Advanced Neighborhoods and Problem Difficulty Measures

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Advanced Neighborhoods and Problem Difficulty Measures

  1. 1. Motivation Outline NK Landscapes Algorithms Problem difficulty measures Experiments Conclusions Advanced Neighborhoods and Problem Difficulty Measures 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, 2011M. Hauschild and M. Pelikan University of Missouri - St. LouisAdvanced Neighborhoods and Problem Difficulty Measures
  2. 2. Motivation Outline NK Landscapes Algorithms Problem difficulty measures Experiments ConclusionsMotivation Important to understand problem difficulty. Identifying more difficult problems. Useful to help choose the algorithm. Many difficulty measures have been proposed. For advanced EAs, measures often do not correlate with actual difficulty. Why?M. Hauschild and M. Pelikan University of Missouri - St. LouisAdvanced Neighborhoods and Problem Difficulty Measures
  3. 3. Motivation Outline NK Landscapes Algorithms Problem difficulty measures Experiments ConclusionsMotivation Type of neighborhoods explored. Hamming distance is often used. Assumes bit flip neighborhood. For advanced EAs, hamming distance might not be a good measure of distance. Extend difficulty measures to nontrivial neighborhoods. Should more closely correlate with advanced EAs Test these new difficulty measures.M. Hauschild and M. Pelikan University of Missouri - St. LouisAdvanced Neighborhoods and Problem Difficulty Measures
  4. 4. Motivation Outline NK Landscapes Algorithms Problem difficulty measures Experiments ConclusionsOutline NK landscapes. Algorithms. GA. Ideal ECGA. hBOA. Deterministic Hill-Climber. Problem difficulty measures. Correlation length. Fitness distance correlation. Neighborhood structures Experiments Conclusions and future work.M. Hauschild and M. Pelikan University of Missouri - St. LouisAdvanced Neighborhoods and Problem Difficulty Measures
  5. 5. Motivation Outline NK Landscapes Algorithms Problem difficulty measures Experiments ConclusionsNK 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 )) (1) i=0M. Hauschild and M. Pelikan University of Missouri - St. LouisAdvanced Neighborhoods and Problem Difficulty Measures
  6. 6. Motivation Outline NK Landscapes Algorithms Problem difficulty measures Experiments ConclusionsNK 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. We examine step = k + 1, fully separable. Enforces strict neighborhoods Bit positions shuffled randomly to increase difficulty.M. Hauschild and M. Pelikan University of Missouri - St. LouisAdvanced Neighborhoods and Problem Difficulty Measures
  7. 7. Motivation Outline NK Landscapes Algorithms Problem difficulty measures Experiments ConclusionsGenetic Algorithm Binary tournament selection Uniform crossover Probability of crossover, pc = 0.6 Bit flip mutation Probability of mutation, pm = 1/nM. Hauschild and M. Pelikan University of Missouri - St. LouisAdvanced Neighborhoods and Problem Difficulty Measures
  8. 8. Motivation Outline NK Landscapes Algorithms Problem difficulty measures Experiments ConclusionsIdeal Extended Compact GA (ECGA) Harik, 1999. Treats groups of string positions as single variables. We consider ideal version of ECGA. Perfect model built from problem instance. Gives us idealized operator. Partitions contain k + 1 bits.M. Hauschild and M. Pelikan University of Missouri - St. LouisAdvanced Neighborhoods and Problem Difficulty Measures
  9. 9. Motivation Outline NK Landscapes Algorithms Problem difficulty measures Experiments Conclusionshierarchical 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 independenceM. Hauschild and M. Pelikan University of Missouri - St. LouisAdvanced Neighborhoods and Problem Difficulty Measures
  10. 10. Motivation Outline NK Landscapes Algorithms Problem difficulty measures Experiments ConclusionsDeterministic Hill-Climber Deterministic hill climber (DHC) used for all runs Performs single-bit changes that lead to maximum improvement. Stops when no single-bit change leads to improvement.M. Hauschild and M. Pelikan University of Missouri - St. LouisAdvanced Neighborhoods and Problem Difficulty Measures
