experiments with stage

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Experiments with STAGEExperiments with STAGE

Wei Wei

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Introduction Introduction

STAGE- Developed by Boyan

Use value function approximation to automatically analyze sample trajectories.

Speed up many local search methods

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Diagram of STAGEDiagram of STAGE

Run to Optimize Obj Hillclimb to

Optimize V

Produces new training data

Produces good start states

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Apply it to SATApply it to SAT

The base algorithm is WalkSAT (modified)

Got results better than pure WalkSAT

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Overview Overview

We need to deal with four aspects of the problem: WalkSAT, STAGE, features, and to make the algorithm Markovian.

Hard to tune; not every combination works.

Marko-vianize

stageWalkSAT

features

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Features Features

%clauses unsatisfied (-)%clauses satisfied by 1 variable (+)%clauses satisfied by 2 variables (-)%critical variables (-)%variables set to naïve setting (~)

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MarkovianizeMarkovianize

S/W1 : patience based, not MarkovianS/W2 : best-so-farS/W3 : epsilon cutoff

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Parameter tuningParameter tuning

Noise 0.25 seems goodPatience 10,000Cutoff 1,000,000Epsilon .0001

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Function approximator V-bar-piFunction approximator V-bar-pi

Quadratic regressionLinear regression

Linear functions perform 25% better, and faster.

Linear functions are coarse approximators.

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resultsresults

algorithm Mean(obj) Time Accept%

WalkSAT 15.2 63min 100

S/W1 5.2 130min 60

S/W2 6.2 112min 58

S/W3 4.5 122min 97

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Results – Hemming Distance Results – Hemming Distance traveled by the V steptraveled by the V stepalgorithm Min Max Average TBN

S/W1 27 5028 2047 90%

S/W2 54 6982 2135 89%

S/W3 1 625 176 99%

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resultsresults

algorithm Linear Quadratic difference

S/HC 21.6 28.3 31%

S/W1 5.2 5.4 4%

S/W2 6.2 5.0 -19%

S/W3 4.4 5.6 27%

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Feature 1 and 2 only Feature 1 and 2 only

algorithm Mean(obj) Time Accept%

WalkSAT 15.2 63min 100

S/W1 8.2 98min 83

S/W2 8.5 96min 85

S/W3 7.3 102min 97

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Added feature: %variables set Added feature: %variables set to true to true

algorithm Mean(obj) Time Accept%

WalkSAT 15.2 63min 100

S/W1 5.4 143min 58

S/W2 5.9 118min 56

S/W3 4.6 135min 95

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Discussion(1)Discussion(1)

Linear regression is very bad approximation is this case, yet it gives better results than quadratic regression. Why?

Hit bottom very oftenLead to long more WalkSAT moves

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Discussion(2)Discussion(2)

Features – coefficients vary a lot among instances. But relatively stable within one instance.

The signs are relatively stable

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Discussion(3)Discussion(3)

Time vs evaluationWhen # of evaluation is fixed, STAGE

performs 3 times better, but time spent is doubled

When time is fixed, the result is 40% better than WalkSAT

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Discussion(4)Discussion(4)

Can it hit the finish line?It does vaguely(?) learn some concepts,

which hopefully can direct WalkSAT to a good place.

Par-? Is a good set of problems to solve?

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One featureOne feature

5 features 1 feature

WalkSAT (15.2)

S/w1 5.2 17.4

S/W2 6.2 18.3

S/W3 4.4 20.9

No improvement over WalkSAT.

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Random restartRandom restart

176 Random flips – Worse than S/W3, still better than WalkSAT

1000 Random flips – Worse than one-run WalkSAT

Complete new start points – similar to the case above.

Parameters: cutoff – 10,000. Restart – 100.

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HanoiHanoi

Parameters not yet carefully tunedIt would be interesting to see whether

Hanoi4 can be solved by carefully tuned S/W3. I ran WalkSAT for 50,000,000 flips, but failed to solve it.

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Hanoi problemsHanoi problems

WalkSAT

GSAT S/W3

Hanoi5 8 18 5

Hanoi4 2 8 1