Performance of Evolutionary Algorithms on NK Landscapes with Nearest Neighbor Interactions and Tunable Overlap

Transcription

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 Download MEDAL Report No

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.

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.

4 Additively Decomposable Problems (ADPs) Additively decomposable problem (ADP) Fitness defined as m f(x 1, X 2,..., X n ) = f i (S i ), i=1 n is the number of bits (variables), m is the number of subproblems, S i 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).

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(X i ) for i-th bit, X i. Subfunction f i defining contribution of X i and Π(X i ). The objective function f nk to maximize is then defined as n 1 f nk (X 0, X 1,..., X n 1 ) = f i (X i, Π(X i )). i=0

6 NK Landscape Exmaple for n = 9 and k = 2:

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:

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(2 k n) evaluations. Multivariate EDAs can solve shuffled EDAs polynomially fast.

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.

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.

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.

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).

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).

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.

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).

16 Results: Flips Until Optimum; hboa; k = 2 and k = 5 Numb Problem size 10 4 k=4, step=1 k=2, step=1 k=4, step=2 k=2, step=2 k=4, step=3 k=2, step=3 k=4, step=4 k=4, step= Number of flips (hboa) Number of flips (hboa) Problem size k=5, step= k=5, step=2 k=5, step=3 k=5, step=4 k=5, step=5 k=5, step= Number of flips (hboa) k=3, step=1 k=3, step=2 k=3, step=3 k=3, step= Problem size Problem size umber of flips (hboa) k=4, step=1 k=4, step=2 k=4, step=3 k=4, step=4 k=4, step= Problem size Problem size umber of flips (hboa) k=5, step=1 k=5, step=2 k=5, step=3 k=5, step=4 k=5, step=5 k=5, step=6 Growth appears to be polynomial w.r.t. problem size, n Performance Figure 1: Average best with number no overlap. of flips for hboa. Besides n, performance depends on both k and step. the effects 10 3 of k on performance of all compared algorithms, figure 6 sh mber of DHC flips with k for hboa and GA on problems of size n = 10 A are not included, because UMDA was incapable of 3 M. Pelikan, K. Sastry, D.E. Goldberg, M.V. Butz, M. Hauschild NK Landscapes with Nearest Neighbors solving and Tunable many Overlap inst

17 Results: Comparison w.r.t. Flips DHC steps (flips) until optimum n k step hboa GA (uniform) GA (twopoint) , , , , , , , , , , , , , , , ,607 35,101 47,576

18 Results: Comparison w.r.t. Evaluations Number of evaluations until optimum n k step hboa GA (uniform) GA (twopoint) ,414 16,519 34, ,011 25,032 56, ,988 30,285 72, ,606 24,016 51, ,307 13,749 26, ,328 6,004 10,949

19 Number o Numb 0.75 Results: Flips Until Optimum; hboa vs. GA; k = Problem size Problem size k=4, step=1 k=4, step=2 k=4, step=3 k=4, step=4 k=4, step=5 Number of flips (GA, uniform) / Number of flips (hboa) k=5, step=1 k=5, step=2 k=5, step=3 k=5, step=4 k=5, step=5 k=5, step= Problem size Problem size Problem size Ratio GA with of the Differences uniform number crossover grow of flips faster for andthan GA hboa. with polynomially twopoint with crossover n. and hboa. Besides n, differences depend on both k and step. DHC flips until optimum GA 5, and (uniform) step GA {1,6}; (twopoint) since UMDA was not capable of solving many o in141,108 practical time, the 220,318 results for UMDA are not included. The figure sho fm. DHC Pelikan, K. flips Sastry, D.E. until Goldberg, optimum M.V. Butz, M. for Hauschild different NK Landscapes percentages with Nearest Neighbors of instances and Tunable Overlap with sm Number Num Number of flips (GA, twopoint) / Number of flips (hboa) 0.75 hboa outperforms both versions of GA k=5, step=1 k=5, step=2 k=5, step=3 k=5, step=4 k=5, step=5 k=5, step= Problem size

20 Results: Correlations Between Algorithms step = 1 (high overlap): step = 6 (separable): I I 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 2 nd 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.

22 hboa (a) hboa (b) GA (uniform) (b) GA (uniform) (c) GA (twopoint) (c) GA ( Results: Flips Until Optimum; hboa vs. GA; k = 5 e 13: Figure Influence 13: of Influence overlapof foroverlap n = 120 forandn = k 120 = 5 and (step k = varies 5 (step withvaries overlap). with o step = 1 (high overlap) step = 6 (separable) (divided by mean) Average number of flips (divided by mean) GA (twpoint) GA (uniform) hboa GA (twpoint) GA (uniform) hboa Signal to noise percentile Signal to noise (% smallest) percentile (% smallest) (a) step = 1(a) step = 1 Average number of flips (divided by mean) GA (tw GA (u hboa Signal to noise percentile Signal to noise (% smallest) percentile (% s (b) step = 6(b) step = 6 For problems with overlap, noise appears insignificant. Figure Influence 14: of Influence signal-to-noise of signal-to-noise ratio on the ratio number on theofnumber flips forof n flips = 120 forandn = k 120 = Average number of flips (divided by mean) For separable problems, noise clearly matters. GA (twpoint) GA (uniform) hboa nowledgments

23 Results: Flips Until Optimum; hboa vs. GA; k = 5 (divided by mean) Average number of flips (divided by mean) step = 1 (high overlap) GA (twopoint) GA (twopoint) GA (uniform) GA (uniform) hboa hboa Average number of flips (divided by mean) step = 6 (separable) Average number of flips (divided by mean) GA (twopoint) GA (twopoi GA (uniform) GA (uniform hboa hboa Signal variance Signal percentile variance (% smallest) percentile (% smallest) Signal variance Signal percentile variance (% percentile smallest) (% small (a) step = (a) 1 step = 1 (b) step = (b) 6 step = 6 For problems with overlap, signal variance appears 15: Figure Influence 15: insignificant. Influence of signal of variance signal variance on the number on the of number flips for of n flips = 120 for n and = 120 k = and 5. ferences s For separable problems, signal variance clearly matters.

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.

25 Acknowledgments Acknowledgments NSF; NSF CAREER grant ECS U.S. Air Force, AFOSR; FA University of Missouri; High Performance Computing Collaboratory sponsored by Information Technology Services; Research Award; Research Board.