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- 1. Performance of Evolutionary Algorithms on NK Landscapes with Nearest Neighbor Interactions and Tunable Overlap Martin Pelikan, Kumara Sastry, David E. Goldberg, Martin V. Butz, and Mark Hauschild Missouri Estimation of Distribution Algorithms Laboratory (MEDAL) University of Missouri, St. Louis, MO http://medal.cs.umsl.edu/ pelikan@cs.umsl.edu Download MEDAL Report No. 2009002 http://medal.cs.umsl.edu/files/2009002.pdf M. Pelikan, K. Sastry, D.E. Goldberg, M.V. Butz, M. Hauschild NK Landscapes with Nearest Neighbors and Tunable Overlap
- 2. Motivation Testing evolutionary algorithms Adversarial problems on the boundary of design envelope. Random instances of important classes of problems. Real-world problems. This work bridges and extends two prior studies on random problems Random additively decomposable problems (rADPs) (Pelikan et al., 2006). NK landscapes (superset of rADPs) (Pelikan et al., 2007). This study Propose the class of polynomially solvable NK landscapes with nearest neighbor interactions and tunable overlap. Generate large number of instances of proposed problem class. Test evolutionary algorithms on the generated instances. Analyze the results. M. Pelikan, K. Sastry, D.E. Goldberg, M.V. Butz, M. Hauschild NK Landscapes with Nearest Neighbors and Tunable Overlap
- 3. Outline 1. Additively decomposable problems NK landscapes. Random additively decomposable problems (rADPs). 2. NK with nearest neighbors and tunable overlap. 3. Experiments. 4. Conclusions and future work. M. Pelikan, K. Sastry, D.E. Goldberg, M.V. Butz, M. Hauschild NK Landscapes with Nearest Neighbors and Tunable Overlap
- 4. Additively Decomposable Problems (ADPs) Additively decomposable problem (ADP) Fitness deﬁned as m f (X1 , X2 , . . . , Xn ) = fi (Si ), i=1 n is the number of bits (variables), m is the number of subproblems, Si is the subset of variables in ith subproblem. ADPs play crucial role in design and analysis of GAs & EDAs. All problems in this work are ADPs. Two prior studies on ADPs serve as starting points Unrestricted NK landscapes. Restricted random ADPs (rADPs). M. Pelikan, K. Sastry, D.E. Goldberg, M.V. Butz, M. Hauschild NK Landscapes with Nearest Neighbors and Tunable Overlap
- 5. NK Landscape NK landscape Proposed by Kauﬀman (1989). Model of rugged landscape and popular test function. An NK landscape is deﬁned by Number of bits, n. Number of neighbors per bit, k. Set of k neighbors Π(Xi ) for i-th bit, Xi . Subfunction fi deﬁning contribution of Xi and Π(Xi ). The objective function fnk to maximize is then deﬁned as n−1 fnk (X0 , X1 , . . . , Xn−1 ) = fi (Xi , Π(Xi )). i=0 M. Pelikan, K. Sastry, D.E. Goldberg, M.V. Butz, M. Hauschild NK Landscapes with Nearest Neighbors and Tunable Overlap
- 6. NK Landscape Exmaple for n = 9 and k = 2: M. Pelikan, K. Sastry, D.E. Goldberg, M.V. Butz, M. Hauschild NK Landscapes with Nearest Neighbors and Tunable Overlap
- 7. Restricted Random ADPs (rADPs) of Bounded Order Order-k rADPs with and without overlap Each subproblem contains k bits. Separable problems contain non-overlapping subproblems: Tight linkage: Shuﬄed: There may be overlap in o bits between neighboring subproblems (may also be shuﬄed): Tight linkage: Shuﬄed: M. Pelikan, K. Sastry, D.E. Goldberg, M.V. Butz, M. Hauschild NK Landscapes with Nearest Neighbors and Tunable Overlap
- 8. Properties of NK Landscapes and rADPs Common properties Additive decomposability. Subproblems are complex (look-up tables). High multimodality, complex structure. Overlap further increases problem diﬃculty. Challenge for most genetic algorithms and local search. NK landscapes NP-completeness (can’t solve worst case in polynomial time). rADPs Using prior knowledge of problem structure, we can exactly solve rADPs in polynomial time (dynamic programming) in O(2k n) evaluations. Multivariate EDAs can solve shuﬄed EDAs polynomially fast. M. Pelikan, K. Sastry, D.E. Goldberg, M.V. Butz, M. Hauschild NK Landscapes with Nearest Neighbors and Tunable Overlap
