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Analyzing Probabilistic Models in Hierarchical BOA on Traps and Spin Glasses
 

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 ...

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 Analyzing Probabilistic Models in Hierarchical BOA on Traps and Spin Glasses Presentation Transcript

    • Analyzing Probabilistic Models in Hierarchical BOA on Traps and Spin Glasses Mark Hauschild, Martin Pelikan, Claudio F. Lima, Kumara Sastry
    • 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
    • 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 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
    • hBOA Bayesian Current New network Selection population population
    • 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 Jij Effects on scalability Using models to understand problems Dynamics How the models change over time Efficiency Enhancement
    • 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
    • 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
    • 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
    • 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, middle, last gen.) Average of the 30 runs for each problem size
    • Trap-5 Results a) Middle snapshot b) End snapshot Ratio of unnecessary dependencies to total dependencies as problem size increases
    • 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.
    • 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.
    • 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
    • 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
    • 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
    • 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
    • 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
    • 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
    • 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
    • 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
    • Future Work Do model analysis on other challenging problems Artificial Real world Develop techniques to automatically exploit this information for EE
    • Any Questions?