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Introduction and Preliminaries Research Directions Fitness-Level Method Drift Analysis
Introduction to the Complexity Analysis ofRandomized Search Heuristics
Dirk Sudholt
CERCIA, University of Birmingham
ThRaSH 2011
Dirk Sudholt Introduction to the Complexity Analysis of Randomized Search Heuristics 1 / 27
Introduction and Preliminaries Research Directions Fitness-Level Method Drift Analysis
Overview
1 Introduction and Preliminaries
2 Research Directions
3 Fitness-Level Method
4 Drift Analysis
Dirk Sudholt Introduction to the Complexity Analysis of Randomized Search Heuristics 2 / 27
Introduction and Preliminaries Research Directions Fitness-Level Method Drift Analysis
Randomized Search Heuristics
Metaheuristics
evolutionary algorithms
simulated annealing
swarm intelligence
artificial immune systems
. . .
Benefits
applicable when problem is not well understood (black-box setting)
lack of time, money, or expertise to design a tailored algorithm
usually easy to implement and easy to apply
robust and often surprisingly successful
Dirk Sudholt Introduction to the Complexity Analysis of Randomized Search Heuristics 3 / 27
Introduction and Preliminaries Research Directions Fitness-Level Method Drift Analysis
Scheme of an Evolutionary Algorithm
Select parents for reproduction
Mutation/Recombination
Selection for new population
Dirk Sudholt Introduction to the Complexity Analysis of Randomized Search Heuristics 4 / 27
Introduction and Preliminaries Research Directions Fitness-Level Method Drift Analysis
Motivation
Goals
understand how metaheuristics work
get to know their capabilities and limitations
solid theoretical foundation
design better metaheuristics
What we are looking for
Bounds on the (expected) time until a metaheuristic finds a globaloptimum for a given problem.
Notion of “time”
number of evaluations of the objective function
number of iterations / generations
Dirk Sudholt Introduction to the Complexity Analysis of Randomized Search Heuristics 5 / 27
Introduction and Preliminaries Research Directions Fitness-Level Method Drift Analysis
Motivation
Goals
understand how metaheuristics work
get to know their capabilities and limitations
solid theoretical foundation
design better metaheuristics
What we are looking for
Bounds on the (expected) time until a metaheuristic finds a globaloptimum for a given problem.
Notion of “time”
number of evaluations of the objective function
number of iterations / generations
Dirk Sudholt Introduction to the Complexity Analysis of Randomized Search Heuristics 5 / 27
Introduction and Preliminaries Research Directions Fitness-Level Method Drift Analysis
Motivation
Goals
understand how metaheuristics work
get to know their capabilities and limitations
solid theoretical foundation
design better metaheuristics
What we are looking for
Bounds on the (expected) time until a metaheuristic finds a globaloptimum for a given problem.
Notion of “time”
number of evaluations of the objective function
number of iterations / generations
Dirk Sudholt Introduction to the Complexity Analysis of Randomized Search Heuristics 5 / 27
Introduction and Preliminaries Research Directions Fitness-Level Method Drift Analysis
Approach
Tools from the analysis of randomized algorithms
tail inequalities (Markov, Chernoff, . . . )
Markov chain theory
random walks, stochastic processes
asymptotic notation
amortized analysis
. . .
Challenge
Metaheuristics often not designed to support an analysis
Perspective
Classical algorithms theory: problem −→ algorithms
Randomized search heuristics: algorithm (paradigm) −→ problems
Dirk Sudholt Introduction to the Complexity Analysis of Randomized Search Heuristics 6 / 27
Introduction and Preliminaries Research Directions Fitness-Level Method Drift Analysis
Approach
Tools from the analysis of randomized algorithms
tail inequalities (Markov, Chernoff, . . . )
Markov chain theory
random walks, stochastic processes
asymptotic notation
amortized analysis
. . .
