Your SlideShare is downloading. ×
0
Analysis of Evolutionary Algorithms on the
               One-Dimensional Spin Glass with Power-Law
                      ...
Motivation

       Testing evolutionary algorithms
               Adversarial problems on the boundary of design envelope....
Outline



          1. Sherrington-Kirkpatrick (SK) spin glass.

          2. Power-law interactions.

          3. Probl...
SK Spin Glass


       SK spin glass (Sherrington & Kirkpatrick, 1978)

            Contains n spins s1 , s2 , . . . , sn ...
Ground States of SK Spin Glasses

       Energy
               Energy of a spin configuration C is given by

              ...
Random Spin Glass Instances

       Generating random spin glass instances
               Generate couplings {Ji,j } using...
Power-Law Interactions

       Power-law interactions
           Spins arranged on a circle.
           Couplings generate...
Power-Law Interactions: Illustration


       Example for n = 10 (normalized)


                           Distance on    ...
Problem Instances
       Parameters
               n = 20 to 150.
               σ ∈ {0.00, 0.55, 0.75, 0.83, 1.00, 1.50, ...
Compared Algorithms

       Basic algorithms
               Hierarchical Bayesian optimization algorithm (hBOA).
         ...
Experimental Setup


       All algorithms
               Bisection determines adequate population size for each
         ...
Results: Evaluations until Optimum




                                                                                   ...
10  2                                                                                   2                                 ...
Analysis of Evolutionary Algorithms on the One-Dimensional Spin Glass with Power-Law Interactions
Analysis of Evolutionary Algorithms on the One-Dimensional Spin Glass with Power-Law Interactions
Analysis of Evolutionary Algorithms on the One-Dimensional Spin Glass with Power-Law Interactions
Analysis of Evolutionary Algorithms on the One-Dimensional Spin Glass with Power-Law Interactions
Analysis of Evolutionary Algorithms on the One-Dimensional Spin Glass with Power-Law Interactions
Analysis of Evolutionary Algorithms on the One-Dimensional Spin Glass with Power-Law Interactions
Upcoming SlideShare
Loading in...5
×

Analysis of Evolutionary Algorithms on the One-Dimensional Spin Glass with Power-Law Interactions

595

Published on

Published in: Technology, Business
0 Comments
0 Likes
Statistics
Notes
  • Be the first to comment

  • Be the first to like this

No Downloads
Views
Total Views
595
On Slideshare
0
From Embeds
0
Number of Embeds
0
Actions
Shares
0
Downloads
5
Comments
0
Likes
0
Embeds 0
No embeds

No notes for slide

Transcript of "Analysis of Evolutionary Algorithms on the One-Dimensional Spin Glass with Power-Law Interactions"

