Transcript Wei Wei Sampling, Counting, and Probabilistic Inference joint work with Bart Selman

Sampling, Counting, and Probabilistic
Inference
Wei Wei
joint work with Bart Selman
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The problem: counting solutions
¬a  b  c
¬a  ¬b
¬b  ¬c
cd
2
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
3
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
Query: Pr(A  B)
A
.1
= Pr(A) * Pr (B|A)
= .1 * .2 = .02
B
B
A
~A
.2
.6
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Complexity

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”.
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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
T
T
null
x2
F
x2
F
solution
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Existing Methods for Counting

CDP (Birnbaum and Lozinskii, 1999)

Relsat (Bayardo and Pehoushek, 2000)
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Existing Methods

1.
2.
cachet (Sang, Beame, and Kautz, 2004)
Component caching
Clause learning
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Conflict Graph
Known Clauses
(p  q  a)
( a   b   t)
(t  x1)
(t  x2)
(t  x3)
(x1  x2  x3  y)
(x2  y)
1-UIP scheme
(t)
x1
p
q
y
a
t
b
Current decisions
p  false
q  false
b  true
x2
false
y
x3
Decision scheme
(p  q   b)
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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 space
2. Pick a variable X in current formula
3. 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 set
5. The number of solutions of the original formula is
the product of all multipliers.
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assignments
X1=T
X1=F
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
Nonergodic
Ergodic
Ergodic
Quickly reach sinks Slow convergence
WalkSat

+
SA
=
Does not satisfy DBC
SampleSat
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
10
104
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WalkSat (50,000,000 runs in total)
Hamming distance
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SampleSat
174 sols, r = 11
Total hits = 5.3m
Average hits = 30.1k
1186 sols, r = 14
Total hits = 17.3m
Average hits = 14.6k
39 sols, r = 7
Total hits = 5.1m
Average hits = 131k
704 sols, r = 14
Total hits = 11.1m
Average hits = 15.8k
24 sols, r = 5
Total hits = 0.6m
Average hits = 25k
192 sols, r = 11
Total hits = 5.7m
Average hits = 29.7k
212 sols, r = 11
Total hits = 2.9m
Average hits = 13.4k
Hamming Distance
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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
b
F
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
a
T
T
T
…
T
T
F
F
F
…
F
T
F
F
F
…
F
F
Proposition 2:
pure RW
reaches this
solution with
exp. small
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prob.
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 space
2. Pick a variable X in current formula
3. 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 set
5. 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)
Cachet
Relsat
CDP
sat/unsat
threshhold
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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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