managing director / cto nutech solutions gmbh / inc. martin-schmeißer-weg 15 d – 44227 dortmund

Managing Director / CTONuTech Solutions GmbH / Inc.

Martin-Schmeißer-Weg 15D – 44227 Dortmund

[email protected]

Tel.: +49 (0) 231 72 54 63-10Fax: +49 (0) 231 72 54 63-29

Thomas Bäck

UPP 2004

Le Mont Saint Michel, September 15, 2004

Problem Solving by Evolution: One of Nature’s UPPs

Full Professor for „Natural Computing“Leiden Institute for Advanced Computer Science (LIACS)Niels Bohrweg 1NL-2333 CA Leiden

[email protected]

Tel.: +31 (0) 71 527 7108Fax: +31 (0) 71 527 6985

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Pharminformatics OtherEC UPP of CAsApplicationsOverview

Overview Optimization and Evolutionary Computation

Genetic Algorithms, Evolution Strategies, Self-Adaptation

Convergence Velocity Theory

Applications: Some Examples

Applications: Programming of CA

Links to Bio- and Pharminformatics Drug Design

Classification

Evolutionary DNA-Computing

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Natural Computing

Computing Paradigms after Natural Models NN, EC, Simulated Annealing, Swarm & Ant Algorithms, DNA

Computing, Quantum Computing, CA, ...

Journals Journal of Natural Computing (Kluwer).

Theoretical Computer Science C (Elsevier).

Book Series on Natural Computing (Springer).

Leiden Center of Natural Computing (NL).

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Unifying Evolutionary Algorithm

t := 0;

initialize(P(t));

evaluate(P(t));

while not terminate do

P‘(t) := mating_selection(P(t));

P‘‘(t) := variation(P‘(t));

evaluate(P‘‘(t));

P(t+1) := environmental_selection(P‘‘(t) Q);

t := t+1;

od

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Real-valued representation

Normally distributed mutations

Fixed recombination rate (= 1)

Deterministic selection

Creation of offspring surplus

Self-adaptation of strategy

parameters:

Variance(s), Covariances

Binary representation

Fixed mutation rate pm (= 1/n)

Fixed crossover rate pc

Probabilistic selection

Identical population size

No self-adaptation

Genetic AlgorithmGenetic Algorithm Evolution StrategiesEvolution Strategies

Genetic Algorithms vs. Evolution Strategies

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Genetic Algorithms: Mutation

0 1 1 1 0 1 0 1 0 0 0 0 0 01

0 1 1 1 0 0 0 1 0 1 0 0 0 01

Mutation by bit inversion with probability pm.

pm identical for all bits.

pm small (e.g., pm = 1/n).

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Genetic Algorithms: Crossover

Crossover applied with probability pc.

pc identical for all individuals.

k-point crossover: k points chosen randomly.

Example: 2-point crossover.

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Genetic Algorithms: Selection Fitness proportional:

f fitness

population size

Tournament selection: Randomly select q << individuals.

Copy best of these q into next generation.

Repeat times.

q is the tournament size (often: q = 2).

1

)(

)(

jj

ii

af

afp

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Evolution Strategies: Mutation

-adaptation by means of

– 1/5-success rule.

– Self-adaptation.

)1,0(iii Nxx

Creation of a new solution:

Convergence speed:

Ca. 10 n down to 5 n is possible.

More complex / powerful strategies:

– Individual step sizes i.

– Covariances.

nx

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Self-Adaptation: Motivation: General search algorithm

Geometric convergence: Arbitrarily slow, if s wrongly controlled !

No deterministic / adaptive scheme for arbitrary functions exists.

Self-adaptation: On-line evolution of strategy parameters.

Various schemes: Schwefel one , n , covariances; Rechenberg MSA.

Ostermeier, Hansen: Derandomized, Covariance Matrix Adaptation.

EP variants (meta EP, Rmeta EP).

Bäck: Application to p in GAs.

tttt vsxx 1

Step size

Direction

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Evolution Strategies: Self-Adaptation

Learning while searching: Intelligent Method.

Different algorithmic approaches, e.g:

• Pure self-adaptation:

• Mutational step size control MSC:

• Derandomized step size adaptation

• Covariance adaptation

))1,0()1,0(exp( iii NN )1,0(iiii Nxx

2/1)1,0( if , /

2/1)1,0( if ,

Uu

Uu

)1,0(iiii Nxx

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Evolution Strategies: Self-Adaptive Mutation

n = 2, n = 1, n = 0

n = 2, n = 2, n = 0

n = 2, n = 2, n = 1

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Self-Adaptation: Dynamic Sphere

Optimum :

Transition time proportionate to n.