- 9. NK Landscapes with Nearest Neighbors & Tunable Overlap NK Landscapes with Nearest Neighbors and Tunable Overlap Neighbors of each bit are restricted to the following k bits. For simplicity, the neighborhoods don’t wrap around. Some subproblems may be excluded to provide a mechanism for tuning the size of overlap. Use parameter step ∈ {1, 2, . . . , k + 1}. Only subproblems at positions i, i mod step = 0 contribute. Bit positions shuﬄed randomly to eliminate tight linkage. M. Pelikan, K. Sastry, D.E. Goldberg, M.V. Butz, M. Hauschild NK Landscapes with Nearest Neighbors and Tunable Overlap
- 10. NK Landscapes with Nearest Neighbors & Tunable Overlap High overlap (k = 2, step = 1): Sequential Shuﬄed Note step = 1 maximizes the amount of overlap between subproblems. M. Pelikan, K. Sastry, D.E. Goldberg, M.V. Butz, M. Hauschild NK Landscapes with Nearest Neighbors and Tunable Overlap
- 11. NK Landscapes with Nearest Neighbors & Tunable Overlap Low overlap (k = 2, step = 2): Sequential Shuﬄed Note step parameter allows tuning of the size of overlap. M. Pelikan, K. Sastry, D.E. Goldberg, M.V. Butz, M. Hauschild NK Landscapes with Nearest Neighbors and Tunable Overlap
- 12. NK Landscapes with Nearest Neighbors & Tunable Overlap No overlap (k = 2, step = 3): Sequential Shuﬄed Note step = k + 1 implies separability (subproblems are independent). M. Pelikan, K. Sastry, D.E. Goldberg, M.V. Butz, M. Hauschild NK Landscapes with Nearest Neighbors and Tunable Overlap
- 13. NK Landscapes with Nearest Neighbors & Tunable Overlap Why? Nearest neighbors enable polynomial solvability Deshuﬄe the string. Use dynamic programming. Parameter step enables tunining the overlap between subproblems: For standard NK landscapes, step = 1. With larger values of step, the amount of overlap between consequent subproblems is reduced. For step = k + 1, the problem becomes separable (the subproblems are fully independent). M. Pelikan, K. Sastry, D.E. Goldberg, M.V. Butz, M. Hauschild NK Landscapes with Nearest Neighbors and Tunable Overlap
- 14. Problem Instances Parameters n = 20 to 120. k = 2 to 5. step = 1 to k + 1 for each k. Variety of instances For each (n, k, step), generate 10,000 random instances. Overall 1,800,000 unique problem instances. M. Pelikan, K. Sastry, D.E. Goldberg, M.V. Butz, M. Hauschild NK Landscapes with Nearest Neighbors and Tunable Overlap
- 15. Compared Algorithms Basic algorithms Hierarchical Bayesian optimization algorithm (hBOA). Genetic algorithm with uniform crossover (GAU). Genetic algorithm with twopoint crossover (G2P). Local search Single-bit-ﬂip hill climbing (DHC) on each solution. Improves performance of all methods. Niching Restricted tournament replacement (niching). M. Pelikan, K. Sastry, D.E. Goldberg, M.V. Butz, M. Hauschild NK Landscapes with Nearest Neighbors and Tunable Overlap
- 16. Num Results: Flips Until Optimum; hBOA; k = 2 and k = 5 2 10 20 40 60 80 100 20 40 60 80 100 Problem size Problem size 4 10 5 10 4 k=4, step=1 k=2, step=1 10 k=5, step=1 k=3, step=1 Number of flips (hBOA) Number of flips (hBOA) k=2, step=2 k=3, step=2 Number of flips (hBOA) k=4, step=2 k=5, step=2 k=4, step=3 k=2, step=3 k=5, step=3 k=3, step=3 k=4, step=4 4 k=5, step=4 k=3, step=4 k=4, step=5 10 k=5, step=5 3 10 k=5, step=6 3 10 3 10 2 2 10 10 20 4020 60 80 40 100 60 80 100 20 40 20 60 80 100 40 60 Problem size Problem size Problem size Problem size 5 10 k=4, step=1 