the probability that ....

Kenneth A. De Jong, William M. Spears and Diana F. Gordon. Using Markov Chains to Analyze GAFOs. Foundations of Genetic Algorithms, Morgan Kaufman, New York.

....for some stopping time s = poly( For this particular problem, however, this implication is unfortunately true. Two points in conclusion: First, the behavior of evolutionary algorithms can be studied numerically provided that the transition matrix of the associated Markov chain is known [38, 39]. But this approach is manageable only for moderately large populations and problem dimensions because of the exponentially growing transition matrices and the inevitable rounding errors during the calculations. Moreover, this approach does not lead to theoretical results as they were presented ....

K. A. De Jong, W. M. Spears, and D. F. Gordon. Using markov chains to analyze GAFOs. In L. D. Whitley and M. D. Vose, editors, Foundations on Genetic Algorithms, 3, pages 115--137. Morgan Kaufmann, San Fransisco (CA), 1995.

....i ) the maximum fitness of any individual x i in X(T ) ffl X 2 fX jX = fx 1 ; x 2 ; x ; xn gg, any state which includes an optimal individual x . Note that if U(T ) is fixed at U(0) T 0, then the system described by f is a finite Markov chain, as described, for example, in [6][7] etc. However, for the systems to be described here, with time varying operators and parameters, the notation of state models is found more convenient. Rather than discussing attainment of a global optimum solution x , we shall use function G(U; X i ; X f ; g; ffl) to denote the ....

K.A. De Jong, W.M. Spears and D.F. Gordon. Using Markov Chains to Analyze GAFOs. Foundations of Genetic Algorithms, Morgan Kaufman, New York.

....The structure of the paper is as follows. In Section 2 we explain the finite Markov chain model for genetic processes. The states of the chain correspond to finite populations. The transition probability between two states is induced by the selection, reproduction, and fitness rules, as in [18,27,13]. Since the evolution from generation to generation is a random process, using finite populations different evolutions may diverge. This is not the case when we consider evolutions of probability density distributions. The idea is to view such processes as corresponding with infinite populations ....

....that the kth generation will be P j given that the (k Gamma 1)st generation is P i , P i ; P j 2 P. A general closed form expression for transition probabilities for simple GA s is derived in [18] and its asymptotics to steady state distributions as population size increases is determined. In [13] it is observed that the mentioned closed form expression allows expression of expected waiting time until global optimum is encountered for the first time , expected waiting time for first optimum within some error tolerance of global optimum , and variance in such measures from run to run , ....

K.A. de Jong, W.M. Spears, and D.F. Gordon, Using Markov chains to analyze GAFOs, pp. 115--137 in: L.D. Whitley and M.D. Vose (Eds.), Proc. Foundations of Genetic Algorithms 3, Morgan Kaufmann, 1995.

.... which uses a similar model; some interesting steady state results have been obtained relating, for instance, to the influence of mutation [19] and to the stability of fixed points (i.e. populations) in an infinite population GA [20] For the case of finite population GAs, De Jong et al. [21] have examined absorption probabilities, waiting times and other traditional Markov chain performance measures. However, once again, computational requirements mean that only very small populations of very short strings can be analyzed by these means. Another perspective is that adopted by Davidor ....

K.A.De Jong, W.M.Spears and D.F.Gordon (1995) Using Markov chains to analyze GAFOs. In D.Whitley and M.Vose (Eds.) (1995) Foundations of Genetic Algorithms 3, Morgan Kaufmann, San Mateo, CA, 115-137.

....The structure of the paper is as follows. In Section 2 we explain the finite Markov chain model for genetic processes. The states of the chain correspond to finite populations. The transition probability between two states is induced by the selection, reproduction, and fitness rules, as in [19,26,14]. Since the evolution from generation to generation is a random process, using finite populations different evolutions may diverge. This is not the case when we consider evolutions of probability density distributions. The idea is to view such processes as corresponding with infinite populations ....

....that the kth generation will be P j given that the (k Gamma 1)st generation is P i (P i ; P j 2 P) A general closed form expression for transition probabilities for simple GA s is derived in [19] and its asymptotics to steady state distributions as population size increases is determined. In [14] it is observed that the mentioned closed form expression allows expression of expected waiting time until global optimum is encountered for the first time , expected waiting time for first optimum within some error tolerance of global optimum , and variance in such measures from run to run , ....

K.A. de Jong, W.M. Spears, and D.F. Gordon, Using Markov chains to analyze GAFOs, pp. 115--137 in: L.D. Whitley and M.D. Vose (Eds.), Proc. Foundations of Genetic Algorithms 3, Morgan Kaufmann, 1995.

