Network Crossover Performance on NK Landscapes and Deceptive Problems

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  • 1. Motivation Outline Network Crossover Algorithms Test Problems Experiments Conclusions Network Crossover Performance on NK Landscapes and Deceptive Problems M. Hauschild1 M. Pelikan1 1 Missouri Estimation of Distribution Algorithms Laboratory (MEDAL) Department of Mathematics and Computer Science University of Missouri - St. Louis Genetic and Evolutionary Computation Conference, 2010 M. Hauschild and M. Pelikan University of Missouri - St. Louis Network Crossover Performance on NK Landscapes and Deceptive Problems
  • 2. Motivation Outline Network Crossover Algorithms Test Problems Experiments Conclusions Motivation Always looking to solve difficult problems with GAs. In a scalable and robust manner. Must respect linkage between bits. Most common variation operators do not do this. Uniform, two-point crossover. One solution is linkage-learning GAs. EDAs respect linkages. Come at the cost of model-building. M. Hauschild and M. Pelikan University of Missouri - St. Louis Network Crossover Performance on NK Landscapes and Deceptive Problems
  • 3. Motivation Outline Network Crossover Algorithms Test Problems Experiments Conclusions Motivation Often have prior information about a problem. Graph-based problems. EDA models on similar problems. What is the best way to exploit this information? Bias EDA model building. Sample directly from a network model. Modify the crossover operator itself. Test this operator against an EDA. M. Hauschild and M. Pelikan University of Missouri - St. Louis Network Crossover Performance on NK Landscapes and Deceptive Problems
  • 4. Motivation Outline Network Crossover Algorithms Test Problems Experiments Conclusions Outline Network crossover Algorithms GA hBOA Deterministic Hill-Climber Test Problems Experiments Trap-5 NK Landscapes Conclusions M. Hauschild and M. Pelikan University of Missouri - St. Louis Network Crossover Performance on NK Landscapes and Deceptive Problems
  • 5. Motivation Outline Network Crossover Algorithms Test Problems Experiments Conclusions Network Crossover Two-parent crossovers start by creating a binary mask. What bits to exchange and what to keep the same. Uniform crossover sets the bits randomly. How to create a mask to respect linkages? Start with a matrix G specifying strongest linkages. M. Hauschild and M. Pelikan University of Missouri - St. Louis Network Crossover Performance on NK Landscapes and Deceptive Problems
  • 6. Motivation Outline Network Crossover Algorithms Test Problems Experiments Conclusions Network Crossover G is often not hard to obtain. Graph problems have this implicitly. MAXSAT and other problems also easy. Trial runs of EDAs. Only requires strongest connections. Does not require perfect knowledge. M. Hauschild and M. Pelikan University of Missouri - St. Louis Network Crossover Performance on NK Landscapes and Deceptive Problems
  • 7. Motivation Outline Network Crossover Algorithms Test Problems Experiments Conclusions Network Crossover To build the mask Choose a random bit Randomized breadth-first search to expand mask Repeat until mask is complete Stop when mask size is n/2. Bits close in G less likely to be disrupted. Bits far from each other more likely to be disrupted. M. Hauschild and M. Pelikan University of Missouri - St. Louis Network Crossover Performance on NK Landscapes and Deceptive Problems
  • 8. Motivation Outline Network Crossover Algorithms Test Problems Experiments Conclusions Network Crossover M. Hauschild and M. Pelikan University of Missouri - St. Louis Network Crossover Performance on NK Landscapes and Deceptive Problems
  • 9. Motivation Outline Network Crossover Algorithms Test Problems Experiments Conclusions Network Crossover M. Hauschild and M. Pelikan University of Missouri - St. Louis Network Crossover Performance on NK Landscapes and Deceptive Problems
  • 10. Motivation Outline Network Crossover Algorithms Test Problems Experiments Conclusions Network Crossover M. Hauschild and M. Pelikan University of Missouri - St. Louis Network Crossover Performance on NK Landscapes and Deceptive Problems