k=5, step=1 4Growth appears to be polynomial w.r.t. problem size, n. umber of flips (hBOA) umber of flips (hBOA) 10 k=4, step=2 k=5, step=2 Performance best with no overlap. for hBOA. k=5, step=3 Figurestep=3 k=4, 1: Average number of ﬂips k=4, step=4 4 k=5, step=4 k=4, step=5 10 Besides n, performance depends on both k and step. step=5 k=5, k=5, step=6 the eﬀects of k on performance of all compared algorithms, ﬁgure 10 3 6 sh umber of DHC ﬂips with k for hBOA and GA on problems of size n = 3 M. are K. Sastry, D.E. Goldberg, M.V. Butz, M. Hauschild was incapable 10 solving many inst DA Pelikan,not included, because UMDA NK Landscapes with Nearest Neighbors and Tunable Overlap of
- 17. Results: Comparison w.r.t. Flips DHC steps (ﬂips) until optimum n k step hBOA GA (uniform) GA (twopoint) 120 5 1 37,155 141,108 220,318 120 5 2 40,151 212,635 353,748 120 5 3 37,480 249,217 443,570 120 5 4 27,411 195,673 310,894 120 5 5 15,589 100,378 145,406 120 5 6 9,607 35,101 47,576 M. Pelikan, K. Sastry, D.E. Goldberg, M.V. Butz, M. Hauschild NK Landscapes with Nearest Neighbors and Tunable Overlap
- 18. Results: Comparison w.r.t. Evaluations Number of evaluations until optimum n k step hBOA GA (uniform) GA (twopoint) 120 5 1 7,414 16,519 34,696 120 5 2 9,011 25,032 56,059 120 5 3 9,988 30,285 72,359 120 5 4 8,606 24,016 51,521 120 5 5 7,307 13,749 26,807 120 5 6 7,328 6,004 10,949 M. Pelikan, K. Sastry, D.E. Goldberg, M.V. Butz, M. Hauschild NK Landscapes with Nearest Neighbors and Tunable Overlap
- 19. 0.75 0.75 Number Num Number Nu Results: Flips Until Optimum; hBOA vs. GA; k = 5 0.5 0.5 20 40 20 60 80 40 100 60 80 100 20 40 60 80 100 Problem size Problem size Problem size k=4, step=1 7 7 k=5, step=1 Number of flips (GA, uniform) / k=5, step=1 6 k=4, step=2 6 k=5, step=2 Number of flips (GA, twopoint) / k=5, step=2 Number of flips (hBOA) k=4, step=3 5 5 k=5, step=3 k=5, step=3 Number of flips (hBOA) k=4, step=4 4 4 k=5, step=4 k=4, step=5 k=5, step=4 k=5, step=5 3 k=5, step=5 3 k=5, step=6 k=5, step=6 2 2 1 1 20 40 20 60 80 40 100 60 80 100 20 40 60 80 100 Problem size Problem size Problem size hBOA outperforms both versions of GA. rRatiowithDiﬀerences grow faster than with twopoint crossover and hBOA. GA of the number of ﬂips for GA polynomially with n. uniform crossover and hBOA. Besides n, diﬀerences depend on both k and step. f DHC ﬂips until optimum =GA and step ∈ GA6}; since UMDA was not capable of solving many o 5, (uniform) {1, (twopoint) s in141,108 time, the results for UMDA are not included. The ﬁgure sho practical 220,318 ofM.DHC K. Sastry, D.E. Goldberg, M.V. Butz, M. Hauschild NK Landscapes with Nearest Neighbors and Tunable Overlap sm Pelikan, ﬂips until optimum for diﬀerent percentages of instances with
- 20. Results: Correlations Between Algorithms step = 1 (high overlap): step = 6 (separable): GA versions more similar than hBOA with GA. Correlations stronger for problems with more overlap/less structure. M. Pelikan, K. Sastry, D.E. Goldberg, M.V. Butz, M. Hauschild NK Landscapes with Nearest Neighbors and Tunable Overlap
- 21. Problem Diﬃculty: Signal-to-Noise and Signal Variance Signal and noise Signal: The diﬀerence between ﬁtness of the best and the 2nd best solutions to a subproblem. Noise: Models contributions of other subproblems. Signal-to-noise ratio Decision making done by GA is stochastic. The larger the signal-to-noise ratio, the easier the decision making. Signal variance Sequential vs. parallel convergence. How much do contributions of diﬀerent subproblems diﬀer? One way to model this is to look at the variance of the signal. M. Pelikan, K. Sastry, D.E. Goldberg, M.V. Butz, M. Hauschild NK Landscapes with Nearest Neighbors and Tunable Overlap