  11. 11. Motivation Outline NK Landscapes Algorithms Problem difficulty measures Experiments ConclusionsFitness landscapes Components of a fitness landscape. A set S of admissible solutions. Fitness function f that assigns a value to each solution in S. Distance measure d defines distance between solutions in S. Distance measure d(x , y ) defines steps to get from x to y . This is what we want to look at. Hamming distance is standard distance measure Doesn’t correspond to advanced EAsM. Hauschild and M. Pelikan University of Missouri - St. LouisAdvanced Neighborhoods and Problem Difficulty Measures
  12. 12. Motivation Outline NK Landscapes Algorithms Problem difficulty measures Experiments ConclusionsCorrelation length Measures ruggedness of landscape. Quantifies strength of relationship between fitness values after taking s steps from a solution x. Simple functions Fitness changes gradually with distance. Difficult functions Fitness can change drastically.M. Hauschild and M. Pelikan University of Missouri - St. LouisAdvanced Neighborhoods and Problem Difficulty Measures
  13. 13. Motivation Outline NK Landscapes Algorithms Problem difficulty measures Experiments ConclusionsCorrelation length Random walk of m − 1 steps Distance measure Fitness values of {ft }t=1...m Random walk correlation function ρ(s) for gap s: m−s 1 ρ(s) = 2 (m − s) (ft − ¯)(ft+s − ¯), f f (2) σF t=1 Correlation length: 1 l =− , (3) ln(|ρ(1)|) Smaller correlation length, harder the problem.M. Hauschild and M. Pelikan University of Missouri - St. LouisAdvanced Neighborhoods and Problem Difficulty Measures
  14. 14. Motivation Outline NK Landscapes Algorithms Problem difficulty measures Experiments ConclusionsFitness distance correlation (FDC) Consider a set of solutions and their distance from a global optimum. FDC quantifies strength of relationship between fitness and distance to nearest global optimum. Simple functions Higher fitness values should be closer to optimum Going to optimum directly should be easy Difficult functions Solutions close to optimum might have low fitness Not easy to go directly to optimumM. Hauschild and M. Pelikan University of Missouri - St. LouisAdvanced Neighborhoods and Problem Difficulty Measures
  15. 15. Motivation Outline NK Landscapes Algorithms Problem difficulty measures Experiments ConclusionsFitness distance correlation (FDC) Consider n solutions. Fitness values: F = {f1 , f2 , . . . , fn }. Distances from optimum: D = {d1 , d2 , ..., dn } Important statistics Standard deviation of F ,D is σF , σD Covariance of F and D is: n 1 cFD = (fi − ¯)(di − d ), f ¯ (4) n i=1 Fitness distance correlation: cFD τ= (5) σF σD FDC takes values from [-1, 1]. Smaller the values of τ , easier the maximization problem. Consider only local optima due to DHC.M. Hauschild and M. Pelikan University of Missouri - St. LouisAdvanced Neighborhoods and Problem Difficulty Measures
  16. 16. Motivation Outline NK Landscapes Algorithms Problem difficulty measures Experiments ConclusionsAdvanced Neighborhoods How do we determine if solutions are close to each other? The neighborhood around each solution. Correlation length Bit flip neighborhood Could be misleading for advanced EAs Consider two additional neighborhood operators Fixed partition Random partitionM. Hauschild and M. Pelikan University of Missouri - St. LouisAdvanced Neighborhoods and Problem Difficulty Measures
  17. 17. Motivation Outline NK Landscapes Algorithms Problem difficulty measures Experiments ConclusionsAdvanced Neighborhoods Fixed partition Solutions that can be reached by all possible permutations of strongly linked bits. For separable NK landscapes, neighborhood is k + 1 bits in size. k bits that are all nearest neighbors to the same bit. Random partition Solutions reachable by permuting x random bits. For this work, x = k + 1 To compare against tightly correlated neighborhoods.M. Hauschild and M. Pelikan University of Missouri - St. LouisAdvanced Neighborhoods and Problem Difficulty Measures