Challenge
Metaheuristics often not designed to support an analysis
Perspective
Classical algorithms theory: problem −→ algorithms
Randomized search heuristics: algorithm (paradigm) −→ problems
Dirk Sudholt Introduction to the Complexity Analysis of Randomized Search Heuristics 6 / 27
Introduction and Preliminaries Research Directions Fitness-Level Method Drift Analysis
Approach
Tools from the analysis of randomized algorithms
tail inequalities (Markov, Chernoff, . . . )
Markov chain theory
random walks, stochastic processes
asymptotic notation
amortized analysis
. . .
Challenge
Metaheuristics often not designed to support an analysis
Perspective
Classical algorithms theory: problem −→ algorithms
Randomized search heuristics: algorithm (paradigm) −→ problems
Dirk Sudholt Introduction to the Complexity Analysis of Randomized Search Heuristics 6 / 27
Introduction and Preliminaries Research Directions Fitness-Level Method Drift Analysis
The (1+1) Evolutionary Algorithm
(1+1) EA for maximization of f : 0, 1n → R
Choose x ∈ 0, 1n uniformly at random.repeat forever
Create y by flipping each bit in x independently with probability 1/n.if f (y) ≥ f (x) then x := y .
Properties:
“population” of size 1, no crossover
stochastic hill-climber
still reflects basic principle of mutation and selection
Dirk Sudholt Introduction to the Complexity Analysis of Randomized Search Heuristics 7 / 27
Introduction and Preliminaries Research Directions Fitness-Level Method Drift Analysis
The (1+1) Evolutionary Algorithm
(1+1) EA for maximization of f : 0, 1n → R
Choose x ∈ 0, 1n uniformly at random.repeat forever
Create y by flipping each bit in x independently with probability 1/n.if f (y) ≥ f (x) then x := y .
Properties:
“population” of size 1, no crossover
stochastic hill-climber
still reflects basic principle of mutation and selection
Dirk Sudholt Introduction to the Complexity Analysis of Randomized Search Heuristics 7 / 27
Introduction and Preliminaries Research Directions Fitness-Level Method Drift Analysis
Overview
1 Introduction and Preliminaries
2 Research Directions
3 Fitness-Level Method
4 Drift Analysis
Dirk Sudholt Introduction to the Complexity Analysis of Randomized Search Heuristics 8 / 27
Introduction and Preliminaries Research Directions Fitness-Level Method Drift Analysis
Runtime Analyses
Design aspects
parent populations
offspring populations
crossover vs. mutation
population diversity
operator bias
coping with obstacles: paths, plateaus, multimodality, . . .
. . .
Areas
multiobjective optimization
hybridization
parallelization
stochastic optimization
Dirk Sudholt Introduction to the Complexity Analysis of Randomized Search Heuristics 9 / 27
Introduction and Preliminaries Research Directions Fitness-Level Method Drift Analysis
One Size Fits All. . .
Evolutionary algorithms like the simple (1+1) EA . . .
. . . solve in expected poly-time
sorting (maximize sortedness) [Scharnow, Tinnefeld, Wegener, 2004]
shortest paths [Scharnow, Tinnefeld, Wegener, 2004]
minimum spanning trees [Neumann and Wegener, 2007]
Matroid optimization [Reichel and Skutella, 2007]
Eulerian cycles [Neumann, 2008 and follow-up work]
. . . are poly-time randomized approximation schemes
maximum matchings [Giel and Wegener, 2003]
PARTITION/makespan scheduling [Witt, 2005]
multiobjective shortest paths [Horoba, 2010]
Dirk Sudholt Introduction to the Complexity Analysis of Randomized Search Heuristics 10 / 27
Introduction and Preliminaries Research Directions Fitness-Level Method Drift Analysis
One Size Fits All. . .
Evolutionary algorithms like the simple (1+1) EA . . .
. . . solve in expected poly-time
sorting (maximize sortedness) [Scharnow, Tinnefeld, Wegener, 2004]
shortest paths [Scharnow, Tinnefeld, Wegener, 2004]
minimum spanning trees [Neumann and Wegener, 2007]
Matroid optimization [Reichel and Skutella, 2007]
Eulerian cycles [Neumann, 2008 and follow-up work]
. . . are poly-time randomized approximation schemes
maximum matchings [Giel and Wegener, 2003]
PARTITION/makespan scheduling [Witt, 2005]
multiobjective shortest paths [Horoba, 2010]
Dirk Sudholt Introduction to the Complexity Analysis of Randomized Search Heuristics 10 / 27
Introduction and Preliminaries Research Directions Fitness-Level Method Drift Analysis
One Size Fits All. . .
