Analyzing Probabilistic Models
     in Hierarchical BOA
 on Traps and Spin Glasses

  Mark Hauschild, Martin Pelikan,
  Cl...
Motivation
 Background
   hBOA can solve difficult problems scalably and reliably
   Much empirical work has been done
   ...
Outline
 Hierarchical BOA (hBOA)
 Models in hBOA
 Experiments
   Trap 5
   2d Ising Spin Glass
 Conclusions
 Future Work
Hierarchical Bayesian
Optimization Algorithm (hBOA)
 Pelikan, Goldberg and Cantu-Paz (2001)
 Uses Bayesian network with lo...
hBOA
                         Bayesian
 Current                              New
                         network
        ...
Bayesian Network
Structure
  Edges determine
  dependence

Parameters
  p(Burglary|Alarm)
What are we looking for?
 Accuracy
   Do the models accurately represent the
   underlying problem structure
             ...
Test Problems
 Selecting our test problems
   Have to know what a good model looks like
   Non-trivial
   Easily scaled up...
Trap-5
 Partition binary string into disjoint
 groups of 5 bits
               5      if ones = 5
 trap (ones) =
         ...
Perfect Model for Trap-5




 Each 5-bit partition should be fully (or close to fully)
 connected (10 edges)
    Necessary...
Trap-5 Experiments
 Problem sizes 15-210
 30 runs for each size
 Bisection was used to determine
 population size
Trap-5 Results




 Necessary and unnecessary dependencies w.r.t problem size
 Three snapshots taken in each run (first, m...
Trap-5 Results




    a) Middle snapshot     b) End snapshot



  Ratio of unnecessary dependencies to total
  dependenci...
Trap-5 Results
               300                                                  600
                                   ...
2D Ising Spin Glass
 Origin in physics
 Spins arranged on a 2D
 grid (periodic boundary
 conditions)

 Each spin (sj) can ...
2D Ising Spin Glass
 Very hard for most optimization techniques
   Extremely large number of local optima
   Any decomposi...
Perfect Model for Spin-Glass
 Not sure what the perfect model is
 To some degree, all positions are
 dependent on all othe...
Spin Glass Experiments
 100 different instances of size 10x10 to 20x20
   Only graphs of 20x20 are shown
   With and witho...
Spin Glass Results
                 800                                        800
  Dependencies




                    ...
Spin Glass Results (No DHC)
                 800                                        800
  Dependencies




           ...
Spin Glass Results
               1800                                                   800
                             ...
Restricting hBOA models
                           5
                          10
                                 stock
 ...
Conclusions
 Studying hBOA models is challenging but
 possible
 hBOA generates accurate models quickly
 Models reveal a lo...
Future Work
 Do model analysis on other challenging
 problems
   Artificial
   Real world
 Develop techniques to automatic...
Any Questions?
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Analyzing Probabilistic Models in Hierarchical BOA on Traps and Spin Glasses

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The hierarchical Bayesian optimization algorithm (hBOA) can solve nearly decomposable and hierarchical problems of bounded difficulty in a robust and scalable manner by building and sampling probabilistic models of promising solutions. This paper analyzes probabilistic models in hBOA on two common test problems: concatenated traps and 2D Ising spin glasses with periodic boundary conditions. We argue that although Bayesian networks with local structures can encode complex probability distributions, analyzing these models in hBOA is relatively straightforward and the results of such analyses may provide practitioners with useful information about their problems. The results show that the probabilistic models in hBOA closely correspond to the structure of the underlying optimization problem, the models do not change significantly in subsequent iterations of BOA, and creating adequate probabilistic models by hand is not straightforward even with complete knowledge of the optimization problem.

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Analyzing Probabilistic Models in Hierarchical BOA on Traps and Spin Glasses