Optimum learned by self-adaptation.

n

Rcopt ,

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Evolution Strategies: Selection()

()

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Possible Selection Operators

(1+1)-strategy: one parent, one offspring.

(1,)-strategies: one parent, offspring.

• Example: (1,10)-strategy.

• Derandomized / self-adaptive / mutative step size control.

(,)-strategies: >1 parents,> offspring

• Example: (2,15)-strategy.

• Includes recombination.

• Can overcome local optima.

(+)-strategies: elitist strategies.

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Vision: Self-adaptive software

Self-adaptation is the ability of an algorithm to iteratively make the solution of a problem more likely.

Software that monitors its performance, improves itself, learns while it interacts with its user(s). [Robertson, Shrobie, Laddaga, 2001]

Self-adaptation in ES: Evolution of solutions and solution search algorithms.

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Robust vs. Fast Optimization:

Global convergence with probability one:

General, but for practical purposes useless.

Convergence velocity:

Local analysis only, specific functions only.

1))(Pr( *lim

tPxt

)))(())1((( maxmax tPftPfE

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GA Convergence Velocity Analysis:

(1+1)-GA, (1,)-GA, (1+)-GA.

For counting ones function:

Convergence velocity:

Mutation rate p, q = 1 – p, kmax = l – fa.

l

iiaaf

1

)(

jfljfl

ij

aifif

i

aa

a

a

k

k

a

a

a

a

qpj

flqp

i

fkp

kafamfkp

kpk

10

0)11(

)(

))())((Pr()(

)(max

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Convergence Velocity Analysis:

Optimum mutation rate ?

Absorption times from transition matrix

in block form, using where

llafp

1

)1)((2

1*

QR

IP

0

Tj

iji ntE )(

1)()( QInN ij

20



p too large:

Exponential

p too small:

Almost constant.

Optimal: O(l ln l) .

p

21



(1,)-GA (kmin = -fa), (1+)-GA (kmin = 0) :

ikk

ikk

i

fl

kk

ppi

ka

''

1)1(

min,

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(1,)-GA, (1+)-GA: (1,)-ES, (1+)-ES:

Conclusion: Unifying, search-space independent theory !?

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(,)-GA (kmin = -fa), (+)-GA (kmin = 0) :

Theory

Experiment

)(1

1)(

min,

kpkafl

kk

jkk

ikk

jikk

ji

ppp

j

i

ikp

'1

'1

'

0

1

1

)1(

1)(

24


Convergence Velocity for Bimodal Function:

A generalized Trap Function (u = number of ones):

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Transition Probabilities for Bimodal Function:

Probability to mutate u1 ones into u2 ones:

Probability that one step of the algorithm changes parent (u1 -> u2):

)())()(()( 10

212120

21 uupuupuupuupi

iii

i

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Convergence Velocity for Bimodal Function:Convergence velocity:

)'())()'(()()()'(,'

uupufufuufufDu

(1+1), z2=100, current position varies (5,20,...).

(1+), z2=100, position 20, lambda varies (1,2,...).

(1+), z2=100, position 35, lambda varies (1,2,...).

Global max. Jump to local max.

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Convergence Velocity for Bimodal Function: New Algorithm: Several mutation rates.

Expands theory to all counting ones functions (including moving ones).

Optimal lower mutation rate: 1/l.

Currently further analyzed / tested on NP-complete problems.

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Optimization Problem:

f: Objective function, can be

Multimodal, with many local optima

Discontinuous

Stochastically perturbed

High-dimensional

Varying over time.

can be heterogenous.

Constraints can be defined over

min)(,: xfMf

nMMMM ...21

)(, xfM

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Optimization Algorithms: Direct optimization algorithm:

Evolutionary Algorithms

First order optimization algorithm:

e.g, gradient method

Second order optimization algorithm:

e.g., Newton method )(xf

)(),( xfxf

)(),(),( 2 xfxfxf

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Applications: General Aspects

Evaluation

EA-Optimizer

Business Process Model

Simulation

215

1

i

iii

i scale

desiredcalculatedweightquality

Function Model from Data

Experiment SubjectiveFunction(s)

...)( yfi

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Dielectric filter design (40-dimensional).

• Quality improvement by factor 2.

Car safety optimization (10-30 dim.)

• 10% improvement.

Traffic control (elevators, planes, cars)

• 3-10% improvement.

Telecommunication

Metal stamping

Nuclear reactors,...

Overview of Examples

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Unconventional Programming ?

„Normal“ EA application:

EA as Programming Paradigm:

EAEA Other AlgorithmOther Algorithm TaskTask

EAEA TaskTask

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UP of CAs (= Inverse Design of CAs)

1D CAs: Earlier work by Mitchell et al., Koza, ...

Transition rule: Assigns each neighborhood configuration a new state.