....the success of the experiment. As in Spears [12] a value of 0.15 yields good aggregation. This can be seen be examining Figure 4. The values p J (t) are computed for t ranging from 1 to 100 generations, for both the aggregated and unaggregated Markov chains, and graphed as 5 See De Jong et al. [2] for a definition of this search space. 0 0.2 0.4 0.6 0.8 1 10 20 30 40 50 60 70 80 90 100 Generations Type 1 Problem No aggregation Aggregation Figure 4: p J (t) where ffl is 0.0 (no aggregation) and 0.15 (aggregation) for N = 2024. The function is Type 1 deceptive. curves. Despite the fact ....

De Jong, K., Spears, W. and Gordon D. (1994) "Using Markov chains to analyze GAFOs". Foundations of Genetic Algorithms, v3, 115--137.

....regarding the use of Markov models to analyze the behavior of evolutionary algorithms. In this paper, we extend and expand on the results we presented in the previous FOGA workshop concerning the use of Markov models to analyze the transient behavior of GAs being used for function optimization (De Jong, Spears and Gordon, 1994). First, we abandon the mathematically convenient lexicographic ordering of the states of the Markov models and explore alternative orderings of states which have interesting semantic properties, and which provide new insight into the transient behavior To appear in the proceedings of the 1996 ....

.... calculate exact state transition probabilities Q i;j , which specify how likely it is that a simple GA in state i (the current population) will be in state j in the next generation: Q i;j = n r Gamma1 Y y=0 M i F OE i jF OE i j j y z j;y z j;y (2) As we showed in our previous paper (De Jong, Spears and Gordon, 1994), the resulting state transition matrix Q can be used in a variety of ways to gain important insights about the 1 For programming convenience we transpose the Z matrix of Nix and Vose (1992) 2 In their paper they assume a standard bit flipping mutation operator and a 1 point crossover which ....

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K. A. De Jong, W. M. Spears, & D. F. Gordon. (1994) Using Markov chains to analyze GAFOs. Proceedings of the Foundations of Genetic Algorithms Workshop. Estes Park, CO: Morgan Kaufmann, 115 - 137.

....to another j, in one generation (time step) The number of states grows extremely fast as the size of the population increases and as the size of individuals increase. The details of the mapping of GAs to Markov chains can be found in [6] Their use in examining transient behavior can be found in [2]. 5 6.1. Accuracy Experiments. The first set of experiments examine the accuracy of the compressed Markov chains by using both Q n u and Q n c to compute the probability distribution p (n) over the states at time n. To answer such questions, Q n u must be combined with a set of initial ....

....yielding negligible error) For N = 969, over 80 of the states have been removed, yielding Q c matrices roughly 3 the size (in terms of memory requirements) of the original Q u matrix. It is also interesting to note that different search spaces are consistently compressed to different 7 See [2] for a definition of these search spaces. 14 W. M. SPEARS Table 6.1 The percentage of states removed when ffl = 0:15. N = 286 N = 455 N = 680 N = 969 Search Space 1 85 88 90 92 Search Space 2 71 76 81 84 Search Space 3 65 73 79 82 Search Space 4 64 73 79 82 degrees. For ....

[Article contains additional citation context not shown here]

K. De Jong, W. Spears, and D. F. Gordon, Using Markov chains to analyze GAFOs. Foundations of GAs Workshop. Morgan Kaufmann, (1994), pp. 115 -- 137.

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K. D. Jong, W. Spears and D. Gordon. Using Markov Chains to Analyze GAFOs. In D. Whitley and M. Vose, editors, FOGA-3. Morgan Kaufmann, 1995.

No context found.

K. A. De Jong, W. M. Spears, and D. F. Gordon, 1995. Using Markov Chains to Analyze GAFOs. In M. D. Vose and L. D. Whitley, editors, Foundations of Genetic Algorithms, 3, pages 115--137. Morgan Kaufmann, San Mateo, CA.

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De Jong, K., Spears, W. and Gordon, D. (1994). Using markov chains to analyze GAFOs. To appear in proceedings of FOGA94.

No context found.

K. A. De Jong, W. M. Spears, and D. F. Gordon. Using Markov chains to analyze GAFOs. In L. D. Whitley and M. D. Vose, editors, Foundations of Genetic Algorithms, volume 3, San Mateo, CA, 1995. Morgan Kaufmann. (To appear).

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