Benchmarking the SMS-EMOA with Self-adaptation on the bbob ... · All55Functions 2-D 5-D 0 1 2 3 4...

Benchmarking the SMS-EMOA withSelf-adaptation on the bbob-biobj Test Suite

Simon Wessing

Chair of Algorithm EngineeringComputer Science Department

Technische Universität Dortmund

16 July 2017

Introduction

I Evolutionary multiobjective optimizationI Continuous decision variablesI (1 + 1)-SMS-EMOA is algorithmically equivalent to

single-objective (1 + 1)-EA⇒ Theory about optimal step size from single-objective

optimization applies

I Situation for (µ+ 1), (µ+ λ) unknownI How to define step size optimality?I How to adapt step size if not with very sophisticated

MO-CMA-ES?

Benchmarking the SMS-EMOA with Self-adaptation 2 / 18

Introduction

I Evolutionary multiobjective optimizationI Continuous decision variablesI (1 + 1)-SMS-EMOA is algorithmically equivalent to

single-objective (1 + 1)-EA⇒ Theory about optimal step size from single-objective

optimization applies

I Situation for (µ+ 1), (µ+ λ) unknownI How to define step size optimality?I How to adapt step size if not with very sophisticated

MO-CMA-ES?


Development of Control Mechanism

I Idea: use self-adaptation from single-objective optimization

I Mutation of genome: y = x + σN (0, I)I Mutation of step size: σ = σ̃ · exp(τN (0, 1))

I Learning parameter τ ∝ 1/√

n

I Not state of the art any moreI Behavior is emergentI Theoretical analysis is difficultI Application to multiobjective optimization is scarce⇒ Experiment to find good parameter configurations



I Idea: use self-adaptation from single-objective optimizationI Mutation of genome: y = x + σN (0, I)I Mutation of step size: σ = σ̃ · exp(τN (0, 1))


n




I Idea: use self-adaptation from single-objective optimizationI Mutation of genome: y = x + σN (0, I)I Mutation of step size: σ = σ̃ · exp(τN (0, 1))


n



Experimental SetupFactor Type Symbol Levels

Number variables observable n {2, 3, 5, 10, 20}Learning param.constant

control c {2−2, 2−1, 20, 21, 22, 23}

Population size control µ {10, 50}Number offspring control λ {1, µ, 5µ}Recombination control {discrete,

intermediate,arithmetic, none}

I Full factorial designI 15 unimodal problems of BBOB-BIOBJ 2016

(only first instance)I Budget: 104n function evaluationsI Assessment: rank-transformed HV values of whole EA runs


Other Factors Held Constant

I Initial mutation strength σinit = 0.025I Repair method for bound violations: Lamarckian reflection

(search space [−100, 100]n, scaled to unit hypercube)I Selection: iteratively removes worst individual, until µ reached

(backward elimination)

⇒ Might have to reconsider in the future


PseudocodeInput: population size µ, initial population P0, number of

offspring λ1: t ← 02: while stopping criterion not fulfilled do3: Ot ← createOffspring(Pt) // create λ offspring4: evaluate(Ot) // calculate objective values5: Qt ← Pt ∪ Ot6: r ← createReferencePoint(Qt)7: while |Qt | > µ do8: {F1, . . . ,Fw} ← nondominatedSort(Qt) // sort in fronts9: x∗ ← argminx∈Fw (∆s(x,Fw , r)) // x∗ with smallest contr.

10: Qt ← Qt \ {x∗} // remove worst individual11: end while12: Pt+1 ← Qt13: t ← t + 114: end while


Main Effect: Learning Parameters τ = c/√

n

c = 2−2 c = 2−1 c = 20 c = 21 c = 22 c = 230

20

40

60

80

100

120

140

Ave

rage

Ran

k

I c = 2−2 is always the worst choice⇒ Exclude c = 2−2 from further analysis


Mutation Strength vs. Generation

100 101 102 103

Generation

10−5

10−4

10−3

10−2

10−1

100A

vg.

step

size

σ̄

(a) τ = 2−2/√

n.

100 101 102 103

Generation

10−5

10−4

10−3

10−2

10−1

100

Avg

.st

epsi

zeσ̄

(b) τ = 20/√

n.

100 101 102 103

Generation

10−5

10−4

10−3

10−2

10−1

100

Avg

.st

epsi

zeσ̄

(c) τ = 22/√

n.

100 101 102 103

Generation

10−5

10−4

10−3

10−2

10−1

100

Avg

.st

epsi

zeσ̄

(d) τ = 23/√

n.Benchmarking the SMS-EMOA with Self-adaptation 8 / 18

Main Effect: Selection Variants

(10 + 1) (10 + 10) (10 + 50) (50 + 1) (50 + 50)(50 + 250)

20

40

60

80

100

Ave

rage

Ran

k


Main and Interaction Effects: Recombination & Selection

arithmetic discrete intermediate none

20

40

60

80

100

Ave

rage

Ran

k

arithmetic discrete intermediate none(10 + 1) 46.97 85.43 82.53 78.95(10 + 10) 51.29 72.55 83.48 68.34(10 + 50) 47.69 62.90 82.25 42.50(50 + 1) 61.93 63.21 84.93 40.95(50 + 50) 58.23 55.88 84.06 30.43(50 + 250) 53.77 51.34 78.82 27.14


