optimization methods morten nielsen department of systems biology, dtu iib-intech, unsam, argentina

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

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)


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Linear function

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Gradient descent


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Linear function

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Gradient descent. Example


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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

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


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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


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Linear function

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l per target


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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

optimization methods morten nielsen department of systems biology, dtu iib-intech, unsam, argentina

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