Evolutionary algorithms like the simple (1+1) EA . . .
. . . solve in expected poly-time
sorting (maximize sortedness) [Scharnow, Tinnefeld, Wegener, 2004]
shortest paths [Scharnow, Tinnefeld, Wegener, 2004]
minimum spanning trees [Neumann and Wegener, 2007]
Matroid optimization [Reichel and Skutella, 2007]
Eulerian cycles [Neumann, 2008 and follow-up work]
. . . are poly-time randomized approximation schemes
maximum matchings [Giel and Wegener, 2003]
PARTITION/makespan scheduling [Witt, 2005]
multiobjective shortest paths [Horoba, 2010]
Dirk Sudholt Introduction to the Complexity Analysis of Randomized Search Heuristics 10 / 27
Introduction and Preliminaries Research Directions Fitness-Level Method Drift Analysis
One Size Fits All (continued)
EAs can mimic behavior of
tailored algorithms
dynamic programming algorithms [Doerr, Eremeev, Horoba,Neumann, Theile, 2009]
greedy algorithms
fixed-parameter tractable algorithms [Kratsch and Neumann, 2009]
Dirk Sudholt Introduction to the Complexity Analysis of Randomized Search Heuristics 11 / 27
Introduction and Preliminaries Research Directions Fitness-Level Method Drift Analysis
Overview
1 Introduction and Preliminaries
2 Research Directions
3 Fitness-Level Method
4 Drift Analysis
Dirk Sudholt Introduction to the Complexity Analysis of Randomized Search Heuristics 12 / 27
Introduction and Preliminaries Research Directions Fitness-Level Method Drift Analysis
Fitness-level Method for the (1+1) EA
A7
A6
A5
A4
A3
A2
A1
fitn
essPr((1+1) EA leaves Ai ) ≥ si
Expected optimization time of (1+1) EA at mostm−1∑i=1
1si
.
Dirk Sudholt Introduction to the Complexity Analysis of Randomized Search Heuristics 13 / 27
Introduction and Preliminaries Research Directions Fitness-Level Method Drift Analysis
Fitness-Level Method: Example
OneMax (x) :=∑n
i=1 xi
Ai := x | OneMax(x) = i.
si ≥ (n − i) · 1
n·(
1− 1
n
)n−1
≥ n − i
en
Bound on the expected optimization time of (1+1) EA
n−1∑i=0
en
n − i= en
n∑i=1
1
i≤ en ln(en)
Dirk Sudholt Introduction to the Complexity Analysis of Randomized Search Heuristics 14 / 27
Introduction and Preliminaries Research Directions Fitness-Level Method Drift Analysis
Fitness-Level Method: Example
OneMax (x) :=∑n
i=1 xi
Ai := x | OneMax(x) = i.
si ≥ (n − i) · 1
n·(
1− 1
n
)n−1
≥ n − i
en
Bound on the expected optimization time of (1+1) EA
n−1∑i=0
en
n − i= en
n∑i=1
1
i≤ en ln(en)
Dirk Sudholt Introduction to the Complexity Analysis of Randomized Search Heuristics 14 / 27
Introduction and Preliminaries Research Directions Fitness-Level Method Drift Analysis
Fitness-Level Method: Example
OneMax (x) :=∑n
i=1 xi
Ai := x | OneMax(x) = i.
si ≥ (n − i) · 1
n·(
1− 1
n
)n−1
≥ n − i
en
Bound on the expected optimization time of (1+1) EA
n−1∑i=0
en
n − i= en
n∑i=1
1
i≤ en ln(en)
Dirk Sudholt Introduction to the Complexity Analysis of Randomized Search Heuristics 14 / 27
Introduction and Preliminaries Research Directions Fitness-Level Method Drift Analysis
Fitness-Level Method: Example
OneMax (x) :=∑n
i=1 xi
Ai := x | OneMax(x) = i.