Interaction Effect: Learning Parameter vs. Recombination

arithmetic discrete intermediate none

2−1/√n 49.96 66.60 79.90 40.82

20/√n 57.01 53.97 83.87 44.49

21/√n 55.65 65.43 82.33 52.42

22/√n 48.70 66.57 80.38 50.98

23/√n 55.25 73.53 86.90 51.54


Comparison with (50+ 250) SBX on bbob-biobj 2016

0 1 2 3 4 5 6 7 8log10 of (# f-evals / dimension)

0.0

0.2

0.4

0.6

0.8

1.0Pro

port

ion o

f fu

nct

ion+

targ

et

pair

s

SBX

ESbbob-biobj - f1-f55, 2-D5, 5 instances

0.0.0


0.0

0.2

0.4

0.6

0.8

1.0

Pro

port

ion o

f fu

nct

ion+

targ

et

pair

s

SBX


0.0.0


0.0

0.2

0.4

0.6

0.8

1.0

Pro

port

ion o

f fu

nct

ion+

targ

et

pair

s

SBX


0.0.0


0.0

0.2

0.4

0.6

0.8

1.0

Pro

port

ion o

f fu

nct

ion+

targ

et

pair

s

SBX


0.0.0


Comparison with (50+ 250) SBX on bbob-biobj 2016


0.0

0.2

0.4

0.6

0.8

1.0

Pro

port

ion o

f fu

nct

ion+

targ

et

pair

s

ES

SBXbbob-biobj - f11, 5-D5, 5 instances

0.0.0

11 sep. Ellipsoid/sep. Ellipsoid


0.0

0.2

0.4

0.6

0.8

1.0

Pro

port

ion o

f fu

nct

ion+

targ

et

pair

s

SBX

ESbbob-biobj - f18, 3-D5, 5 instances

0.0.0

18 sep. Ellipsoid/Schwefel

I SBX is better/competitive on separable problems


Discussion

I Self-adaptive step size adaptation works in both directions(increasing/decreasing)

I Best configuration for budget of 104n:I No recombinationI τ = 20/

√n

I (50 + 250)-selectionI Surprisingly similar to single-objective caseI Only arithmetic and no recombination seem to be worth

investigating further


Application to bbob-biobj 2017

Modifications to previous experiments:I Initialization in [0.475, 0.525]n (normalized), corresponding to

[−5, 5]n in original problem spaceI Budget of 105nI Comparison to (µ+ 1)-SMS-EMOA from bbob-biobj 2016

I DE variationI SBX/PM variation


Some Results 5-Dseparable-separable separable-moderate


0.0

0.2

0.4

0.6

0.8

1.0P

ropo

rtion

of f

unct

ion+

targ

et p

airs

SMS-DE

SMS-PM

SMS-ES

best 2016bbob-biobj - f1, f2, f11, 5-D58 targets in 1..-1.0e-410 instances

v2.1, hv-hash=ff0e71e8cd978373


0.0

0.2

0.4

0.6

0.8

1.0

Pro

porti

on o

f fun

ctio

n+ta

rget

pai

rs

SMS-PM

SMS-DE

SMS-ES

best 2016bbob-biobj - f3, f4, f12, f13, 5-D58 targets in 1..-1.0e-410 instances


multimodal-multimodal multimodal-weakstructure


0.0

0.2

0.4

0.6

0.8

1.0

Pro

porti

on o

f fun

ctio

n+ta

rget

pai

rs

SMS-ES

SMS-PM

SMS-DE

best 2016bbob-biobj - f46, f47, f50, 5-D58 targets in 1..-1.0e-410 instances



0.0

0.2

0.4

0.6

0.8

1.0

Pro

porti

on o

f fun

ctio

n+ta

rget

pai

rs

SMS-ES

SMS-PM

SMS-DE

best 2016bbob-biobj - f48, f49, f51, f52, 5-D58 targets in 1..-1.0e-410 instances



All 55 Functions2-D 5-D


0.0

0.2

0.4

0.6

0.8

1.0P

ropo

rtion

of f

unct

ion+

targ

et p

airs

SMS-PM

SMS-DE

SMS-ES

best 2016bbob-biobj - f1-f55, 2-D58 targets in 1..-1.0e-410 instances



0.0

0.2

0.4

0.6

0.8

1.0

Pro

porti

on o

f fun

ctio

n+ta

rget

pai

rs

SMS-PM

SMS-DE

SMS-ES



10-D 20-D


0.0

0.2

0.4

0.6

0.8

1.0

Pro

porti

on o

f fun

ctio

n+ta

rget

pai

rs

SMS-PM

SMS-DE

SMS-ES




0.0

0.2

0.4

0.6

0.8

1.0

Pro

porti

on o

f fun

ctio

n+ta

rget

pai

rs

SMS-PM

SMS-DE

SMS-ES




Conclusions and OutlookConclusions:

I Self-adaptive variation better than SBX in all testeddimensions, also on multimodal problems

I But not better than DE on multimodal problemsI Not a good anytime algorithmI Restarts?

Outlook:I Separate step size for each decision variable?I Exploit knowledge that dominated solutions need higher

mutation strength?I More sophisticated recombination variants?I Does variation interact with backward/forward greedy

selection?


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Benchmarking the SMS-EMOA with Self-adaptation on the bbob ... · All55Functions 2-D 5-D 0 1 2 3 4...

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