  1. 1. Analyzing Probabilistic Models in Hierarchical BOA on Traps and Spin Glasses Mark Hauschild, Martin Pelikan, Claudio F. Lima, Kumara Sastry
  2. 2. Motivation Background hBOA can solve difficult problems scalably and reliably Much empirical work has been done efficiency enhancement (EE) scalability Relatively little studying of the models themselves Purpose Understand models in hBOA Design problem-specific EE Design automated EE techniques Educate researchers about their problems
  3. 3. Outline Hierarchical BOA (hBOA) Models in hBOA Experiments Trap 5 2d Ising Spin Glass Conclusions Future Work
  4. 4. Hierarchical Bayesian Optimization Algorithm (hBOA) Pelikan, Goldberg and Cantu-Paz (2001) Uses Bayesian network with local structures to model promising solutions Bayesian network Acyclic directed Graph String positions are the nodes Edges represent conditional dependencies Where there is no edge, implicit independence Sampled to create new population each gen. Niching to maintain diversity
  5. 5. hBOA Bayesian Current New network Selection population population
  6. 6. Bayesian Network Structure Edges determine dependence Parameters p(Burglary|Alarm)
  7. 7. What are we looking for? Accuracy Do the models accurately represent the underlying problem structure Jij Effects on scalability Using models to understand problems Dynamics How the models change over time Efficiency Enhancement
  8. 8. Test Problems Selecting our test problems Have to know what a good model looks like Non-trivial Easily scaled up Diverse Test Problems Selected Trap-5 2D Ising Spin Glass
  9. 9. Trap-5 Partition binary string into disjoint groups of 5 bits 5 if ones = 5 trap (ones) = 4 ones otherwise Total fitness is sum of single traps Optimum: String 1111…1
  10. 10. Perfect Model for Trap-5 Each 5-bit partition should be fully (or close to fully) connected (10 edges) Necessary dependencies Bits between partitions should be independent Unnecessary dependencies
  11. 11. Trap-5 Experiments Problem sizes 15-210 30 runs for each size Bisection was used to determine population size
  12. 12. Trap-5 Results Necessary and unnecessary dependencies w.r.t problem size Three snapshots taken in each run (first, middle, last gen.) Average of the 30 runs for each problem size
  13. 13. Trap-5 Results a) Middle snapshot b) End snapshot Ratio of unnecessary dependencies to total dependencies as problem size increases
  14. 14. Trap-5 Results 300 600 Total Total + New Dep + New Dep 250 500 − Old Dep − Old Dep Dependencies Dependencies 200 400 150 300 100 200 50 100 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 5 10 15 20 25 Generations Generations a) N = 100 bits. b) N = 200 bits. Model dynamics for two different problem sizes. Dependencies were counted as the same even if their direction changed.
  15. 15. 2D Ising Spin Glass Origin in physics Spins arranged on a 2D grid (periodic boundary conditions) Each spin (sj) can have two values: +1 or -1 E = E (c ) si J i , j s j Set of connection weights i, j Jij specifies an instance.
  16. 16. 2D Ising Spin Glass Very hard for most optimization techniques Extremely large number of local optima Any decomposition of bounded order is insufficient The problem is solvable in polynomial time by analytical techniques hBOA does solve it in polynomial time A deterministic hill-climber (DHC) is used to improve the quality of all evaluated solutions Single bit flips until no improvements
  17. 17. Perfect Model for Spin-Glass Not sure what the perfect model is To some degree, all positions are dependent on all others If we consider all, it will not scale We would expect the interactions should decrease in magnitude as their distance increases
  18. 18. Spin Glass Experiments 100 different instances of size 10x10 to 20x20 Only graphs of 20x20 are shown With and without DHC 5 runs of each instance Scaling as we restrict models Restrict models to short order dependencies
  19. 19. Spin Glass Results 800 800 Dependencies Dependencies 600 600 400 400 200 200 0 0 0 10 20 0 10 20 Distance Distance a) First generation b) Second snapshot 800 800 Dependencies Dependencies 600 600 400 400 200 200 0 0 0 10 20 0 10 20 Distance Distance c) Third snapshot d) Last generation
  20. 20. Spin Glass Results (No DHC) 800 800 Dependencies Dependencies 600 600 400 400 200 200 0 0 0 10 20 0 10 20 Distance Distance a) First generation b) Second snapshot 800 800 Dependencies Dependencies 600 600 400 400 200 200 0 0 0 10 20 0 10 20 Distance Distance c) Third snapshot d) Last generation
  21. 21. Spin Glass Results 1800 800 Total Total + New Dep + New Dep 1500 − Old Dep − Old Dep 600 Dependencies Dependencies 1200 900 400 600 200 300 0 0 1 2 3 4 5 6 7 8 9 101112131415 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Generations Generations a) All dependencies b) Neighbor dependencies One example instance (rest similar) Dependencies were counted the same even if their direction changed
  22. 22. Restricting hBOA models 5 10 stock d(2) Fitness Evaluations d(1) 4 10 3 10 100 144 196 256 324400 625 900 Problem Size Evaluations varying by different restrictions Experiments without DHC had similar scaling
  23. 23. Conclusions Studying hBOA models is challenging but possible hBOA generates accurate models quickly Models reveal a lot of information about the problem Models do not change rapidly over time Information about models can help us design EE
  24. 24. Future Work Do model analysis on other challenging problems Artificial Real world Develop techniques to automatically exploit this information for EE
  25. 25. Any Questions?

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