- 22. hBOA (a) hBOA (b) GA (uniform) (uniform) (b) GA (c) GA (twopoint) ( (c) GA Results: Flips Until Optimum; hBOA vs. GA; k = 5 re 13:Figure 13: of overlap of overlap for n = 1205 and k = 5 (step varies with o Inﬂuence Inﬂuence for n = 120 and k = (step varies with overlap). step = 1 (high overlap) step = 6 (separable) 1.075 1.075 1.075 1.075 GA (twpoint) GA (twpoint) GA (twpoint) GA (tw GA (uniform) GA (uniform) GA (uniform) GA (u Average number of flips Average number of flips Average number of flips 1.05 1.05 hBOA hBOA 1.05 1.05 hBOA hBOA (divided by mean) (divided by mean) (divided by mean) (divided by mean) 1.025 1.025 1.025 1.025 1 1 1 1 0.975 0.975 0.975 0.975 0.95 0.95 0.95 0.95 0.1 0.2 0.3 0.4 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.5 0.6 0.7 0.8 0.9 1 Signal to noise percentilenoise percentile (% smallest) Signal to (% smallest) Signal to noise percentilenoise percentile (% s Signal to (% smallest) (a) step = 1(a) step = 1 (b) step = 6(b) step = 6 For separable problems, noise clearly matters. For problems with overlap, noise appears insigniﬁcant. :Figure 14: of signal-to-noise ratio on the number ofnumber of ﬂips for n = 120 Inﬂuence Inﬂuence of signal-to-noise ratio on the ﬂips for n = 120 and k = cknowledgments M.V. Butz, M. Hauschild edgments D.E. Goldberg, M. Pelikan, K. Sastry, NK Landscapes with Nearest Neighbors and Tunable Overlap
- 23. Results: Flips Until Optimum; hBOA vs. GA; k = 5 step = 1 (high overlap) step = 6 (separable) 1.1 1.1 1.1 1.1 GA (twopoint) GA (twopoint) GA (twopoint) (twopoi GA GA (uniform) GA (uniform) GA (uniform)GA (uniform Average number of flips 1.075 1.075 Average number of flips Average number of flips 1.075 1.075 hBOA hBOA hBOA hBOA (divided by mean) (divided by mean) (divided by mean) (divided by mean) 1.05 1.05 1.05 1.05 1.025 1.025 1.025 1.025 1 1 1 1 0.975 0.975 0.975 0.975 0.95 0.95 0.95 0.95 0.1 0.2 0.3 0.40.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.8 0.9 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Signal variance percentile (% smallest) (% smallest) Signal variance percentile Signal variance percentile (%percentile (% small Signal variance smallest) (a) step = (a) step = 1 1 (b) step = 6 step = 6 (b) For separable problems, signal variance clearly matters. For problems with overlap, signal variance appears e 15: Inﬂuence Inﬂuence of signal variance on the of ﬂips for n = 120 n = 120 and Figure 15: of signal variance on the number number of ﬂips for and k = 5. insigniﬁcant. eferences es M. Pelikan, K. Sastry, D.E. Goldberg, M.V. Butz, M. Hauschild NK Landscapes with Nearest Neighbors and Tunable Overlap
- 24. Conclusions and Future Work Summary and conclusions Considered subset of NK landscapes as class of random test problems with tunable subproblem size and overlap. All proposed instances solvable in polynomial time. Generated a broad range of problem instances. Analyzed results using hybrids of GEAs. Future work Use generated problems to test other algorithms. Relate performance to other measures of problem diﬃculty. Develop/test new tools for understanding of problem diﬃculty. Wrap subproblems around. Use other distributions for generating look-up tables. M. Pelikan, K. Sastry, D.E. Goldberg, M.V. Butz, M. Hauschild NK Landscapes with Nearest Neighbors and Tunable Overlap
- 25. Acknowledgments Acknowledgments NSF; NSF CAREER grant ECS-0547013. U.S. Air Force, AFOSR; FA9550-06-1-0096. University of Missouri; High Performance Computing Collaboratory sponsored by Information Technology Services; Research Award; Research Board. M. Pelikan, K. Sastry, D.E. Goldberg, M.V. Butz, M. Hauschild NK Landscapes with Nearest Neighbors and Tunable Overlap

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