EMO = evolutionary algorithms and ... - sop.tik.ee.ethz.ch · Computer Engineering and Networks Laboratory Two Decades of EMO: A Glance Back and A Look Ahead Eckart Zitzler IEEE Symposium

Computer EngineeringComputer Engineeringand and NetworksNetworks LaboratoryLaboratory

TwoTwo DecadesDecades of EMO:of EMO:A A GlanceGlance Back and A Look Back and A Look AheadAhead

Eckart ZitzlerEckart Zitzler

IEEE Symposium on CI in MCDMIEEE Symposium on CI in MCDM, 5 April 20075 April 2007

Two Decades of EMO 2© Eckart Zitzler ETH Zurich

Evolutionary Multi-Criterion Optimization (EMO)

Key issues:How to formalize what a goodPareto set approximation is?

How to search for a goodPareto set approximation?

How to use the informationprovided by an approximation?

f2

f1

EMO = evolutionary algorithms and other randomized search heuristics

... applied to problems involving multiple objectives (in general)

... used to approximate the Pareto-optimal set (mainly)


A Brief History of EMO Research

1984

1990

1995

2000

2007

first EMO approaches

dominance-based EMO algorithms with diversity preservation techniques

elitist EMO algorithms

quantitative performance assessment

attainment functions

EMO algorithms based on set quality measures

preference articulation convergence proofs

running time analyses quality measure designuncertainty and robustness

statistical performance assessment

test problem design

high-dimensional objective spaces

multiobjectivization

dominance-based population ranking


EMO: A Fast Growing Field

Statistics of the EMO repositorymaintained by C. A. Coello Coello

Overall: 2615 references by 11/2006

http://www.lania.mx/~ccoello/EMOO/EMOOstatistics.html


The EMO Community

The EMO conference series:

EMO2001 EMO2003 EMO2005 EMO2007Zurich Faro Guanajuato Matsushima

Switzerland Portugal Mexico Japan

45 / 87 56 / 100 59 / 115 65 / 124

Many further activities:special sessions, special issues, workshops, tutorials, ...


A Personal View: Four Main Lessons Learned

Lesson 1: EMO provides information about a problem(search space exploration)

Lesson 2: EMO can help in single-objective scenarios(multiobjectivization)

Lesson 3: EMO is part of the decision making process(preference articulation)

Lesson 4: EMO for large n is different from n = 2(high-dimensional objective spaces)














Lesson 1: EMO provides information about a problem

The question:Why at all should one try to approximate the entire Pareto-optimal set?

An answer:Because it provides useful information about the problem...

and

...we know how to do that for a small number of objectives!

ProblemProblem

DecisionMaking

DecisionMaking

EMOEMO

ModelModel

SolutionSolution


Note: good in terms of set quality = good in terms of search?

A General Scheme of a Dominance-Based MOEA

(archiv)population offspring

environmental selection (greedy heuristic)environmental selection (greedy heuristic)

mating selection (stochastic)mating selection (stochastic) fitness assignmentpartitioning into

dominance classes

rank refinement withindominance classes

fitness assignmentpartitioning into

dominance classes

rank refinement withindominance classes

+


Ranking of the Population Using Dominance

... goes back to a proposal by David Goldberg in 1989.

... is based on pairwise comparisons of the individuals only.

dominance rank: by howmany individuals is anindividual dominated?MOGA, NPGAdominance count: how manyindividuals does an individualdominate?SPEA, SPEA2dominance depth: at whichfront is an individual located?NSGA, NSGA-II

f2

f1

dominancecount

dominancerank

dominance depth


Refinement of Dominance Rankings

Goal: rank incomparable solutions within a dominance class

Density information (good for search)

Quality indicator (good for set quality): later...

ff

f

Kernel method

density =function of the

distances

k-th nearest neighbor

density =function of distance

to k-th neighbor

Histogram method

density =number of elements

within box


EMO and Population Synergies

One MOEA run can be more effective than aggregation + multiple runs

MOEAs

Theoretical evidence:[Laumanns et al. 2004]

Empirical evidence:

[Zitzler, Thiele 1999]


Application: Design Space Exploration

Cost

Latency Power

SpecificationSpecification OptimizationOptimization ImplementationImplementation

EnvironmentalSelection

EnvironmentalSelectionMutation

Mutation

x2

x1

f

MatingSelection

MatingSelectionRecombination

Recombination

EvaluationEvaluation


Application: Design Space Exploration

Cost

Latency Power

SpecificationSpecification OptimizationOptimization ImplementationImplementation

EnvironmentalSelection

EnvironmentalSelectionMutation

Mutation

x2

x1

f

MatingSelection

MatingSelectionRecombination

Recombination

EvaluationEvaluation

Water resourcemanagement[Siegfried et al. 2006]