  • 11. Motivation Outline Network Crossover Algorithms Test Problems Experiments Conclusions Network Crossover M. Hauschild and M. Pelikan University of Missouri - St. Louis Network Crossover Performance on NK Landscapes and Deceptive Problems
  • 12. Motivation Outline Network Crossover Algorithms Test Problems Experiments Conclusions Network Crossover M. Hauschild and M. Pelikan University of Missouri - St. Louis Network Crossover Performance on NK Landscapes and Deceptive Problems
  • 13. Motivation Outline Network Crossover Algorithms Test Problems Experiments Conclusions Network Crossover M. Hauschild and M. Pelikan University of Missouri - St. Louis Network Crossover Performance on NK Landscapes and Deceptive Problems
  • 14. Motivation Outline Network Crossover Algorithms Test Problems Experiments Conclusions Genetic Algorithm Three crossover operators used. Network crossover Two-point Uniform Probability of crossover, pc = 0.6 Probability of mutation, pm = 1/n M. Hauschild and M. Pelikan University of Missouri - St. Louis Network Crossover Performance on NK Landscapes and Deceptive Problems
  • 15. Motivation Outline Network Crossover Algorithms Test Problems Experiments Conclusions hierarchical Bayesian Optimization Algorithm (hBOA) Pelikan, Goldberg, and Cantú-Paz; 2001 Uses Bayesian network with local structures to model solutions Acyclic directed Graph String positions are the nodes Edges represent conditional dependencies Where there is no edge, implicit independence M. Hauschild and M. Pelikan University of Missouri - St. Louis Network Crossover Performance on NK Landscapes and Deceptive Problems
  • 16. Motivation Outline Network Crossover Algorithms Test Problems Experiments Conclusions Deterministic Hill-Climber Deterministic hill climber (DHC) used for all runs Performs single-bit changes that lead to maximum performance Stops when no single-bit change leads to improvement Originally considered not using DHC Dramatically improved performance M. Hauschild and M. Pelikan University of Missouri - St. Louis Network Crossover Performance on NK Landscapes and Deceptive Problems
  • 17. Motivation Outline Network Crossover Algorithms Test Problems Experiments Conclusions Trap-5 Partition binary string into disjoint groups of 5 bits 5 if ones = 5 trap5 (ones) = , (1) 4 − ones otherwise Total fitness is sum of single traps Global Optimum: String 1111...1 Local Optimum: 00000 in any partition G has all bits in the same partition connected M. Hauschild and M. Pelikan University of Missouri - St. Louis Network Crossover Performance on NK Landscapes and Deceptive Problems
  • 18. Motivation Outline Network Crossover Algorithms Test Problems Experiments Conclusions NK Landscapes Popular test function developed by Kaufmann (1989). Gives a model of a tunable rugged landscape. An NK fitness landscape is defined by Number of bits, n. Number of neighbors per bit, k . Set of k neighbors (Xi ) for i-th bit, Xi . Subfunction fi defining contribution of Xi and (Xi ). The objective function fnk to maximize is defined as n−1 fnk (X0 , . . . , Xn−1 ) = fi (Xi , (Xi )) (2) i=0 M. Hauschild and M. Pelikan University of Missouri - St. Louis Network Crossover Performance on NK Landscapes and Deceptive Problems
  • 19. Motivation Outline Network Crossover Algorithms Test Problems Experiments Conclusions NK landscapes Nearest neighbor NK landscapes. Bits are arranged in a circle. Neighbors of each bit restricted to the following k bits. Parameter step ∈ {1, 2, . . . , k + 1} used to control overlap. For step = 1, maximum overlap. For step = k + 1, fully separable. Bit positions shuffled randomly to increase difficulty. M. Hauschild and M. Pelikan University of Missouri - St. Louis Network Crossover Performance on NK Landscapes and Deceptive Problems
  • 20. Motivation Outline Network Crossover Algorithms Test Problems Experiments Conclusions NK landscapes Unrestricted NK landscapes. NP-complete for k > 1 Branch and bound algorithm used to find optima. Nearest neighbor NK landscapes. Polynomial solvability. Dynamic programming used to find optima. G connects all neighboring bits. M. Hauschild and M. Pelikan University of Missouri - St. Louis Network Crossover Performance on NK Landscapes and Deceptive Problems
