Optimization methods

Morten NielsenDepartment of Systems Biology,

DTU

Outline

• Optimization procedures – Gradient decent– Monte Carlo

• Overfitting – cross-validation

• Method evaluation

Linear methods. Error estimate

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w1 w2

Linear function

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Gradient decent (from wekipedia)

Gradient descent is based on the observation that if the real-valued function F(x) is defined and differentiable in a neighborhood of a point a, then F(x) decreases fastest if one goes from a in the direction of the negative gradient of F at a. It follows that, if

for > 0 a small enough number, then F(b)<F(a)

Gradient decent (example)

Gradient decent

Gradient decent

Weights are changed in the opposite direction of the gradient of the error

Gradient decent (Linear function)

Weights are changed in the opposite direction of the gradient of the error

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Gradient decent

Weights are changed in the opposite direction of the gradient of the error

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Linear function

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Gradient decent. Example

Weights are changed in the opposite direction of the gradient of the error

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Linear function

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Gradient decent. Example

Weights are changed in the opposite direction of the gradient of the error

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Gradient decent. Doing it your selfWeights are changed in the opposite direction of the gradient of the error

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W1=0.1 W2=0.1

Linear function

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What are the weights after 2 forward (calculate predictions) and backward (update weights) iterations with the given input, and has the error decrease (use =0.1, and t=1)?

Fill out the table

itr W1 W2 O

0 0.1 0.1

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2

What are the weights after 2 forward/backward iterations with the given input, and has the error decrease (use =0.1, t=1)?

1 0

W1=0.1 W2=0.1

Linear function

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Monte Carlo

Because of their reliance on repeated computation of random or pseudo-random numbers, Monte Carlo methods are most suited to calculation by a computer. Monte Carlo methods tend to be used when it is unfeasible or impossible to compute an exact result with a deterministic algorithmOr when you are too stupid to do the math yourself?

Monte Carlo (Minimization)

dE<0dE>0

Gibbs sampler. Monte Carlo simulations

RFFGGDRGAPKRGYLDPLIRGLLARPAKLQVKPGQPPRLLIYDASNRATGIPA GSLFVYNITTNKYKAFLDKQ SALLSSDITASVNCAK GFKGEQGPKGEPDVFKELKVHHANENI SRYWAIRTRSGGITYSTNEIDLQLSQEDGQTIE

RFFGGDRGAPKRGYLDPLIRGLLARPAKLQVKPGQPPRLLIYDASNRATGIPAGSLFVYNITTNKYKAFLDKQ SALLSSDITASVNCAK GFKGEQGPKGEPDVFKELKVHHANENI SRYWAIRTRSGGITYSTNEIDLQLSQEDGQTIE

E1 = 5.4 E2 = 5.7

E2 = 5.2

dE>0; Paccept =1

dE<0; 0 < Paccept < 1

Note the sign. Maximization

Monte Carlo Temperature

• What is the Monte Carlo temperature?

• Say dE=-0.2, T=1

• T=0.001

MC minimization

Monte Carlo - Examples

• Why a temperature?

Local minima

• A prediction method contains a very large set of parameters

– A matrix for predicting binding for 9meric peptides has 9x20=180 weights

• Over fitting is a problem

Data driven method training

yearsTe

mperature

ALAKAAAAMALAKAAAANALAKAAAARALAKAAAATALAKAAAAVGMNERPILTGILGFVFTMTLNAWVKVVKLNEPVLLLAVVPFIVSVMRSGRVHAVVRFNIDETPANYIGQDGLAELCGDPGDQTRAVADGKGRPVPAAHPMTAQWWLDAFARGVVHVILQRELTRLQAVAEEMTKS

Evaluation of predictive performance• Train PSSM on raw data

– No pseudo counts, No sequence weighting– Fit 9*20 parameters to 9*10 data points

• Evaluate on training data–PCC = 0.97–AUC = 1.0

• Close to a perfect prediction method

Bin

ders

Non

e B

ind

ers

AAAMAAKLAAAKNLAAAAAKALAAAARAAAAKLATAALAKAVAAAIPELMRTNGFIMGVFTGLNVTKVVAWLLEPLNLVLKVAVIVSVPFMRSGRVHAVVRFNIDETPANYIGQDGLAELCGDPGDQTRAVADGKGRPVPAAHPMTAQWWLDAFARGVVHVILQRELTRLQAVAEEMTKS

Evaluation of predictive performance• Train PSSM on Permuted (random) data

– No pseudo counts, No sequence weighting– Fit 9*20 parameters to 9*10 data points

• Evaluate on training data–PCC = 0.97–AUC = 1.0

• Close to a perfect prediction method AND• Same performance as one the original data

Bin

ders

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e B

ind

ers

Repeat on large training data (229 ligands)

Cross validation

Cross validation

Train on 4/5 of dataTest/evaluate on 1/5=>Produce 5 different methods each with a different prediction focus

Model over-fitting

2000 MHC:peptide binding dataPCC=0.99

Evaluate on 600 MHC:peptide binding dataPCC=0.80

Model over-fitting (early stopping)

Evaluate on 600 MHC:peptide binding dataPCC=0.89

Stop training

What is going on?

years

Temperature

5 fold training

Which method to choose?

5 fold training

Method evaluation

• Use cross validation• Evaluate on concatenated data and not

as an average over each cross-validated performance

Method evaluation

Which prediction to use?

Method evaluation

SMM - Stabilization matrix method

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Per target:

Global:

Sum over weights

Sum over data points

SMM - Stabilization matrix method

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l per target

SMM - Stabilization matrix method

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SMM training

Evaluate on 600 MHC:peptide binding dataL=0: PCC=0.70L=0.1 PCC = 0.78

SMM - Stabilization matrix methodMonte Carlo

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Global:

• Make random change to weights

• Calculate change in “global” error

• Update weights if MC move is accepted

Note difference between MC and GD in the use of “global” versus “per target” error

Training/evaluation procedure

• Define method• Select data• Deal with data redundancy

– In method (sequence weighting)– In data (Hobohm)

• Deal with over-fitting either– in method (SMM regulation term) or– in training (stop fitting on test set

performance)• Evaluate method using cross-validation

A small doit script/usr/opt/www/pub/CBS/courses/27623.algo/exercises/code/SMM/doit_ex

#! /bin/tcsh foreach a ( `cat allelefile` )

mkdir -p $cd $a

foreach l ( 0 1 2.5 5 10 20 30 )

mkdir -p l.$lcd l.$l

foreach n ( 0 1 2 3 4 )

smm -nc 500 -l $l train.$n > mat.$npep2score -mat mat.$n eval.$n > eval.$n.pred

end

echo $a $l `cat eval.?.pred | grep -v "#" | gawk '{print $2,$3}' | xycorr`

cd ..

end

cd ..

end