Water resourcemanagement[Siegfried et al. 2006]


Application: Trade-Off Analysis

Module identification from biological data [Calonder et al. 2006]

Find group of genes w.r.t.different data types:

similarity of geneexpression profiles

overlap of proteininteraction partners

metabolic pathwaymap distances


Application: Approximation Set Analysis

Multiple disk clutch brake design [Deb, Srinivasan 2006]


Lesson 2: EMO Helps in Single-Objective Scenarios

Have seen: EMO can help in single-objective aggregation scenariosEven better: EMO can help in general in single-objective scenarios

Multiobjectivization [Knowles et al. 2001]:1. Transform a single-objective problem into a multiobjective one2. Solve it using an MOEA

Types of multiobjectivization:Approach I:

Decompose objective function into several functionsApproach II:

Leave problem as it is, but add further objective functions

(in principle combination possible)


Approach I: Decomposition Into Multiple Objectives

Empirical studies:

converting constraints into objectives[Coello 1999]transforming the H-IFF problem into a biobjective problem[Knowles et al. 2001]

Theoretical studies:

shortest path problem[Scharnow et al.2004]decomposing the spanning tree problem into two objectives[Neumann, Wegener 2005]

In all of the above studies, the MOEA outperformed its single-objective counterpart.


Approach II: Together Is Easier

Running time analysis results:[Brockhoff et al. 2007]

(1+1)-EA on both functions:

Θ(n3)

Simple MOEA on biobjectiveproblem:

Θ(n2logn)

Note: search space and Pareto-optimal set unchanged

Problem:


Approach II: Multiobjective Genetic Programming

Problem: trees grow rapidlypremature convergenceoverfitting of training data

Common methods:constraint(tree size limitation)penalty term(parsimony pressure) objective ranking(size post-optimization)

structure-based (ADF, etc.)

Multiobjective method:Optimize both error and size[Ekart, Nemeth 2001; deJong et al. 2001; Bleuler et al 2001]

Keep small trees(diversity)

error

tree size


Multiobjective GP: Results

Algorithm:SPEA2 without density estimation

Benchmark:even-parity problem

Runs

Siz

e (#

of e

dges

)

SPEA2

Runs

Siz

e (#

of e

dges

)

Constant Parsimony


Multiobjective GP: Why Does It Work?


Lesson 3: EMO is based on preferences

What we thought: EMO is preference-less

What we learnt: EMO just uses weaker preference information

⇒ (almost) all MOEAs implicitly implement specific preferences

[Zitzler 1999]

A

B

preferable?environmentalselection

3 out of 6


What is the Quality of a Pareto Set Approximation?

Problem: incomparability does not give a search directionNeeded: total ordering of the set of Pareto set approximations

One possibility: Quality indicators I: Ωm → ℜ

unary indicator: assign each approximation a real number I(A)binary indicator: assigns each approximation pair a real number I(A,B)

Example: unary indicators combined

AB

hypervolume 432.34 420.13distance 0.3308 0.4532diversity 0.3637 0.3463spread 0.3622 0.3601cardinality 6 5

A B

“A better”


Set Quality Measures: Examples

Unary (absolute)Hypervolume indicator

Binary (relative)Coverage indicator

I(A)A

B

A

I(A) = 60%I(A,B) = 25%I(B,A) = 75%


What Are Good Set Quality Measures?

There are three aspects [Zitzler et al. 2000]:

Wrong! [Zitzler et al. 2003]:

f2

f1

An infinite number of unary set measures is needed to detectin general whether A is better than B


Order Compliance + Strict Monotonicity

Order preserving: the preference is refined and not violated

Strictly monotonic: sensitive to Pareto dominance

⇒ Uniqueness of optimum: Pareto front achieves maximum value

Bad news: the hypervolume is currently the only known unary setmeasure with these properties

Good news: preferences can be integrated into the hypervolumeindicator [Zitzler et al. 2007]


Problems With Non-Compliant Indicators


Incorporation of Preferences During Search

Refine/modify dominance relation, e.g.:

using goals, priorities, constraints[Fonseca, Fleming 1998]using different types of cones[Branke 2000]

Use quality indicators, e.g.:

based on reference points [Deb, Sundar 2006]based on the hypervolume indicator (later)based on binary quality indicators (now)

f2

f1


Preference-Adaptive Search

Given:Preference information in terms of a binary quality indicator I,here binary epsilon indicator(≡ continuous extension of dominance relation)

Optimization goal:Find Pareto set approximation A such that

I(A, S)

is minimum (S = Pareto-optimal set)

Question:How to assign fitness values?