  1. 1. Analysis of Evolutionary Algorithms on the One-Dimensional Spin Glass with Power-Law Interactions Martin Pelikan and Helmut G. Katzgraber 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. 2009004 http://medal.cs.umsl.edu/files/2009004.pdf Martin Pelikan and Helmut G. Katzgraber Analysis of EAs on 1D Spin Glass with Power-Law Interactions
  2. 2. Motivation Testing evolutionary algorithms Adversarial problems on the boundary of design envelope. Random instances of important classes of problems. Real-world problems. This study Use one-dimensional spin glass with power-law interactions. This allows the user to tune the effective range of interactions. Short-range to long-range interactions. Generate large number of instances of proposed problem class. Solve all instances with branch and bound and hybrids. Test evolutionary algorithms on the generated instances. Analyze the results. Martin Pelikan and Helmut G. Katzgraber Analysis of EAs on 1D Spin Glass with Power-Law Interactions
  3. 3. Outline 1. Sherrington-Kirkpatrick (SK) spin glass. 2. Power-law interactions. 3. Problem instances. 4. Experiments. 5. Conclusions and future work. Martin Pelikan and Helmut G. Katzgraber Analysis of EAs on 1D Spin Glass with Power-Law Interactions
  4. 4. SK Spin Glass SK spin glass (Sherrington & Kirkpatrick, 1978) Contains n spins s1 , s2 , . . . , sn . Ising spin can be in two states: +1 or −1. All pairs of spins interact. Interaction of spins si and sj specified by real-valued coupling Ji,j . Spin glass instance is defined by set of couplings {Ji,j }. Spin configuration is defined by the values of spins {si }. Martin Pelikan and Helmut G. Katzgraber Analysis of EAs on 1D Spin Glass with Power-Law Interactions
  5. 5. Ground States of SK Spin Glasses Energy Energy of a spin configuration C is given by H(C) = − Ji,j si sj i<j Ground states are spin configurations that minimize energy. Finding ground states of SK instances is NP-complete. Compare with other standard spin glass types 2D: Spin interacts with only 4 neighbors in 2D lattice. 3D: Spin interacts with only 6 neighbors in 3D lattice. SK: Spin interacts with all other spins. 2D is polynomially solvable; 3D and SK are NP-complete. Martin Pelikan and Helmut G. Katzgraber Analysis of EAs on 1D Spin Glass with Power-Law Interactions
  6. 6. Random Spin Glass Instances Generating random spin glass instances Generate couplings {Ji,j } using a specific distribution. Study the properties of generated spin glasses. Example study Find ground states and analyze their properties. Example coupling distributions Each coupling is generated from N (0, 1). Each coupling is +1 or -1 with equal probability. Each coupling is generated from a power-law distribution. Martin Pelikan and Helmut G. Katzgraber Analysis of EAs on 1D Spin Glass with Power-Law Interactions
  7. 7. Power-Law Interactions Power-law interactions Spins arranged on a circle. Couplings generated according to i,j Ji,j = c(σ) σ , ri,j i,j are generated according to N (0, 1), c(σ) is a normalization constant, σ > 0 is a parameter to control effective range of interactions, ri,j = n sin(π|i − j|/n)/π is geometric Figure 1: One-dimensional spin glass of size n = 10 ar distance between si and sj Magnitude ofwhere ǫi,j are generated decreases with their distance. zero spin-spin couplings according to normal distribution with Effects of distance on magnitude of couplings increase withparameter t is a normalization constant, σ > 0 is the user-specified σ. interactions, and ri,j = n sin(π|i − j|/n)/π denotes the geometric d figure 1). The magnitude of spin-spin couplings decreases with th discussed shortly, the effects EAsdistance on the magnitude of coupli Martin Pelikan and Helmut G. Katzgraber Analysis of of on 1D Spin Glass with Power-Law Interactions
  8. 8. Power-Law Interactions: Illustration Example for n = 10 (normalized) Distance on Coupling variance circle σ = 0.0 σ = 0.5 σ = 2.0 1 1.00 1.00 1.00 2 1.00 0.73 0.28 3 1.00 0.62 0.15 4 1.00 0.57 0.11 5 1.00 0.56 0.10 Martin Pelikan and Helmut G. Katzgraber Analysis of EAs on 1D Spin Glass with Power-Law Interactions
  9. 9. Problem Instances Parameters n = 20 to 150. σ ∈ {0.00, 0.55, 0.75, 0.83, 1.00, 1.50, 2.00}. σ = 0 denotes standard SK spin glass with N(0,1) couplings. σ = 2 enforces short-range interactions. Variety of instances For each n and σ, generate 10,000 random instances. Overall 610,000 unique problem instances. Finding optima Small instances solved using branch and bound. For large instances, use heuristic methods to find reliable (but not guaranteed) optima. Martin Pelikan and Helmut G. Katzgraber Analysis of EAs on 1D Spin Glass with Power-Law Interactions