si ≥ (n − i) · 1
n·(
1− 1
n
)n−1
≥ n − i
en
Bound on the expected optimization time of (1+1) EA
n−1∑i=0
en
n − i= en
n∑i=1
1
i≤ en ln(en)
Dirk Sudholt Introduction to the Complexity Analysis of Randomized Search Heuristics 14 / 27
Introduction and Preliminaries Research Directions Fitness-Level Method Drift Analysis
(1+1) EA for Minimum Spanning Trees
Given a weighted graph G = (V ,E ,w) with n := |V |,m := |E |.x ∈ 0, 1m encodes selection of edges.
f (x) := (#components(x)− 1) · n3wmax+nwmax
∣∣∣∣∣n − 1−m∑i=1
xi
∣∣∣∣∣+m∑i=1
wixi
Theorem (Neumann and Wegener, 2007)
The expected time until the (1+1) EA constructs a minimum spanningtree is O(m2(log n + log wmax)).
Analysis of Typical Runs
Phase 1: Find some connected graph O(m log m).Phase 2: Find some spanning tree O(m log m).Phase 3: Find a minimum spanning tree by suitable 2-bit flips withguaranteed average weight decrease.
Dirk Sudholt Introduction to the Complexity Analysis of Randomized Search Heuristics 15 / 27
Introduction and Preliminaries Research Directions Fitness-Level Method Drift Analysis
(1+1) EA for Minimum Spanning Trees
Given a weighted graph G = (V ,E ,w) with n := |V |,m := |E |.x ∈ 0, 1m encodes selection of edges.
f (x) := (#components(x)− 1) · n3wmax+nwmax
∣∣∣∣∣n − 1−m∑i=1
xi
∣∣∣∣∣+m∑i=1
wixi
Theorem (Neumann and Wegener, 2007)
The expected time until the (1+1) EA constructs a minimum spanningtree is O(m2(log n + log wmax)).
Analysis of Typical Runs
Phase 1: Find some connected graph O(m log m).Phase 2: Find some spanning tree O(m log m).Phase 3: Find a minimum spanning tree by suitable 2-bit flips withguaranteed average weight decrease.
Dirk Sudholt Introduction to the Complexity Analysis of Randomized Search Heuristics 15 / 27
Introduction and Preliminaries Research Directions Fitness-Level Method Drift Analysis
(1+1) EA for Minimum Spanning Trees
Given a weighted graph G = (V ,E ,w) with n := |V |,m := |E |.x ∈ 0, 1m encodes selection of edges.
f (x) := (#components(x)− 1) · n3wmax+nwmax
∣∣∣∣∣n − 1−m∑i=1
xi
∣∣∣∣∣+m∑i=1
wixi
Theorem (Neumann and Wegener, 2007)
The expected time until the (1+1) EA constructs a minimum spanningtree is O(m2(log n + log wmax)).
Analysis of Typical Runs
Phase 1: Find some connected graph O(m log m).Phase 2: Find some spanning tree O(m log m).Phase 3: Find a minimum spanning tree by suitable 2-bit flips withguaranteed average weight decrease.
Dirk Sudholt Introduction to the Complexity Analysis of Randomized Search Heuristics 15 / 27
Introduction and Preliminaries Research Directions Fitness-Level Method Drift Analysis
(1+1) EA for Minimum Spanning Trees
Given a weighted graph G = (V ,E ,w) with n := |V |,m := |E |.x ∈ 0, 1m encodes selection of edges.
f (x) := (#components(x)− 1) · n3wmax+nwmax
∣∣∣∣∣n − 1−m∑i=1
xi
∣∣∣∣∣+m∑i=1
wixi
Theorem (Neumann and Wegener, 2007)
The expected time until the (1+1) EA constructs a minimum spanningtree is O(m2(log n + log wmax)).
Analysis of Typical Runs
Phase 1: Find some connected graph O(m log m).
Phase 2: Find some spanning tree O(m log m).Phase 3: Find a minimum spanning tree by suitable 2-bit flips withguaranteed average weight decrease.