  • 21. Motivation Outline Network Crossover Algorithms Test Problems Experiments Conclusions Experimental Setup Trap-5 Problem sizes from n = 100 to n = 300. Bisection used, 10 out of 10 independent runs. 10 independent bisection runs performed. Some experiments cut short at extreme problem sizes. Unrestricted NK landscapes Problem sizes of n ∈ {20, 22, . . . , 38}. k =5 1000 random problem instances for each setting. M. Hauschild and M. Pelikan University of Missouri - St. Louis Network Crossover Performance on NK Landscapes and Deceptive Problems
  • 22. Motivation Outline Network Crossover Algorithms Test Problems Experiments Conclusions Experimental Setup Nearest neighbor NK landscapes Problem sizes of n ∈ {30, 60, . . . , 210}. Two step sizes considered, step ∈ {1, 5}. k =5 1000 instances for each combination of n, k , step. M. Hauschild and M. Pelikan University of Missouri - St. Louis Network Crossover Performance on NK Landscapes and Deceptive Problems
  • 23. Motivation Outline Network Crossover Algorithms Test Problems Experiments Conclusions Experimental Setup Two replacement techniques considered. Restricted Tournament Replacement(RTR) Niching, replaces similar solutions. Window size set to w = min{n, N/5}. Elitism Keeps a portion of the best individuals each generation. 50% of the most fit individuals kept. Examined three measures Evaluations Local search steps Execution time M. Hauschild and M. Pelikan University of Missouri - St. Louis Network Crossover Performance on NK Landscapes and Deceptive Problems
  • 24. Motivation Outline Network Crossover Algorithms Test Problems Experiments Conclusions Trap-5 Evaluations, RTR DHC flips, RTR 10 netx 10 netx uniform uniform Number of flips Evaluations hboa hboa 2−point 2−point 5 10 5 10 100 150 200 250 300 100 150 200 250 300 Problem Size Problem Size Execution Time, RTR netx Execution Time uniform hboa 2−point 0 10 100 150 200 250 300 Problem Size M. Hauschild and M. Pelikan University of Missouri - St. Louis Network Crossover Performance on NK Landscapes and Deceptive Problems
  • 25. Motivation Outline Network Crossover Algorithms Test Problems Experiments Conclusions Trap-5 Evaluations, elitism DHC flips, elitism netx netx Number of flips uniform uniform Evaluations hboa hboa 5 10 5 10 100 150 200 250 300 100 150 200 250 300 Problem Size Problem Size Execution Time, elitism netx Execution Time uniform hboa 0 10 100 150 200 250 300 Problem Size M. Hauschild and M. Pelikan University of Missouri - St. Louis Network Crossover Performance on NK Landscapes and Deceptive Problems
  • 26. Motivation Outline Network Crossover Algorithms Test Problems Experiments Conclusions Nearest neighbor NK, step = 5 Evaluations, RTR DHC flips, RTR netx netx Number of flips uniform uniform Evaluations 5 4 hboa 10 hboa 10 2p 2p 30 60 90 120 150 210 30 60 90 120 150 210 Problem Size Problem Size Execution Time, RTR netx Execution Time uniform 0 hboa 10 2p 30 60 90 120 150 210 Problem Size M. Hauschild and M. Pelikan University of Missouri - St. Louis Network Crossover Performance on NK Landscapes and Deceptive Problems
  • 27. Motivation Outline Network Crossover Algorithms Test Problems Experiments Conclusions Nearest neighbor NK, step = 5 Evaluations, RTR DHC flips, RTR netx netx Number of flips uniform uniform Evaluations 5 4 hboa 10 hboa 10 2p 2p 30 60 90 120 150 210 30 60 90 120 150 210 Problem Size Problem Size Execution Time, RTR netx Execution Time uniform 0 hboa 10 2p 30 60 90 120 150 210 Problem Size M. Hauschild and M. Pelikan University of Missouri - St. Louis Network Crossover Performance on NK Landscapes and Deceptive Problems
  • 28. Motivation Outline Network Crossover Algorithms Test Problems Experiments Conclusions Nearest neighbor NK, step = 1 Evaluations, RTR DHC flips, RTR netx netx Number of flips uniform uniform Evaluations 4 hboa 5 10 hboa 10 2p 2p 30 60 90 120 150 210 30 60 90 120 150 210 Problem Size Problem Size Execution Time, RTR netx Execution Time uniform hboa 2p 0 10 30 60 90 120 150 210 Problem Size M. Hauschild and M. Pelikan University of Missouri - St. Louis Network Crossover Performance on NK Landscapes and Deceptive Problems