  10. 10. 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). Martin Pelikan and Helmut G. Katzgraber Analysis of EAs on 1D Spin Glass with Power-Law Interactions
  11. 11. Experimental Setup All algorithms Bisection determines adequate population size for each instance. Ensure 10 successful runs out of 10 independent runs. In RTR, use window size w = min{N/20, n}. GA Probability of crossover, pc = 0.6. Probability of bit-flip in mutation, pm = 1/n. Martin Pelikan and Helmut G. Katzgraber Analysis of EAs on 1D Spin Glass with Power-Law Interactions
  12. 12. Results: Evaluations until Optimum Number of evaluations (GA, twopoint) Number of evaluations (GA, twopoint) Number of evaluations (GA, twopoint) Number of evaluations (GA, twopoint) Number of evaluations (GA, twopoint) Number of evaluations (GA, twopoint) 5 5 5 10 10 10 σ=2.00 10 10 10 σ=2.00 10 10 10 σ=2.00 5 5 5 σ=2.00 σ=2.00 σ=2.00 Number of evaluations (hBOA) 5 5 5 σ=2.00 σ=2.00 σ=2.00 Number of evaluations (hBOA) Number of evaluations (hBOA) σ=1.50 σ=1.50σ=1.50 σ=1.50 σ=1.50 σ=1.50 σ=1.50 σ=1.50 σ=1.50 4 σ=1.00 4 4 σ=1.00 σ=1.00 4 σ=1.00 4 4 σ=1.00σ=1.00 4 4 σ=1.00 σ=1.00 4 σ=1.00 10 10 10 10 10 10 10 10 10 σ=0.83 σ=0.83σ=0.83 σ=0.83 σ=0.83 σ=0.83 σ=0.83 σ=0.83 σ=0.83 σ=0.75 σ=0.75σ=0.75 σ=0.75 σ=0.75 σ=0.75 σ=0.75 σ=0.75 σ=0.75 3 3 3 3 3 3 3 3 3 10 10 10 σ=0.55 σ=0.55σ=0.55 10 10 10 σ=0.55 σ=0.55 σ=0.55 10 10 10 σ=0.55 σ=0.55 σ=0.55 σ=0.00 σ=0.00σ=0.00 σ=0.00 σ=0.00 σ=0.00 σ=0.00 σ=0.00 σ=0.00 2 2 2 2 2 2 2 2 2 10 10 10 10 10 10 10 10 10 1 1 1 1 1 1 1 1 1 10 10 10 10 10 10 10 10 10 16 16 16 32 32 32 64 64 64 128 128 128 16 16 16 32 32 32 64 64 64 128 128 128 16 16 16 32 32 32 64 64 64 128128 128 Problem size Problem size size Problem Problem size Problem size size Problem Problem sizesize Problem Problem size (a) hBOA (a)(a) hBOA hBOA (b) GA (twopoint) (b) GA (twopoint) (b) GA (twopoint) (c)(c) GA (uniform) GA (uniform) (c) GA (uniform) Scalability of hBOA and GA with twopoint crossover better forFigure 2:2: 2: Growththe the numberevaluations withwith problem size. short-range interactions. of of evaluations problem size. Figure Growth ofof of number of evaluations with problem size. Figure Growth the number 6 6 σ=2.00 σ=2.00 10 10 10 σ=2.00 Linkage tightens 10 σ=2.00grows. as σ 10 10 σ=2.00 σ=2.00 σ=2.00 σ=2.00 10 10 10 σ=2.00 Number of flips (GA, twopoint) Number of flips (GA, twopoint) Number of flips (GA, twopoint) Number of flips (GA, twopoint) Number of flips (GA, twopoint) Number of flips (GA, twopoint) 6 6 6 6 6 6 6 σ=1.50σ=1.50 σ=1.50 σ=1.50σ=1.50 σ=1.50 σ=1.50 σ=1.50 σ=1.50 Number of flips (hBOA) Number of flips (hBOA) Tighter linkage makes problem easier (if good recombination). Number of flips (hBOA) 5 5 σ=1.00 σ=1.00 5 σ=1.00 σ=1.00σ=1.00 σ=1.00 5 σ=1.00 σ=1.00 σ=1.00 5 5 5 5 5 10 10 10 10 10 10 10 10 10 σ=0.83σ=0.83 σ=0.83 σ=0.83 4 4 4 σ=0.83 σ=0.75σ=0.75 σ=0.75 Twopoint crossoverσ=0.75 respects tight linkage. σ=0.75σ=0.83 σ=0.75 σ=0.83 σ=0.83 σ=0.75 4 σ=0.83 4 σ=0.75 σ=0.75 4 4 4 4 σ=0.55σ=0.55 10 10 10 σ=0.55 σ=0.55 σ=0.55 10 10 10 σ=0.55 σ=0.55 σ=0.55 10 10 10 σ=0.55 3 σ=0.00 10 103 10 3 σ=0.00 σ=0.00 GA with uniform 10 10 10σ=0.00σ=0.00 with shorter-range σ=0.00 gets σ=0.00 worse 3 σ=0.00 10 10 10 3 interactions. σ=0.00 3 3 3 3 2 2 2 2 2 2 2 2 2 10 10 10 10 10 10 10 10 10 16 16 16 32 32 32 64 64 64 128 128 128 16 16 16 32 32 32 64 64 64 128 128 128 16 16 16 32 32 32 64 64 64 128128 128 Problem size size Problem Problem size Problem size size Problem Problem size Problem sizesize Problem size Problem (a) hBOA (a)(a) hBOA hBOA (b) GA (twopoint) (b) GA (twopoint) (b) GA (twopoint) (c)(c) GA (uniform) GA (uniform) (c) GA (uniform) Martin Pelikan and Helmut G. Katzgraber Analysis of EAs on 1D Spin Glass with Power-Law Interactions