Dirk Sudholt Introduction to the Complexity Analysis of Randomized Search Heuristics 15 / 27
Introduction and Preliminaries Research Directions Fitness-Level Method Drift Analysis
(1+1) EA for Minimum Spanning Trees
Given a weighted graph G = (V ,E ,w) with n := |V |,m := |E |.x ∈ 0, 1m encodes selection of edges.
f (x) := (#components(x)− 1) · n3wmax+nwmax
∣∣∣∣∣n − 1−m∑i=1
xi
∣∣∣∣∣+m∑i=1
wixi
Theorem (Neumann and Wegener, 2007)
The expected time until the (1+1) EA constructs a minimum spanningtree is O(m2(log n + log wmax)).
Analysis of Typical Runs
Phase 1: Find some connected graph O(m log m).Phase 2: Find some spanning tree O(m log m).
Phase 3: Find a minimum spanning tree by suitable 2-bit flips withguaranteed average weight decrease.
Dirk Sudholt Introduction to the Complexity Analysis of Randomized Search Heuristics 15 / 27
Introduction and Preliminaries Research Directions Fitness-Level Method Drift Analysis
(1+1) EA for Minimum Spanning Trees
Given a weighted graph G = (V ,E ,w) with n := |V |,m := |E |.x ∈ 0, 1m encodes selection of edges.
f (x) := (#components(x)− 1) · n3wmax+nwmax
∣∣∣∣∣n − 1−m∑i=1
xi
∣∣∣∣∣+m∑i=1
wixi
Theorem (Neumann and Wegener, 2007)
The expected time until the (1+1) EA constructs a minimum spanningtree is O(m2(log n + log wmax)).
Analysis of Typical Runs
Phase 1: Find some connected graph O(m log m).Phase 2: Find some spanning tree O(m log m).Phase 3: Find a minimum spanning tree by suitable 2-bit flips withguaranteed average weight decrease.
Dirk Sudholt Introduction to the Complexity Analysis of Randomized Search Heuristics 15 / 27
Introduction and Preliminaries Research Directions Fitness-Level Method Drift Analysis
Populations
A7
A6
A5
A4
A3
A2
A1
fitn
ess
Phase i .1: wait until fraction χ(i) of the population is in Ai .Phase i .2: try to find improvement from there.
Fitness-level method for populations
si := probability bound when fraction χ(i) of individuals is in Ai .
m−1∑i=1
(1
si+ time for population takeover to fraction χ(i)
)
Dirk Sudholt Introduction to the Complexity Analysis of Randomized Search Heuristics 16 / 27
Introduction and Preliminaries Research Directions Fitness-Level Method Drift Analysis
Populations
A7
A6
A5
A4
A3
A2
A1
fitn
ess
Phase i .1: wait until fraction χ(i) of the population is in Ai .Phase i .2: try to find improvement from there.
Fitness-level method for populations
si := probability bound when fraction χ(i) of individuals is in Ai .
m−1∑i=1
(1
si+ time for population takeover to fraction χ(i)
)
Dirk Sudholt Introduction to the Complexity Analysis of Randomized Search Heuristics 16 / 27
Introduction and Preliminaries Research Directions Fitness-Level Method Drift Analysis
Populations
A7
A6
A5
A4
A3
A2
A1
fitn
ess
Phase i .1: wait until fraction χ(i) of the population is in Ai .Phase i .2: try to find improvement from there.
Fitness-level method for populations
si := probability bound when fraction χ(i) of individuals is in Ai .
