Towards Efficient Sampling: Exploiting Random Walk Strategy

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Towards Efficient Sampling Exploiting Random
Walk Strategy

• Wei Wei, Jordan Erenrich, and Bart Selman

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Motivations

• Recent years have seen tremendous improvements in
  SAT solving. Formulas with up to 300 variables
  (1992) to formulas with one million variables.
• Various techniques for answering
• does a satisfying assignment exist for a
  formula?
• But there are harder questions to be answered .
• how many satisfying assignments does a formula
  have? Or closely related can we sample from the
  satisfying assignments of a formula?

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Complexity

• SAT is NP-complete. 2-SAT is solvable in linear
  time.
• Counting assignments (even for 2cnf) is
  P-complete, and is NP-hard to approximate
  (Valiant, 1979).
• Approximate counting and sampling are equivalent
  if the problem is downward self-reducible.

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Challenge

• Can we extend SAT techniques to solve harder
  counting/sampling problems?
• Such an extension would lead us to a wide range
  of new applications.

SAT testing
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Standard Methods for Sampling - MCMC

• Based on setting up a Markov chain with a
  predefined stationary distribution.
• Draw samples from the stationary distribution by
  running the Markov chain for sufficiently long.
• Problem for interesting problems, Markov chain
  takes exponential time to converge to its
  stationary distribution

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Simulated Annealing

• Simulated Annealing uses Boltzmann distribution
  as the stationary distribution.
• At low temperature, the distribution concentrates
  around minimum energy states.
• In terms of satisfiability problem, each
  satisfying assignment (with 0 cost) gets the same
  probability.
• Again, reaching such a stationary distribution
  takes exponential time for interesting problems.
  shown in a later slide.

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Standard Methods for Counting

• Current solution counting procedures extend DPLL
  methods with component analysis.
• Two counting precedures are available. relsat
  (Bayardo and Pehoushek, 2000) and cachet (Sang,
  Beame, and Kautz, 2004). They both count exact
  number of solutions.

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• Question Can state-of-the-art local search
  procedures be used for SAT sampling/counting? (as
  alternatives to standard Monte Carlo Markov Chain
  and DPLL methods)

Yes! Shown in this talk
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Our approach biased random walk

• Biased random walk greedy bias pure random
  walk. Example WalkSat (Selman et al, 1994),
  effective on SAT.
• Can we use it to sample from solution space?

• Does WalkSat reach all solutions?

• How uniform is the sampling?

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WalkSat
Hamming distance
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Probability Ranges in Different Domains
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Improving the Uniformity of Sampling
WalkSat
SA

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

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Comparison Between WalkSat and SampleSat
WalkSat
SampleSat
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SampleSat
Hamming Distance
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Analysis
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Property of F

• Proposition 1 SA with fixed temperature takes
  exponential time to find a solution of F
• This shows even for some simple formulas in 2cnf,
  SA cannot reach a solution in poly-time

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Analysis, cont.
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SampleSat

• In SampleSat algorithm, we can devide the search
  into 2 stages. Before SampleSat reaches its first
  solution, it behaves like WalkSat.

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SampleSat, cont.

• After reaching the solution, random walk
  component is turned off because all clauses are
  satisfied. SampleSat behaves like SA.
• Proposition 3 SA at zero temperature samples all
  solutions within a cluster uniformly.
• This 2-stage model explains why SampleSat samples
  more uniformly than random walk algorithms alone.

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Verification on Larger formulas - ApproxCount

• Small formulas -gt Figures, solution frequencies.
  How to verify on large formulas? ApproxCount.
• ApproxCount approximates the number of solutions
  of Boolean formulas, based on SampleSat
  algorithm.
• Besides using it to justify the accuracy of our
  sampling approach, ApproxCount is interesting on
  its own right.

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Algorithm

• The algorithm works as follows (Jerrum and
  Valiant, 1986)
• Pick a variable X in current formula
• Draw K samples from the solution space
• Set variable X to its most sampled value t, and
  the multiplier for X is K/(Xt). Note
  1 ? multiplier ? 2
• Repeat step 1-3 until all variables are set
• The number of solutions of the original formula
  is the product of all multipliers.

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Accumulation of Errors
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Within the Capacity of Exact Counters

• We compare the results of approxcount with those
  of the exact counters.

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And beyond

• We developed a family of formulas whose solutions
  are hard to count
• The formulas are based on SAT encodings of the
  following combinatorial problem
• If one has n different items, and you want to
  choose from the n items a list (order matters) of
  m items (mltn). Let P(n,m) represent the number
  of different lists you can construct. P(n,m)
  n!/(n-m)!

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Hard Instances

• Encoding of P(20,10) has only 200 variables, but
  neither cachet or Relsat was able to count it in
  5 days in our experiments.
• On the other hard, ApproxCount is able to finish
  in 2 hours, and estimates the solutions of even
  larger instances.

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Summary

• Small formulas -gt complete analysis of the
  search space
• Larger formulas -gt compare ApproxCount results
  with results of exact counting procedures
• Harder formulas -gt handcraft formulas compare
  with analytic results

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Conclusion and Future Work

• Shows good opportunity to extend SAT solvers to
  develop algorithms for sampling and counting
  tasks.
• Next step Use our methods in probabilistic
  reasoning and Bayesian inference domains.