  • 29. Motivation Outline Network Crossover Algorithms Test Problems Experiments Conclusions Nearest neighbor NK, step = 5 Evaluations, elitism DHC flips, elitism netx netx Number of flips uniform uniform Evaluations hboa hboa 5 10 2p 2p 5 10 30 60 90 120 150 210 30 60 90 120 150 210 Problem Size Problem Size Execution Time, elitism netx Execution Time uniform hboa 2p 0 10 30 60 90 120 150 210 Problem Size M. Hauschild and M. Pelikan University of Missouri - St. Louis Network Crossover Performance on NK Landscapes and Deceptive Problems
  • 30. Motivation Outline Network Crossover Algorithms Test Problems Experiments Conclusions Nearest neighbor NK, step = 1 Evaluations, elitism DHC flips, elitism netx netx Number of flips uniform uniform Evaluations 5 10 hboa hboa 2p 5 2p 10 30 60 90 120 150 210 30 60 90 120 150 210 Problem Size Problem Size Execution Time, elitism netx Execution Time uniform hboa 2p 0 10 30 60 90 120 150 210 Problem Size M. Hauschild and M. Pelikan University of Missouri - St. Louis Network Crossover Performance on NK Landscapes and Deceptive Problems
  • 31. Motivation Outline Network Crossover Algorithms Test Problems Experiments Conclusions Nearest neighbor NK by difficulty n = 120, step = 5 n = 120, step = 1 1 1 Number of flips/mean Number of flips/mean 0.8 0.8 0.6 0.6 netx netx 0.4 uniform 0.4 uniform hboa hboa 0.2 0.2 0 0.5 1 0 0.5 1 Percent easiest netx instances Percent easiest netx instances M. Hauschild and M. Pelikan University of Missouri - St. Louis Network Crossover Performance on NK Landscapes and Deceptive Problems
  • 32. Motivation Outline Network Crossover Algorithms Test Problems Experiments Conclusions Unrestricted NK landscapes Evaluations, RTR DHC flips, RTR netx netx Number of flips uniform uniform Evaluations hboa hboa 2p 3 2p 10 2 10 20 22 26 30 34 38 20 22 26 30 34 38 Problem Size Problem Size Execution Time, RTR netx Execution Time uniform hboa −2 2p 10 20 22 26 30 34 38 Problem Size M. Hauschild and M. Pelikan University of Missouri - St. Louis Network Crossover Performance on NK Landscapes and Deceptive Problems
  • 33. Motivation Outline Network Crossover Algorithms Test Problems Experiments Conclusions Unrestricted NK landscapes Evaluations, elitism DHC flips, elitism netx netx Number of flips uniform uniform Evaluations hboa hboa 2p 2p 3 10 2 10 20 22 26 30 34 38 20 22 26 30 34 38 Problem Size Problem Size Execution Time, elitism netx Execution Time uniform hboa −2 2p 10 20 22 26 30 34 38 Problem Size M. Hauschild and M. Pelikan University of Missouri - St. Louis Network Crossover Performance on NK Landscapes and Deceptive Problems
  • 34. Motivation Outline Network Crossover Algorithms Test Problems Experiments Conclusions Unrestricted NK by difficulty n = 38, RTR n = 38, elitism 1 1 Number of flips/mean Number of flips/mean netx netx 0.8 uniform 0.8 uniform hboa hboa 0.6 2p 0.6 2p 0.4 0.4 0.2 0.2 0 0.5 1 0 0.5 1 Percent easiest netx instances Percent easiest netx instances M. Hauschild and M. Pelikan University of Missouri - St. Louis Network Crossover Performance on NK Landscapes and Deceptive Problems
  • 35. Motivation Outline Network Crossover Algorithms Test Problems Experiments Conclusions Conclusions Compared GA with network crossover against GA with uniform and two-point crossover. hBOA, a state of the art EDA. On nearest neighbor NK landscapes and trap5 Network crossover had the best execution time through all settings. Niching with RTR outperformed elitism. hBOA had the least variance in instance difficulty. On unrestricted NK landscapes Results less clear. hBOA had the best scalability. RTR and elitism results were mixed. M. Hauschild and M. Pelikan University of Missouri - St. Louis Network Crossover Performance on NK Landscapes and Deceptive Problems
  • 36. Motivation Outline Network Crossover Algorithms Test Problems Experiments Conclusions Conclusions Future Work Test on more diverse problems. Use trial runs of an EDA to learn the crossover network. Test other network based crossovers. Test against a version of hBOA that takes into account problem structure M. Hauschild and M. Pelikan University of Missouri - St. Louis Network Crossover Performance on NK Landscapes and Deceptive Problems
  • 37. Motivation Outline Network Crossover Algorithms Test Problems Experiments Conclusions Any Questions? M. Hauschild and M. Pelikan University of Missouri - St. Louis Network Crossover Performance on NK Landscapes and Deceptive Problems