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Effects of a Deterministic Hill climber on hBOA

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Hybridization of global and local search algorithms is a well-established technique for enhancing the efficiency of search algorithms. Hybridizing estimation of distribution algorithms (EDAs) has been …

Hybridization of global and local search algorithms is a well-established technique for enhancing the efficiency of search algorithms. Hybridizing estimation of distribution algorithms (EDAs) has been repeatedly shown to produce better performance than either the global or local search algorithm alone. The hierarchical Bayesian optimization algorithm (hBOA) is an advanced EDA which has previously been shown to benefit from hybridization with a local searcher. This paper examines the effects of combining hBOA with a deterministic hill climber (DHC). Experiments reveal that allowing DHC to find the local optima makes model building and decision making much easier for hBOA. This reduces the minimum population size required to find the global optimum, which substantially improves overall performance.

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  • 1. Effects of a Deterministic Hill Climber on hBOA Elizabeth Radetic, Martin Pelikan, and David E. Goldberg Missouri Estimation of Distribution Algorithms Laboratory (MEDAL) University of Missouri, St. Louis, MO http://medal.cs.umsl.edu/ err26c@umsl.edu pelikan@cs.umsl.edu Illinois Genetic Algorithms Laboratory University of Illinois at Urbana-Champaign, Urbana, IL http://www-illigal.ge.uiuc.edu/ deg@uiuc.edu Elizabeth Radetic, Martin Pelikan, and David E. Goldberg Effects of a Deterministic Hill Climber on hBOA
  • 2. Background & Motivation Hybrid Estimation of Distribution Algorithms (EDAs) For very large or complex problems, scalability may not be sufficient; additional efficiency enhancement may be required Hybridization is a popular method of efficiency enhancement for EDAs and other GEAs A hybrid combines a local search algorithm with a global searcher such as an EDA Hybrid hBOA The hierarchical Bayesian optimization algorithm (hBOA) is frequently hybridized This study focuses on hBOA combined with a deterministic hill climber (DHC) Examine the effects of varying the parameters of DHC Study the effects of variance reduction on model building Elizabeth Radetic, Martin Pelikan, and David E. Goldberg Effects of a Deterministic Hill Climber on hBOA
  • 3. Outline 1. Algorithms 2. Goals & Methodology 3. Test Problems 4. Results 5. Discussion of Results 6. Summary and conclusions. Elizabeth Radetic, Martin Pelikan, and David E. Goldberg Effects of a Deterministic Hill Climber on hBOA
  • 4. Algorithms Hierarchical Bayesian Optimization Algorithm (hBOA) Randomly initialize a population of binary strings Select promising solutions from the population Build a Bayesian network from the selected solutions Generate new solutions from the constructed model Incorporate new solutions into the population using restricted tournament replacement Elizabeth Radetic, Martin Pelikan, and David E. Goldberg Effects of a Deterministic Hill Climber on hBOA
  • 5. Algorithms Deterministic Hill Climber (DHC) Flip one bit in the string that will produce the most fitness improvement Repeat until a specified quality has been reached, a maximum number of flips has been performed, or no more improvement is possible hBOA has frequently been combined with DHC to improve performance (e.g., Pelikan & Goldberg, 2003; Pelikan & Hartmann, 2006; Pelikan, Katzgraber, & Kobe, 2008) Elizabeth Radetic, Martin Pelikan, and David E. Goldberg Effects of a Deterministic Hill Climber on hBOA
  • 6. Goals & Methodology Experiment Goals Examine the effects of DHC on performance of hBOA Determine optimal settings for DHC Methodology Tune the frequency of local search, i.e. the proportion of strings in the population on which DHC operates Tune the duration of local search, i.e. the maximum number of flips per string Compare Population size Number of generations until optimum Elizabeth Radetic, Martin Pelikan, and David E. Goldberg Effects of a Deterministic Hill Climber on hBOA
  • 7. Test Problems Test Problems Trap-5 2-D Ising Spin Glass 3-CNF MAXSAT Elizabeth Radetic, Martin Pelikan, and David E. Goldberg Effects of a Deterministic Hill Climber on hBOA
  • 8. Test Problems Trap-5 Concatenation of non-overlapping subproblems of 5 bits The fitness of each subproblem is determined as 5 if u = 5, ftrap5 (u) = , 4−u otherwise where u is the number of ones in the subproblem All subproblems are added together to determine the string’s fitness Elizabeth Radetic, Martin Pelikan, and David E. Goldberg Effects of a Deterministic Hill Climber on hBOA