  13. 13. 10 2 2 2 1010 10 10 1010 2 2 2 Number Numbe 10 10 Number of Number of Number Number of Number of Number of Number of Results: LS Steps until Optimum (Flips) 10 1 10 10 16 1 16 16 1 32 32 32 64 64 64 Problem size 128 128 128 1010 10 1 1 1616 16 1 32 32 32 64 64 64 128 128 Problem size 128 10 1010 16 1616 1 1 1 32 3232 64 6464 128128 128 Problem size size Problem Problem size size Problem Problem sizesize Problem size Problem (a) hBOA (a) (a) hBOA hBOA (b) GA (twopoint) (b) GA GA (twopoint) (b) (twopoint) (c)(c) GA (uniform) GA GA (uniform) (c) (uniform) Figure 2: Growth ofofof the numberevaluations with problem size. Figure 2: 2: Growththe number ofofof evaluations with problem size. Figure Growth the number evaluations with problem size. 6 σ=2.00 σ=2.00 6 6 6 σ=2.00 σ=2.00 6 6 σ=2.00 σ=2.00 Number of flips (GA, twopoint) Number of flips (GA, twopoint) 10 10 σ=2.00 1010 10σ=2.00 10 1010σ=2.00 6 Number of flips (GA, twopoint) Number of flips (GA, twopoint) Number of flips (GA, twopoint) Number of flips (GA, twopoint) 6 6 10 σ=1.50 σ=1.50 σ=1.50 σ=1.50 σ=1.50 σ=1.50 σ=1.50 σ=1.50 σ=1.50 Number of flips (hBOA) Number of flips (hBOA) Number of flips (hBOA) 5 5 5 σ=1.00 σ=1.00 σ=1.00 5 σ=1.00 5 5 σ=1.00σ=1.00 5 5 σ=1.00 σ=1.00 5 σ=1.00 10 10 10 σ=0.83 1010 10 σ=0.83 10 1010 σ=0.83 σ=0.83 σ=0.83 σ=0.83 σ=0.83 σ=0.83 σ=0.83 σ=0.75 σ=0.75 σ=0.75 σ=0.75 σ=0.75 σ=0.75 σ=0.75 σ=0.75 σ=0.75 4 4 4 4 4 4 4 4 4 10 σ=0.55 σ=0.55 10 10 σ=0.55 σ=0.55 1010 10σ=0.55 σ=0.55 σ=0.55 10 1010σ=0.55 σ=0.55 σ=0.00 σ=0.00 σ=0.00 σ=0.00 σ=0.00 σ=0.00 σ=0.00 σ=0.00 σ=0.00 3 3 3 3 3 3 3 3 3 10 10 10 1010 10 10 1010 2 2 2 2 2 2 2 2 2 10 10 10 1010 10 10 1010 16 16 16 32 32 32 64 64 64 128 128 128 1616 16 32 32 32 64 64 64 128 128 128 16 1616 32 3232 64 6464 128128 128 Problem size Problem size size Problem Problem size Problem size size Problem Problem sizesize Problem size Problem (a) (a) hBOA hBOA (a) hBOA (b) GA GA (twopoint) (b) (twopoint) (b) GA (twopoint) (c)(c) GA (uniform) GA GA (uniform) (c) (uniform) Scalability 3:3:Growth ofofof the numberflipsflips with problem size. better Figure of hBOA and GA with twopoint crossover Figure 3: Growththe number ofofof with problem size. Figure Growth the number flips with problem size. for short-range interactions. and how thethe effects σuniform gets worse the algorithm under consideration; this is the topic and how theeffects of ofchange depending on the algorithm under consideration; this is is the topic effects of change depending on the algorithm under consideration; this the topic and howGA with σ σ change depending on with shorter-range interactions. discussed in thethe following few paragraphs. discussed in in following few paragraphs. discussed the following few paragraphs. Based on thethe definitionthe the 1D spin glass with power-law interactions,the the value of σ grows, Based onon definition of of 1D spin glass with power-law interactions, asas the value of grows, Based the definition of the 1D spin glass with power-law interactions, as value of σ σ grows, thethe rangethethe most significant interactions is reduced. With reduction of the range of interactions, therange of of most significant interactions isis reduced. With reduction of the range of interactions, range of the most significant interactions reduced. With reduction of the range of interactions, thethe problem should become easier both for selectorecombinative GAs capablelinkage learning, theproblem should become easier both for selectorecombinative GAs capable of ofof linkage learning, problem should become easier both for selectorecombinative GAs capable linkage learning, such as hBOA, as well as for for selectorecombinative GAs which rarely break interactionsbetween such as as hBOA, as well as selectorecombinative GAs which rarely break interactions between such hBOA, as well as for selectorecombinative GAs which rarely break interactions between closely located bits, such as GAGA with twopoint crossover. This isis clearlydemonstrated by the closely located bits, such asas with twopoint crossover. This is clearly demonstrated by the the closely located bits, such GA with twopoint crossover. This clearly demonstrated by results for for these two algorithms presentedfigures 2 2 2 andAlthough for many problem sizes, the the results forthese two algorithms presented ininin figures and 3.3. Although for many problem sizes, results these two algorithms presented figures and 3. Although for many problem sizes, the Martinabsolute number evaluations and the the number flips are of EAs on smaller Glassfor larger valuesσ, σ, absolute number of G. Katzgraber Pelikan and Helmut of evaluations and number of of flips are factfact smaller larger values of of Analysis in in 1D Spin for with Power-Law Interactions
  1. A particular slide catching your eye?

    Clipping is a handy way to collect important slides you want to go back to later.

×