One rule can be expressed by bits.

There are rules for a binary 1D CA.

1 0 0 0 0 1 1 0 1 0 1 0 1 0 0

Neighborhood(radius r = 2)

1,01,0: 12 r

122 r

1222r

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UP of CAs (rule encoding)

Assume r=1: Rule length is 8 bits

Corresponding neighborhoods

1 0 0 0 0 1 1 0

000 001 010 011 100 101 110 111

35


Inverse Design of CAs: 1D

Time evolution diagram:

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Majority problem:

Particle-based rules.

Fitness values:

0.76, 0.75, 0.76, 0.73

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Don‘t care about initial state rules

Block expanding rules

Particle communication based rules

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Inverse Design of CAs: 1D Majority Records

Gacs, Kurdyumov, Levin 1978 (hand-written): 81.6%

Davis 1995 (hand-written): 81.8%

Das 1995 (hand-written): 82.178%

David, Forrest, Koza 1996 (GP): 82.326%

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Inverse Design of Cas: 2D

Generalization to 2D (nD) CAs ?

Von Neumann vs. Moore neighborhood (r = 1)

Generalization to r > 1 possible (straightforward)

Search space size for a GA: vs.

10

0

1

1 10

0

1

1

0

0

1

1

52 92

522922

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Inverse Design of CAs

Learning an AND rule.

Input boxes are defined.

Some evolution plots:

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Learning an XOR rule.

Input boxes are defined.

Some evolution plots:

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Learning the majority task.

84/169 in a), 85/169 in b).

Fitness value: 0.715

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Learning pattern compression tasks.

44


Current Drug Targets:

2%2%

11%

5% 7%

45%

28%

receptors enzymeshormones & factors DNAnuclear receptors ion channelsunknown

GPCR

http://www.gpcr.org/

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Goals (in Cooperation with LACDR): CI Methods:

Automatic knowledge extraction from biological databases.

Automatic optimisation of structures – evolution strategies.

Exploration for Drug Discovery,

De novo Drug Design.

Initialisation

Final (optimized)

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Clustering GPCRs: New Ways

SOM based on sequence homology, family clusters marked.

Overlay with phylogenetic (sub-)tree.

Class A amine dopamine trace amine peptide angiotensin chemokine CC other melanocortin viral (rhod)opsin vertebrate other unclassifiedClass B corticotropic releasing factor

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Evolutionary DNA-Computing (with IMB): DNA-Molecule = Solution candidate !

Potential Advantage: > 1012 candidate solutions in parallel.

Biological operators: Cutting, Splicing.

Ligating.

Amplification.

Mutation.

Current approaches very limited.

Our approach: Suitable NP-complete problem.

Modern technology.

Scalability (n > 30).

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Evolutionary DNA-Computing: Example: Maximum Clique Problem

Problem Instance: Graph

Feasible Solution: V‘ such that

Objective Function: Size |V‘| of clique V‘

Optimal Solution: Clique V‘ that maximizes |V‘| .

Example:

VVEnVEVG ;,...,2,1;),(

1:', ijeVji

1 2 3 4 5

6 7 8

{2,3,6,7}: Maximum Clique (01100110)

{4,5,8}: Clique. (00011001)

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DNA-Computing: Classical Approach

1: X := randomly generate DNA strands representing all candidates;

2: Remove the set Y of all non-cliques from X: C = X – Y;

3: Identify with smallest length (largest clique);Cx *

Based on filtering out the optimal solution.

Fails for large n (exponential growth).

Applied in the lab for n=6 (Ouyang et el., 1997); limited to n=36 (nanomole operations).

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DNA-Computing: Evolutionary Approach1: Generate an initial random population P, ;

2: while not terminate do

3: P := amplify and mutate P;

4: Remove the set Y of all non-cliques from P: P := P - Y;

5: P‘ := select shortest DNA strands from P;

6: od

Based on evolving an (near-) optimal solution.

Also applicable for large n.

Currently tested in the lab (Leiden, IMB).

nP 2

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Scalability Issues:Maximum Clique: Simulation Results (1,)-GA (best of 10)Problem n =10 100 1000 10000 Opt.

brock200_1 200 14 17 17 19 21

brock200_2 200 6 9 8 9 12

brock200_3 200 10 11 12 12 15

brock200_4 200 11 12 13 14 17

hamming8-4 256 --- 12 12 16 16

p_hat300-1 300 --- 6 7 7 8

p_hat300-2 300 --- 19 19 20 25

Averages (not shown here) confirm trends.

Theory for large (NOT infinite) population sizes (other than c) ?

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Questions ?

Thank you very much for your time !

managing director / cto nutech solutions gmbh / inc. martin-schmeißer-weg 15 d – 44227 dortmund

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