  • 9. Test Problems Trap-5 Properties Separable, non-overlapping subproblems Subproblems are fully deceptive Difficult for local search, simple GA Parameters Problem sizes of 100-350 bits Population size determined empirically using bisection method 10 independent bisections of 10 runs each Elizabeth Radetic, Martin Pelikan, and David E. Goldberg Effects of a Deterministic Hill Climber on hBOA
  • 10. Test Problems 2-D Ising Spin Glass Spins are arranged on a 2-D grid Each spin si gets a value from +1, −1 Each edge between neighboring spins si and sj is assigned a real value Ji,j The task is to find a configuration of spins to minimize the energy, specified as n H(s) = − Jij si sj i,j=0 Spins are represented as bits in the binary string, 1 for +1, 0 for −1 Elizabeth Radetic, Martin Pelikan, and David E. Goldberg Effects of a Deterministic Hill Climber on hBOA
  • 11. Test Problems 2-D Ising Spin Glass Properties More complex fitness landscape than trap Not fully decomposable into subproblems of bounded order, substructures overlap Solvable in polynomial time Parameters Problem sizes 8x8, 10x10, 12x12, 14x14, 16x16 (64 - 256 bits) 1000 instances For each instance, Jij randomly set to +1 or −1 Population size determined using bisection method, 10 successful runs per instance Elizabeth Radetic, Martin Pelikan, and David E. Goldberg Effects of a Deterministic Hill Climber on hBOA
  • 12. Test Problems MAXSAT Instances consist of 3-CNF formulas: conjunctions of disjunctions of 3 literals The task is to find an assignment of Boolean variables to satisfy the maximum number of clauses Bits in the string represent the variables, with 1 representing true and 0 representing false Elizabeth Radetic, Martin Pelikan, and David E. Goldberg Effects of a Deterministic Hill Climber on hBOA
  • 13. Test Problems MAXSAT Properties Complex fitness landscape Not fully decomposable into subproblems of bounded order, substructures overlap Not solvable in polynomial time Parameters Problem sizes of 50 and 75 variables 100 randomly-generated instances Population size determined using bisection method, 10 successful runs per instance Elizabeth Radetic, Martin Pelikan, and David E. Goldberg Effects of a Deterministic Hill Climber on hBOA
  • 14. Experimental Results Trap-5 (350 bits) Reduction factor, speed-up: ratio of values without DHC to those with DHC 60 1.8 Population Size Reduction Factor 1.7 50 Generations Speed-Up 1.6 40 1.5 1.4 30 1.3 1.2 20 1.1 10 1 0.9 0 0.8 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Max. Flips (% of Prob. Size) Max. Flips (% of Prob. Size) pdhc pdhc 0.1 0.5 0.9 0.1 0.5 0.9 0.2 0.6 1 0.2 0.6 1 0.3 0.7 0.3 0.7 0.4 0.8 0.4 0.8 Elizabeth Radetic, Martin Pelikan, and David E. Goldberg Effects of a Deterministic Hill Climber on hBOA
  • 15. Experimental Results Trap-5 (350 bits) 100 7 90 6 Evaluations Speed-Up 80 Flips per Evaluation 70 5 60 4 50 40 3 30 2 20 1 10 0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Max. Flips (% of Prob. Size) Max. Flips (% of Prob. Size) pdhc pdhc 0.1 0.5 0.9 0.1 0.5 0.9 0.2 0.6 1 0.2 0.6 1 0.3 0.7 0.3 0.7 0.4 0.8 0.4 0.8 Elizabeth Radetic, Martin Pelikan, and David E. Goldberg Effects of a Deterministic Hill Climber on hBOA
  • 16. Experimental Results Trap-5 100000 100 No DHC No DHC With DHC With DHC Population Size Generations 10000 10 1000 100 1 100 150 200 250 300 350 100 150 200 250 300 350 Problem size (# of bits) Problem size (# of bits) Elizabeth Radetic, Martin Pelikan, and David E. Goldberg Effects of a Deterministic Hill Climber on hBOA
  • 17. Experimental Results Ising Spin Glass (16x16, 256 bits) 16 4.5 Population Reduction Factor 14 4 Generation Speed-up 12 3.5 10 3 8 2.5 6 2 4 2 1.5 0 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Max. Flips (% of Prob. Size) Max. Flips (% of Prob. Size) pdhc pdhc 0.1 0.5 0.9 0.1 0.5 0.9 0.2 0.6 1.0 0.2 0.6 1.0 0.3 0.7 0.3 0.7 0.4 0.8 0.4 0.8 Elizabeth Radetic, Martin Pelikan, and David E. Goldberg Effects of a Deterministic Hill Climber on hBOA
  • 18. Experimental Results Ising Spin Glass (16x16, 256 bits) 70 9 60 8 7 Evaluation Speedup Flips per Evaluation 50 6 40 5 30 4 3 20 2 10 1 0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Max. Flips (% of Prob. Size) Max. Flips (% of Prob. Size) pdhc pdhc 0.1 0.5 0.9 0.1 0.5 0.9 0.2 0.6 1.0 0.2 0.6 1.0 0.3 0.7 0.3 0.7 0.4 0.8 0.4 0.8 Elizabeth Radetic, Martin Pelikan, and David E. Goldberg Effects of a Deterministic Hill Climber on hBOA