  18. 18. Motivation Outline NK Landscapes Algorithms Problem difficulty measures Experiments ConclusionsAdvanced Neighborhoods Fitness distance correlation Hamming distance Could be misleading for advanced EAs Partition distance Measures distance by amount of differing partitions in strings. Total distance is sum of number of different partitions they have. Partitions considered as a single variable.M. Hauschild and M. Pelikan University of Missouri - St. LouisAdvanced Neighborhoods and Problem Difficulty Measures
  19. 19. Motivation Outline NK Landscapes Algorithms Problem difficulty measures Experiments ConclusionsExperimental Setup NK landscapes Problem size of n = 120. Two step sizes considered, step ∈ {4, 6}. Corresponding k were considered, k ∈ {3, 5}. 1000 instances for each combination of n and step. DHC used on all solutions during evaluations. Difficulty of instances ranked by number of DHC flips.M. Hauschild and M. Pelikan University of Missouri - St. LouisAdvanced Neighborhoods and Problem Difficulty Measures
  20. 20. Motivation Outline NK Landscapes Algorithms Problem difficulty measures Experiments ConclusionsExperimental Setup Correlation length For each instance, 100 random walks of 1000 steps each Repeated for all 3 neighborhood step operators Used to estimate correlation length Fitness distance correlation For each instance, 100 samples of 1000 solutions each Repeated for both neighborhood operators Used to estimate fitness distance correlationM. Hauschild and M. Pelikan University of Missouri - St. LouisAdvanced Neighborhoods and Problem Difficulty Measures
  21. 21. Motivation Outline NK Landscapes Algorithms Problem difficulty measures Experiments ConclusionsCorrelation length results NK instances solved with GA of n = 120, k = 3 desc. of DHC steps for correlation length correlation length correlation length instances GA using bit flip using random partition using set partition 10% easiest 3130.5 27.722(1) 13.882(1) 29.415(5) 25% easiest 3577.6 27.668(3) 13.851(3) 29.424(1) 50% easiest 4165 27.674(2) 13.855(2) 29.420(3) all instances 5655.3 27.603(4) 13.818(4) 29.417(4) 50% hardest 7145.6 27.531(5) 13.781(5) 29.413(6) 25% hardest 8545.6 27.519(6) 13.770(6) 29.407(7) 10% hardest 10556 27.479(7) 13.740(7) 29.422(2) Bit flip and random partition show strong relationship. Set partition seems to have no relation to actual difficulty.M. Hauschild and M. Pelikan University of Missouri - St. LouisAdvanced Neighborhoods and Problem Difficulty Measures
  22. 22. Motivation Outline NK Landscapes Algorithms Problem difficulty measures Experiments ConclusionsCorrelation length results NK instances solved with hBOA of n = 120, k = 3 desc. of DHC steps for correlation length correlation length correlation length instances hBOA using bit flip using random partition using set partition 10% easiest 4371.6 27.817(1) 13.922(1) 29.417(4) 25% easiest 4645.4 27.719(2) 13.874(2) 29.419(2) 50% easiest 4890.6 27.685(3) 13.855(3) 29.414(6) all instances 5540.4 27.603(4) 13.818(4) 29.417(5) 50% hardest 6190.3 27.520(6) 13.781(6) 29.419(1) 25% hardest 6798 27.504(7) 13.777(7) 29.418(3) 10% hardest 7828.5 27.556(5) 13.800(5) 29.412(7) Bit flip and random partition show weak relationship. Set partition seems to have no relation to actual difficulty.M. Hauschild and M. Pelikan University of Missouri - St. LouisAdvanced Neighborhoods and Problem Difficulty Measures
  23. 23. Motivation Outline NK Landscapes Algorithms Problem difficulty measures Experiments ConclusionsCorrelation length results NK instances solved with ECGAperfect of n = 120, k = 3 desc. of DHC steps for correlation length correlation length correlation length instances ECGAperfect using bit flip using random partition using set partition 10% easiest 1487.8 27.581(4) 13.804(5) 29.427(1) 25% easiest 1569.9 27.556(7) 13.794(7) 29.426(2) 50% easiest 1669.3 27.581(5) 13.805(4) 29.426(3) all instances 1873 27.603(3) 13.818(3) 29.417(5) 50% hardest 2076.7 27.624(2) 13.831(2) 29.407(7) 25% hardest 2201.5 27.577(6) 13.803(6) 29.411(6) 10% hardest 2316.3 27.745(1) 13.875(1) 29.421(4) Bit flip and random partition show no relation to actual difficulty. Set partition is no better.M. Hauschild and M. Pelikan University of Missouri - St. LouisAdvanced Neighborhoods and Problem Difficulty Measures
