Sampling combinatorial space using biased random walks
Download
1 / 10

Sampling Combinatorial Space Using Biased Random Walks - PowerPoint PPT Presentation


  • 88 Views
  • Uploaded on

Sampling Combinatorial Space Using Biased Random Walks. Jordan Erenrich, Wei Wei and Bart Selman Dept. of Computer Science Cornell University. Many forms of probabilistic reasoning can be effectively reduced to sampling satisfying assignments from a Boolean formula (an instance of SAT).

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about 'Sampling Combinatorial Space Using Biased Random Walks' - paul


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
Sampling combinatorial space using biased random walks

Sampling Combinatorial Space Using Biased Random Walks

Jordan Erenrich, Wei Wei and Bart Selman

Dept. of Computer Science

Cornell University


  • Many forms of probabilistic reasoning can be effectively reduced to sampling satisfying assignments from a Boolean formula (an instance of SAT).

  • Question: Can state-of-the-art local search procedures for SAT sample effectively from the solution space? (as an alternative to standard Monte Carlo Markov Chain methods)


Characteristics of solution space solution clustering
Characteristics of Solution space: Solution Clustering reduced to sampling satisfying assignments from a Boolean formula (an instance of SAT).

  • Visualization with multi-dimensional scaling (MDS)

    • Solutions to specific 75 variable, 325 clause 3-SAT instances

    • 75 dimensional solution projected to two dimensions

    • Distance between points approximates hamming distance


Solution probability
Solution Probability reduced to sampling satisfying assignments from a Boolean formula (an instance of SAT).

  • Consider a simple 2-SAT problem

    • x OR y

  • Consider a simple SAT heuristic

    • Starts with a random bit assignment

    • Randomly flip a bit until a solution is found

  • Consider the probability of finding each solution


Solution probability using walksat algorithm
Solution Probability Using WalkSat Algorithm reduced to sampling satisfying assignments from a Boolean formula (an instance of SAT).

  • Empirically determined each solution’s probability (uf75-01 - 75 variable, 325 clause 3-SAT instance)

  • WalkSat finds every solution, but with very large range of probabilities (1:104)

  • Probability Clusters


Probability ranges in different domains
Probability Ranges in Different Domains reduced to sampling satisfying assignments from a Boolean formula (an instance of SAT).


Improving the uniformity of sampling
Improving the Uniformity of Sampling reduced to sampling satisfying assignments from a Boolean formula (an instance of SAT).

Mixed sampling strategy

  • To reduce the range of probabilities, we propose a hybrid local search algorithm:

    • With probability p, the algorithm makes a biased random walk move

    • With probability 1-p, the algorithm makes a SA (simulated annealing) move

  • In our experiment, we used

    • 50% WalkSat + 50% SA at a fixed temperature


Results of the hybrid approach
Results of the Hybrid Approach reduced to sampling satisfying assignments from a Boolean formula (an instance of SAT).

Our key figure.


Solution clusters
Solution Clusters reduced to sampling satisfying assignments from a Boolean formula (an instance of SAT).

Results on a random 3-SAT instance (70 vars, 301 clauses, 2531 solutions).


Summary
Summary reduced to sampling satisfying assignments from a Boolean formula (an instance of SAT).

Proposal: Use SAT solvers to sample solutions from a

combinatorial space.

Findings:

  • WalkSAT does sample all solutions.

  • But, sampling can be highly biased.

  • Using a new hybrid strategy, we can obtain

  • effective near-uniform sampling.

  • Lesson: Hybrid of SA and biased walk, is a

  • promising alternative to MCMC methods

  • for sampling.


ad