  • 19. Experimental Results Ising Spin Glass 10000 100 No DHC No DHC With DHC With DHC Population Size Generations 1000 10 100 10 1 64 100 144 196 256 64 100 144 196 256 Problem size (# of bits) Problem size (# of bits) Elizabeth Radetic, Martin Pelikan, and David E. Goldberg Effects of a Deterministic Hill Climber on hBOA
  • 20. Experimental Results MAXSAT (75 bits) Population Reduction Factor (original/reduced) 120 2.8 2.6 100 2.4 Generation Speed-up 2.2 80 2 1.8 60 1.6 40 1.4 1.2 20 1 0.8 0 0.6 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Max. Flips (% of Prob. Size) Max. Flips (% of Prob. Size) pdhc pdhc 0.1 0.5 0.9 0.1 0.5 0.9 0.2 0.6 1.0 0.2 0.6 1.0 0.3 0.7 0.3 0.7 0.4 0.8 0.4 0.8 Elizabeth Radetic, Martin Pelikan, and David E. Goldberg Effects of a Deterministic Hill Climber on hBOA
  • 21. Experimental Results MAXSAT (75 bits) 300 6 250 5 Evaluation speedups Flips per Evaluation 200 4 150 3 100 2 50 1 0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Max. Flips (% of Prob. Size) Max. Flips (% of Prob. Size) pdhc pdhc 0.1 0.5 0.9 0.1 0.5 0.9 0.2 0.6 1.0 0.2 0.6 1.0 0.3 0.7 0.3 0.7 0.4 0.8 0.4 0.8 Elizabeth Radetic, Martin Pelikan, and David E. Goldberg Effects of a Deterministic Hill Climber on hBOA
  • 22. Discussion of Results Effects of DHC DHC improved performance on each test problem, for all sizes Substantial benefits were obtained despite the fact that DHC by itself performs poorly on each of these problems Benefits of DHC are mainly due to variance reduction Reduced variance makes model building easier Also makes decision making easier Elizabeth Radetic, Martin Pelikan, and David E. Goldberg Effects of a Deterministic Hill Climber on hBOA
  • 23. Discussion of Results Effects of DHC on Model Building in hBOA: Trap-5 Model quality Without DHC, the perfect model for trap-5 requires connections between each variable within the partition, 10 links in total With DHC, each partition contains either 11111 or 00000, so fewer dependencies (4) are required Elizabeth Radetic, Martin Pelikan, and David E. Goldberg Effects of a Deterministic Hill Climber on hBOA
  • 24. Discussion of Results Effects of DHC on Model Building in hBOA: Trap-5 (200 bits) Model quality: Proportion of correct dependencies Without DHC, much larger populations are required to obtain the 10 necessary dependencies With DHC, smaller populations are required to find the 4 necessary dependencies First generation Middle of the run proportion of correct dependencies proportion of correct dependencies 1 1 0.9 No DHC 0.9 No DHC, 10th generation 0.8 With DHC 0.8 With DHC, 5th generation 0.7 0.7 0.6 0.6 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 0 50 10 20 40 80 16 32 0 64 0 12 0 25 00 51 00 10 00 20 40 40 80 50 10 20 40 80 16 32 0 64 0 12 0 25 00 51 00 10 00 20 40 0 0 0 0 0 0 0 8 6 2 2 4 0 96 0 0 0 0 0 0 0 0 8 6 2 2 48 0 00 00 Population Size Population Size Elizabeth Radetic, Martin Pelikan, and David E. Goldberg Effects of a Deterministic Hill Climber on hBOA
  • 25. Discussion of Results Effects of DHC on Model Building in hBOA: Trap-5 (200 bits) Model quality: proportion of incorrect dependencies Without DHC, very many incorrect (spurious) dependencies are discovered for all but very large populations With DHC, very few spurious dependencies are discovered, except for very large populations First generation Middle of the run proportion of incorrect dependencies proportion of incorrect dependencies 0.035 0.04 No DHC 0.035 With DHC, 5th generation No DHC, 10th 0.03 With DHC generation 0.025 0.03 0.025 0.02 0.02 0.015 0.015 0.01 0.01 0.005 0.005 0 0 50 10 200 400 800 160 3200 6400 1200 25800 51600 10200 20240 40480 50 10 20 40 80 16 32 0 64 0 12 0 25 00 51 00 10 00 20 40 96 0 0 0 0 0 0 0 0 8 6 2 2 48 0 00 00 0 Population Size Population Size Elizabeth Radetic, Martin Pelikan, and David E. Goldberg Effects of a Deterministic Hill Climber on hBOA
  • 26. Summary & Conclusions DHC’s Effects on hBOA DHC significantly improves the performance of hBOA on each of the test problems In these experiments the maximum improvement was attained by allowing DHC to operate on the full population and run until it found the local optimum DHC helps by reducing the variance Model building is easier Decision making is also easier Elizabeth Radetic, Martin Pelikan, and David E. Goldberg Effects of a Deterministic Hill Climber on hBOA
  • 27. 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. Elizabeth Radetic, Martin Pelikan, and David E. Goldberg Effects of a Deterministic Hill Climber on hBOA