  24. 24. Motivation Outline NK Landscapes Algorithms Problem difficulty measures Experiments ConclusionsCorrelation length results NK instances solved with GA of n = 120, k = 5 desc. of DHC steps for correlation length correlation length correlation length instances GA using bit flip using random partition using set partition 10% easiest 13328 19.139(2) 6.399(1) 19.435(7) 25% easiest 16877 19.134(4) 6.395(2) 19.448(1) 50% easiest 21211 19.128(5) 6.395(3) 19.446(5) all instances 33457 19.137(3) 6.395(4) 19.447(4) 50% hardest 45703 19.145(1) 6.394(5) 19.448(2) 25% hardest 56967 19.124(6) 6.387(6) 19.447(3) 10% hardest 71993 19.073(7) 6.373(7) 19.438(6) Bit flip and set partition show no relationship. Random partition correctly ranks the instances.M. Hauschild and M. Pelikan University of Missouri - St. LouisAdvanced Neighborhoods and Problem Difficulty Measures
  25. 25. Motivation Outline NK Landscapes Algorithms Problem difficulty measures Experiments ConclusionsCorrelation length results NK instances solved with hBOA of n = 120, k = 5 desc. of DHC steps for correlation length correlation length correlation length instances hBOA using bit flip using random partition using set partition 10% easiest 7482.9 19.207(1) 6.416(1) 19.444(5) 25% easiest 7945.3 19.203(2) 6.414(2) 19.440(7) 50% easiest 8399.8 19.179(3) 6.408(3) 19.447(1) all instances 9311.6 19.137(4) 6.395(4) 19.447(2) 50% hardest 10223 19.095(6) 6.381(6) 19.447(3) 25% hardest 10900 19.066(7) 6.374(7) 19.444(4) 10% hardest 11806 19.119(5) 6.390(5) 19.441(6) Bit flip and random partition show a weak relationship. Set partition is ineffective.M. Hauschild and M. Pelikan University of Missouri - St. LouisAdvanced Neighborhoods and Problem Difficulty Measures
  26. 26. Motivation Outline NK Landscapes Algorithms Problem difficulty measures Experiments ConclusionsCorrelation length results NK instances solved with ECGAperfect of n = 120, k = 5 desc. of DHC steps for correlation length correlation length correlation length instances ECGAperfect using bit flip using random partition using set partition 10% easiest 2128.3 19.157(2) 6.396(3) 19.437(7) 25% easiest 2273.6 19.129(5) 6.390(5) 19.443(6) 50% easiest 2428.9 19.112(7) 6.386(6) 19.445(5) all instances 2770.3 19.137(4) 6.395(4) 19.447(4) 50% hardest 3111.7 19.162(1) 6.403(1) 19.449(3) 25% hardest 3415.5 19.153(3) 6.399(2) 19.452(2) 10% hardest 3869.8 19.121(6) 6.383(7) 19.452(1) Bit flip and random partition are ineffective. Set partition shows an inverse relationship.M. Hauschild and M. Pelikan University of Missouri - St. LouisAdvanced Neighborhoods and Problem Difficulty Measures
  27. 27. Motivation Outline NK Landscapes Algorithms Problem difficulty measures Experiments ConclusionsFitness distance correlation results NK instances solved with GA of n = 120, k = 3 desc. of DHC for FDC FDC instances GA bit distance partition 10% easiest 3130.5 -0.65915(7) -0.67115(6) 25% easiest 3577.6 -0.65378(6) -0.67190(7) 50% easiest 4165 -0.64500(5) -0.66583(5) all instances 5655.3 -0.62917(4) -0.65771(4) 50% hardest 7145.6 -0.61334(3) -0.64959(3) 25% hardest 8545.6 -0.60603(2) -0.64778(2) 10% hardest 10556 -0.59750(1) -0.64301(1) Bit distance correctly ranks instances. Partition distance also shows strong relationship.M. Hauschild and M. Pelikan University of Missouri - St. LouisAdvanced Neighborhoods and Problem Difficulty Measures
  28. 28. Motivation Outline NK Landscapes Algorithms Problem difficulty measures Experiments ConclusionsFitness distance correlation results NK instances solved with hBOA of n = 120, k = 3 desc. of DHC for FDC FDC instances hBOA bit distance partition 10% easiest 4371.6 -0.65842(7) -0.68031(7) 25% easiest 4645.4 -0.64346(6) -0.67007(6) 50% easiest 4890.6 -0.63894(5) -0.66447(5) all instances 5540.4 -0.62917(4) -0.65771(4) 50% hardest 6190.3 -0.61940(3) -0.65095(3) 25% hardest 6798 -0.61141(2) -0.64588(2) 10% hardest 7828.5 -0.60218(1) -0.64098(1) Bit distance correctly ranks instances. Partition distance correctly ranks instances.M. Hauschild and M. Pelikan University of Missouri - St. LouisAdvanced Neighborhoods and Problem Difficulty Measures
