* Adapted from slides by Chen Kaeasar, Ben-Gurion University

Optimization methods

Morten NielsenDepartment of Systems biology,

DTUIIB-INTECH, UNSAM, Argentina

*Adapted from slides by Chen Kaeasar, Ben-Gurion University

The path to the closest local minimum = local minimization

Minimization

*Adapted from slides by Chen Kaeasar, Ben-Gurion University

The path to the closest local minimum = local minimization

Minimization

The path to the global minimum

*Adapted from slides by Chen Kaeasar, Ben-Gurion University

Minimization

Outline

• Optimization procedures – Gradient descent– Monte Carlo

• Overfitting – cross-validation

• Method evaluation

Linear methods. Error estimate

I1 I2

w1 w2

Linear function

o

Gradient descent (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 descent (example)

Gradient descent

Gradient descent

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

Gradient descent (Linear function)

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

I1 I2

w1 w2

Linear function

o

Gradient descent

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

I1 I2

w1 w2

Linear function

o

Gradient descent. Example

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

I1 I2

w1 w2

Linear function

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Gradient descent. Example

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

I1 I2

w1 w2

Linear function

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

1 0

W1=0.1 W2=0.1

Linear function

o

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

1

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

o

Fill out the table

itr W1 W2 O

0 0.1 0.1 0.1

1 0.19 0.1 0.19

2 0.27 0.1 0.27

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

o

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?

Example: Estimating Π by Independent

Monte-Carlo SamplesSuppose we throw darts randomly (and uniformly) at the square:

Algorithm:For i=[1..ntrials] x = (random# in [0..r]) y = (random# in [0..r]) distance = sqrt (x^2 + y^2) if distance ≤ r hits++EndOutput:

Adapted from course slides by Craig Douglas

http://www.chem.unl.edu/zeng/joy/mclab/mcintro.html

Estimating P

Monte Carlo (Minimization)

dE<0dE>0

The Traveling Salesman

Adapted from www.mpp.mpg.de/~caldwell/ss11/ExtraTS.pdf

Adapted from www.mpp.mpg.de/~caldwell/ss11/ExtraTS.pdf

Adapted from www.mpp.mpg.de/~caldwell/ss11/ExtraTS.pdf

Adapted from www.mpp.mpg.de/~caldwell/ss11/ExtraTS.pdf

Adapted from www.mpp.mpg.de/~caldwell/ss11/ExtraTS.pdf

Adapted from www.mpp.mpg.de/~caldwell/ss11/ExtraTS.pdf

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

Stabilization matrix method

• 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

mpe

rature

Regression methods. The mathematics

y = ax + b2 parameter model

Good description, poor fit

y = ax6+bx5+cx4+dx3+ex2+fx+g

7 parameter modelPoor description, good fit

Model over-fitting

Stabilization matrix method (Ridge regression). The mathematics

y = ax + b2 parameter model

Good description, poor fit

y = ax6+bx5+cx4+dx3+ex2+fx+g

7 parameter modelPoor description, good fit

SMM training

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

Stabilization matrix method.The analytic solution

Each peptide is represented as 9*20 number (180)H is a stack of such vectors of 180 valuest is the target value (the measured binding)l is a parameter introduced to suppress the effect of noise in the experimental data and lower the effect of overfitting

SMM - Stabilization matrix method

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Linear function

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Sum over weights

Sum over data points

SMM - Stabilization matrix method

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Linear function

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Per target error:

Global error:

Sum over weights

Sum over data points

SMM - Stabilization matrix methodDo it yourself

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Linear function

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l per target

SMM - Stabilization matrix method

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Linear function

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l per target

SMM - Stabilization matrix method

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Linear function

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SMM - Stabilization matrix methodMonte Carlo

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Linear function

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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//home/user1/bin/doit_ex

#! /bin/tcsh foreach a ( `cat allelefile` )mkdir -p $cd $aforeach l ( 0 1 2.5 5 10 20 30 )mkdir -p l.$lcd l.$lforeach n ( 0 1 2 3 4 )smm -nc 500 -l $l train.$n > mat.$npep2score -mat mat.$n eval.$n > eval.$n.predendecho $a $l `cat eval.?.pred | grep -v "#" | gawk '{print $2,$3}' | xycorr`cd ..endcd ..end