Causal Models, Learning Algorithms and their Application to Performance Modeling

Jan LemeireParallel Systems labNovember 15th 2006

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Overview

  I. Causal Models  II. Learning Algorithms  III. Performance Modeling  IV. Extensions

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I. Multivariate Analysis

  Variables

Probabilistic model of joint distribution?Relational information?A priori unknown relations

  Experimental data

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A. Representation of distributions

Factorization

Reduction of factorization complexity

Bayesian Network

A, B, C, DOrdering 1 Ordering 2

A

B C

D

C D B

A C B

A, D, B, C

A

D B

C

C D B

A C B

A D

P(A, B, C, D)=P(A).P(B|A).P(C|A, B).P(D|A, B, C)

P(C|A, B)=P(C|B) ó A C B

A

B C

D A

D B

C

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Conditional independence

Qualitative property: P(rain|quality of speech)=P(rain)?

Markov condition in graphVariable becomes independent from all its non-descendants by

conditioning on its direct parents.

– graphical d-separation criterion

B. Representation of Independencies

D

CBA

P(A|B, C) = P(A|B) ó A C B

B d-separates A from CA is d-separated from DA is not d-separated from D, given B

A C BA D

A D B

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Faithfulness

Faithfulness:Joint Distribution Directed Acyclic Graph

Conditional independencies d-separation

Theorem:

if a faithful graph exists, it is the minimal factorization.

Independence-map:

All independencies in the Bayesian network appear in the distribution

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Definition through interventions

causal model + Conditional Probability Distributions + Causal Markov Condition = Bayesian network

C. Representation of Causal Mechanisms

Model of the underlying physical mechanisms

A B

do(A=a)

P(B|A=a)

A B

do(A=a)

P(B)

A B A B

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Reductionism Causal modeling = reductionism

Canonical representation: unique, minimal, independent

Building block = P(Xi|parents(Xi))Whole theory is based on this modularity

Intervention = change of block

X1 X2

X3 X4

X5

X1 X2

X3 X4

X5

do(X3=a) =a

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Ultimate motivation for causality

Model = canonical representation able to explain all qualitative properties (independencies) close to reality

If causal mechanisms are unrelated model is faithful

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II. Learning Algorithms

Two types: Constraint-basedbased on the independencies

Scoring-basedsearches set of all models, give a score of how good they

represent distribution

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Step 1: Adjacency search

Property: adjacent nodes do not become independent

Algorithm: start with full-connected graphcheck for marginal independenciescheck for conditional independencies

C

A B

DC

A B

D

A D

C

A B

D

A C B

C D B

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Step 2: Orientation

Property: V-structure can be recognized

Algorithm: look for v-structures derived rules

C

A B

DC

A B

D

A D

A D B

C

A B

D

A C

A C B

A C

B

A B C

A B C

A C

B

A C

A C B

A C

A C B

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Assumptions

General statistical assumptions: No selection bias Random sample Sufficient data for correctness of statistical tests

Underlying network is faithful

Causal sufficiency No unknown common causes

A C

B

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Criticism

Definition causality?About predicting the effect of changes to the system

Faithfulness assumption Eg.: accidental cancellation

Causal Markov Condition “All relations are causal”

Learning algorithms are not robustStatistical tests make mistakes

X

Y

VU

X Y

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Part III: Performance Analysis

  High-Performance computing

1 processor

parallel system

Performance Questions:Performance prediction

System-dependency?

Parameter-dependency? Reasons of bad performance?

Effect of Optimizations?

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Causal modeling (cf. COMO lab, VUB) Representation form Close to reality Learning algorithms TETRAD tool (open-source, java)

PhD??

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Performance Models

Aim performance analysis Support software developer High-performance applications

Expected properties offer insight into causes performance degradation prediction estimate effect of optimizations reusable submodels

separate application and system-dependency

reason under uncertainty

causal models

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Integrated in statistical analysis

Statistical characteristics Regression analysisProbability table compressionOutlier detection

Iterative process1. Perform additional experiments2. Extract additional characteristics3. Indicate exceptions4. Analyze the divergences of the

data points with the current hypotheses

Experiments

Profiling

Causal Model

User Inspection

ModelConstruction

Application

Curve Fitting

Analytical Model

Database

DivergencesExc

eptio

ns

1

2

34

CPT compression

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A. Model construction

Model of computation

time of LU decom- position algorithm

elementsize (redundant variable) is sufficient for influence datatype -> cache misses regression analysis on submodels X=f(parents) analysis of parameters

#op

fclock

datatypeTcomp

n

Cop

L1Mop

elementsize

#instrop

L2Mop

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B. Detection of unexpected dependencies

Point-to-point communication performance

background communication

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C. Finding explanations for outliers

Exceptional data in communication performance measurements

Probability table compressionX P(X=1)

}X0

X1

X2

X3

00

1

1 }

Y

Y0

Y0

Y1

Y1

=> derived variableInteresting features

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IV. Complexity of Performance Data

Mixture discrete and continuous variables Mutual Information & Kernel Density Estimation

Non-linear relations Mutual Information & Kernel Density Estimation

Deterministic relations Augmented models & Complexity criterion

Context variables Work in progress

Context-specific independencies Work in progress

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A. Information-theoretic Dependency

  Entropy of random variable X   Discretized entropy for continuous variable

  Mutual Information

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B. Kernel Density Estimation

  See applets

Trade-off maximal entropy <> typicalness

Conclusions Limited number data points needed Discretization of continuous data justified Form-free dependency measure

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C. Deterministic relations

  Y=f(X)

Y becomes independent from Z conditioned on X~ violation of the intersection condition (Pearl ’88)Not faithfully describable

Solution: augmented causal model- add regularity to model- adapt inference algorithms

YX Z

YX Z

X YZ XY ZY Z

XY Z

X Z

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The Complexity Criterion

Select simplest relation

X & Y contain equivalent information about Z

Complexity(Y-Z) < Complexity(X-Z)

X Y

Z

X Y

Z

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Augmented causal model

Consistent models underComplexity Increase assumption

Compl(X-Z) ≥ Compl(X-Y)Compl(X-Z) ≥ Compl(Y-Z)

X Y Z

X YZS

Restrict conditional independenciesGeneralize d-separation

Reestablish faithfulness

X YZeq

S

eq{

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Theory works!

Deterministic

A

B Probabilistic

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Conclusions

  Benefit of the integration of statistical techniques

  Causal modeling is a challenge– wants to know the inner from the outer

  More information– http://parallel.vub.ac.be– http://parallel.vub.ac.be/~jan