  29. 29. Motivation Outline NK Landscapes Algorithms Problem difficulty measures Experiments ConclusionsFitness distance correlation results NK instances solved with ECGAperfect of n = 120, k = 3 desc. of DHC for FDC FDC instances ECGAp bit distance partition 10% easiest 1487.8 -0.64460(7) -0.67060(7) 25% easiest 1569.9 -0.63868(6) -0.66548(6) 50% easiest 1669.3 -0.63145(5) -0.66049(5) all instances 1873 -0.62917(3) -0.65771(3) 50% hardest 2076.7 -0.62689(2) -0.65493(2) 25% hardest 2201.5 -0.62623(1) -0.65466(1) 10% hardest 2316.3 -0.63102(4) -0.65841(4) Both unable to differentiate between the harder instances.M. Hauschild and M. Pelikan University of Missouri - St. LouisAdvanced Neighborhoods and Problem Difficulty Measures
  30. 30. Motivation Outline NK Landscapes Algorithms Problem difficulty measures Experiments ConclusionsFitness distance correlation results NK instances solved with GA of n = 120, k = 5 desc. of DHC for FDC FDC instances GA bit dist. partition 10% easiest 13328 -0.40403(7) -0.44834(7) 25% easiest 16877 -0.39321(6) -0.44414(6) 50% easiest 21211 -0.38170(5) -0.44055(5) all instances 33457 -0.36075(4) -0.43371(4) 50% hardest 45703 -0.33981(3) -0.42687(3) 25% hardest 56967 -0.32773(2) -0.42079(2) 10% hardest 71993 -0.30975(1) -0.41416(1) FDC with both distance measures able to correctly rank all instances.M. Hauschild and M. Pelikan University of Missouri - St. LouisAdvanced Neighborhoods and Problem Difficulty Measures
  31. 31. Motivation Outline NK Landscapes Algorithms Problem difficulty measures Experiments ConclusionsFitness distance correlation results NK instances solved with hBOA of n = 120, k = 5 desc. of DHC for FDC FDC instances hBOA bit distance partition 10% easiest 7482.9 -0.37490(7) -0.45194(7) 25% easiest 7945.3 -0.37354(6) -0.44697(6) 50% easiest 8399.8 -0.36649(5) -0.43962(5) all instances 9311.6 -0.36075(4) -0.43371(4) 50% hardest 10223 -0.35501(3) -0.42780(3) 25% hardest 10900 -0.35022(2) -0.42226(2) 10% hardest 11806 -0.34699(1) -0.41737(1) FDC with both distance measures able to correctly rank all instances.M. Hauschild and M. Pelikan University of Missouri - St. LouisAdvanced Neighborhoods and Problem Difficulty Measures
  32. 32. Motivation Outline NK Landscapes Algorithms Problem difficulty measures Experiments ConclusionsFitness distance correlation results NK instances solved with ECGAperfect of n = 120, k = 5 desc. of DHC for FDC FDC instances ECGAp bit distance partition 10% easiest 2128.3 -0.36917(7) -0.443970(7) 25% easiest 2273.6 -0.36651(6) -0.438740(6) 50% easiest 2428.9 -0.36419(5) -0.435860(5) all instances 2770.3 -0.36075(4) -0.433710(4) 50% hardest 3111.7 -0.35731(3) -0.431560(3) 25% hardest 3415.5 -0.35088(2) -0.426320(2) 10% hardest 3869.8 -0.34526(1) -0.421470(1) FDC with both distance measures able to correctly rank all instances.M. Hauschild and M. Pelikan University of Missouri - St. LouisAdvanced Neighborhoods and Problem Difficulty Measures
  33. 33. Motivation Outline NK Landscapes Algorithms Problem difficulty measures Experiments ConclusionsConclusions Modified difficulty measures. Take into account more complex neighborhoods. Different difficulty measures used on NK landscapes. Compared to actual computation requirements. GA, ECGAperfect , hBOA. Correlation length. Hamming distance initially worked for GA, hBOA. Correlation weaker as problems got harder. Advanced neighborhoods did not improve results.M. Hauschild and M. Pelikan University of Missouri - St. LouisAdvanced Neighborhoods and Problem Difficulty Measures
  34. 34. Motivation Outline NK Landscapes Algorithms Problem difficulty measures Experiments ConclusionsConclusions Fitness distance correlation. Noisy results for easier problems. Was able to rank difficulty on harder problems. Advanced neighborhoods did not improve results. Future Work Look at additional measures of problem difficulty. Signal-to-noise ratio. Scaling. Fluctuating crosstalk. Develop new neighborhood operators.M. Hauschild and M. Pelikan University of Missouri - St. LouisAdvanced Neighborhoods and Problem Difficulty Measures
  35. 35. Motivation Outline NK Landscapes Algorithms Problem difficulty measures Experiments Conclusions Any Questions?M. Hauschild and M. Pelikan University of Missouri - St. LouisAdvanced Neighborhoods and Problem Difficulty Measures

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