Indicator-Based Fitness Assignment

Idea: measure for “loss in quality” if x1 is removed

Fitness A:

...corresponds to continuous extension of dominance rank(MOGA, Fonseca & Fleming 1993)

...blurrs influence of dominating and dominated individuals

Fitness B:

... parameter κ is problem- and indicator-dependent

... no additional diversity preservation mechanism


Lesson 4: Two Objectives Are Not Many

Search:What are the effects of multiple objectives on the search spacestructure?How to guide the search towards the Pareto-optimal set (efficiently)?

Decision making:

Can objectives be omitted, is there redundancy in the set of objectives?Which are the most important objectives?

Most EMO publications focused on two or three objectives –what about many objectives?


What Happens If Objectives Are Added?

Here: graph-based representation of dominance relation

Adding objectives can only remove edges, i.e.,(i) comparable → incomparable; or indifferent → (in)comparable

4 separate objectives 4 objectives combined


The Effect of Adding Objectives

Observation:The number of incomparable solutions may increase with thenumber of objectives.

Winkler (1985): For random orders with n points and k dimensions, the exptected number of incomparable solutions is between

and

But: In general, adding objectives can have good and bad effects:

Neumann and Wegener (2006), Scharnow et al. (2002):Theoretical examples where more objectives helpBrockhoff et al. (2007):Adding an objective can have positive and/or negative effects


Third Objective Makes Problem Easier / Harder

0

500

1000

1500

2000

2500

3000

3500

4000

100 150 200 250 300 350 400 450 500

runt

ime

[gen

erat

ions

]

bitstring length

Making indifferent solutions comparable

LOTZ with additional function (i)2-dimensional LOTZ

LOTZ with additional function (ii)

Average runtimes for 10 IBEA runs with population size 200


The Problem of Cycling Behavior

Observation: current density-based EMO algorithms fail for n > 3[Wagner et al. 2007]

Example:20-objective problemSPEA2 for 1000 generations

⇒ all solutions visited during the run are incomparable

Explanation:cyclic behavior, i.e.,preference informationill-defined


Hypervolume ResearchEmpirical performance assessment:

(Zitzler, Thiele: 1998, 1999)many more...

Theoretical investigations of properties:(Knowles, Corne: 2002)(Fleischer: 2003)(Zitzler et al.: 2003, 2007)

Algorithm design:(Knowles et al.: 2003)(Zitzler, Künzli: 2004)(Emmerich et al.: 2005)(Igel et al.: 2007)

Computational issues:(While et al.: 2005, 2006)(Beume, Rudolph: 2006)(Fonseca et al.: 2006)

How to assign fitness values?

Fitness = loss in hypervolumeif individual is removed

How to assign fitness values?

Fitness = loss in hypervolumeif individual is removed

How to make the calculation fast?How to make the calculation fast?


Hypervolume-Based Search: Proof of Principle


Dimensionality Reduction

Key question: Are some objectives redundant or less important?

→ decision making easier→ search computationally less expensive

Objective reduction approaches:reduction based on correlation (PCA) [Deb, Saxena 2005]reduction based on ε-dominance [Brockhoff, Zitzler 2006]

2200

2400

2600

2800

3000

3200

3400

2019181716151413121110987654321

valu

es

objectives

2500

2600

2700

2800

2900

3000

3100

3200

1511851

valu

es

objectives

?


Objective Reduction: Example

values values

omit

still the same relations

Key question: Can objectives be omitted without loosing too much?

all objectives pairwisely conflicting


Approximation of efficient set computed by evolutionary algorithmused as

for and for

Objective reduction of 50% possible for various test problems

Dimensionality Reduction: What Is Possible?

Num

bero

f obj

ectiv

esin

min

imal

set

problemsDTLZ2 DTLZ5 DTLZ7 KP100 KP250 KP500

k=15k=25


So Far So Good – And Now?

My main conclusion: EMO is part of the decision making process

1. Throw everything in2. Run your EMO tool3. Analyze results and learn about the problem4. Refine problem / preferences in a guided or automated fashion5. Go to 2

Many further research topics:Uncertainty and robustnessExpensive objective function evaluationsHybridization: EMO and OR methodsMulti-multiobjective problems...

Documents

EMO = evolutionary algorithms and ... - sop.tik.ee.ethz.ch · Computer Engineering and Networks Laboratory Two Decades of EMO: A Glance Back and A Look Ahead Eckart Zitzler IEEE Symposium