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Sampling, Counting, and Probabilistic Inference

Wei Weijoint work with Bart Selman

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The problem: counting solutions

¬a b c¬a ¬b

¬b ¬c

c d

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Motivation

Consider the standard logical inference

iff ( ) is unsat

there doesn’t exist a model in in which is true.

in all models of , query holds

holds with absolute certainty

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Degree of belief

Natural generalization: degree of belief of is defined as P( | ) (Roth, 1996)

In absence of statistical information, degree of belief can be calculated as

M( ) / M( )

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Bayesian Nets to Weighted Counting (Sang, Beame, and Kautz, 2004)

Introduce new vars so all internal vars are deterministic

A

B

A ~A

B .2 .6

A .1Query: Pr(A B)

= Pr(A) * Pr (B|A)

= .1 * .2 = .02

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SAT is NP-complete. 2-SAT is solvable in linear time.

Counting assignments (even for 2cnf, Horn logic, etc) is #P-complete, and is NP-hard to approximate to a factor within ( (Valiant 1979, Roth 1996).

Approximate counting and sampling are equivalent if the problem is “downward self-reducible”.

Complexity

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(Roth, 1996)

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Existing method: DPLL(Davis, Logemann and Loveland, 1962)

(x1 x2 x3) (x1 x2 x3) (x1 x2)

DPLL was first proposed as a basic depth-first tree

search.

x1

x2

FT

T

null

F

solution

x2

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Existing Methods for Counting

CDP (Birnbaum and Lozinskii, 1999)

Relsat (Bayardo and Pehoushek, 2000)

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Existing Methods

cachet (Sang, Beame, and Kautz, 2004)

1. Component caching

2. Clause learning

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Conflict Graph

Decision scheme(p q b)

1-UIP scheme(t)

p

q

b

a

x1

x2

x3

y

yfalset

Known Clauses(p q a)

( a b t)(t x1)(t x2)(t x3)

(x1 x2 x3 y)(x2 y)

Current decisionsp falseq falseb true

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Existing Methods

Pro: get exact count

Cons: 1. Cannot predict execution time

2. Cannot halt execution to get an approximation

3. Cannot handle large formulas

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Our proposal: counting by sampling

The algorithm works as follows (Jerrum and Valiant, 1986):

1. Draw K samples from the solution space2. Pick a variable X in current formula3. Set variable X to its most sampled value t, and

the multiplier for X is K/#(X=t). Note 1 multiplier 2

4. Repeat step 1-3 until all variables are set5. The number of solutions of the original formula is

the product of all multipliers.

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X1=TX1=F

assignments

models

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Research issues

how well can we estimate each multiplier?

    we'll see that sampling works quite well.      

how do errors accumulate? (note formula can have  hundreds of variables; could potentially be very bad)

surprisingly, we will see that errors often cancel each other out.

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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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Question: Can state-of-the-art local search procedures be used for SAT sampling? (as alternatives to standard Monte Carlo Markov Chain)

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 (50,000,000 runs in total)

visited 500,000 times

visited 60 times

Hamming distance

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Probability Ranges in Different Domains

Instance Runs Hits Rarest

Hits Common

Common-to -Rare Ratio

Random 50 106

53 9 105 1.7 104

Logistics planning

1 106 84 4 103 50

Verif. 1 106 45 318 7

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Improving the Uniformity of Sampling

SampleSat:– With probability p, the algorithm makes a

biased random walk move– With probability 1-p, the algorithm makes a

SA (simulated annealing) move

WalkSat

Nonergodic

Quickly reach sinks

Ergodic

Slow convergence

Ergodic

Does not satisfy DBC

SA = SampleSat+

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Comparison Between WalkSat and SampleSat

WalkSat SampleSat

104

10

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WalkSat (50,000,000 runs in total)

Hamming distance

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SampleSat

Hamming Distance

174 sols, r = 11Total hits = 5.3mAverage hits = 30.1k

704 sols, r = 14Total hits = 11.1mAverage hits = 15.8k

39 sols, r = 7Total hits = 5.1mAverage hits = 131k

212 sols, r = 11Total hits = 2.9mAverage hits = 13.4k

192 sols, r = 11Total hits = 5.7mAverage hits = 29.7k

24 sols, r = 5Total hits = 0.6mAverage hits = 25k

1186 sols, r = 14Total hits = 17.3mAverage hits = 14.6k

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Instance Runs Hits Rarest

Hits Common

Common-to -Rare Ratio

WalkSat

Ratio SampleSat

Random 50 106

53 9 105 1.7 104 10

Logistics planning

1 106 84 4 103 50 17

Verif. 1 106 45 318 7 4

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Analysis

c1 c2 c3 … cn a bF F F … F F F

F F F … F F T

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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.

c1 c2 c3 … cn aT T T … T T

F F F … F T

F F F … F F

Proposition 2: pure RW reaches this solution with exp. small prob.

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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.

instance WalkSat SampleSat SA

random 382 677 24667

logistics 5.7 104 15.5 105 > 109

verification 36 65 10821

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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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Back to Counting: ApproxCount

The algorithm works as follows (Jerrum and Valiant, 1986):

1. Draw K samples from the solution space2. Pick a variable X in current formula3. Set variable X to its most sampled value t, and

the multiplier for X is K/#(X=t). Note 1 multiplier 2

4. Repeat step 1-3 until all variables are set5. The number of solutions of the original formula is

the product of all multipliers.

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Random 3-SAT, 75 Variables(Sang, Beame, and Kautz, 2004)

sat/unsat threshhold

CDP

Relsat

Cachet

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Within the Capacity of Exact Counters We compare the results of approxcount with those of the exact

counters.

instances #variables Exact count

ApproxCount Average Error per step

prob004-log-a 1790 2.6 1016

1.4 1016 0.03%

wff.3.200.810 200 3.6 1012

3.0 1012 0.09%

dp02s02.shuffled 319 1.5 1025

1.2 1025 0.07%

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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 (m<=n). Let P(n,m) represent the number of different lists you can construct. P(n,m) = n!/(n-m)!

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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.

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The end.

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