m−1∑i=1
(1
si+ time for population takeover to fraction χ(i)
)Dirk Sudholt Introduction to the Complexity Analysis of Randomized Search Heuristics 16 / 27
Introduction and Preliminaries Research Directions Fitness-Level Method Drift Analysis
Offspring Populations and Parallel EAs
Create λ independent offspring: si −→ 1− (1− si )λ
Ai
Ai−3
Ai−1
Ai−2
Ai−1
Ai
p+
Ai
Expected parallel time with µ islands [Lassig and Sudholt, 2010]
Ring:m−1∑i=1
3
(p+si )1/2Torus:
m−1∑i=1
10
p2/3+ s
1/3i
Kµ:4m
p++
4
µ
m−1∑i=1
1
si
Dirk Sudholt Introduction to the Complexity Analysis of Randomized Search Heuristics 17 / 27
Introduction and Preliminaries Research Directions Fitness-Level Method Drift Analysis
Offspring Populations and Parallel EAs
Create λ independent offspring: si −→ 1− (1− si )λ
Ai
Ai−3
Ai−1
Ai−2
Ai−1
Ai
p+
Ai
Expected parallel time with µ islands [Lassig and Sudholt, 2010]
Ring:m−1∑i=1
3
(p+si )1/2Torus:
m−1∑i=1
10
p2/3+ s
1/3i
Kµ:4m
p++
4
µ
m−1∑i=1
1
si
Dirk Sudholt Introduction to the Complexity Analysis of Randomized Search Heuristics 17 / 27
Introduction and Preliminaries Research Directions Fitness-Level Method Drift Analysis
Offspring Populations and Parallel EAs
Create λ independent offspring: si −→ 1− (1− si )λ
Ai
Ai−3
Ai−1
Ai−2
Ai−1
Ai
p+
Ai
Expected parallel time with µ islands [Lassig and Sudholt, 2010]
Ring:m−1∑i=1
3
(p+si )1/2Torus:
m−1∑i=1
10
p2/3+ s
1/3i
Kµ:4m
p++
4
µ
m−1∑i=1
1
si
Dirk Sudholt Introduction to the Complexity Analysis of Randomized Search Heuristics 17 / 27
Introduction and Preliminaries Research Directions Fitness-Level Method Drift Analysis
Offspring Populations and Parallel EAs
Create λ independent offspring: si −→ 1− (1− si )λ
Ai
Ai−3
Ai−1
Ai−2
Ai−1
Ai
p+
Ai
Expected parallel time with µ islands [Lassig and Sudholt, 2010]
Ring:m−1∑i=1
3
(p+si )1/2Torus:
m−1∑i=1
10
p2/3+ s
1/3i
Kµ:4m
p++
4
µ
m−1∑i=1
1
si
Dirk Sudholt Introduction to the Complexity Analysis of Randomized Search Heuristics 17 / 27
Introduction and Preliminaries Research Directions Fitness-Level Method Drift Analysis
Offspring Populations and Parallel EAs
Create λ independent offspring: si −→ 1− (1− si )λ
Ai
Ai−3
Ai−1
Ai−2
Ai−1
Ai
p+
Ai
Expected parallel time with µ islands [Lassig and Sudholt, 2010]
Ring:m−1∑i=1
3
(p+si )1/2Torus:
m−1∑i=1
10
p2/3+ s
1/3i
Kµ:4m
p++
4
µ
m−1∑i=1
1
si
Dirk Sudholt Introduction to the Complexity Analysis of Randomized Search Heuristics 17 / 27
Introduction and Preliminaries Research Directions Fitness-Level Method Drift Analysis
Offspring Populations and Parallel EAs
Create λ independent offspring: si −→ 1− (1− si )λ
Ai
Ai−3
Ai−1
Ai−2
Ai−1
Ai
p+
Ai
Expected parallel time with µ islands [Lassig and Sudholt, 2010]
Ring:m−1∑i=1
3
(p+si )1/2
Torus:m−1∑i=1
10
p2/3+ s
1/3i
Kµ:4m
p++
4
µ
m−1∑i=1
1
si
Dirk Sudholt Introduction to the Complexity Analysis of Randomized Search Heuristics 17 / 27
Introduction and Preliminaries Research Directions Fitness-Level Method Drift Analysis
Offspring Populations and Parallel EAs
Create λ independent offspring: si −→ 1− (1− si )λ
Ai
Ai−3
Ai−1
Ai−2
Ai−1
Ai
p+
Ai
Expected parallel time with µ islands [Lassig and Sudholt, 2010]
Ring:m−1∑i=1
3
(p+si )1/2Torus:
m−1∑i=1
10
p2/3+ s
1/3i
Kµ:4m
p++
4
µ
m−1∑i=1
1
si
Dirk Sudholt Introduction to the Complexity Analysis of Randomized Search Heuristics 17 / 27
Introduction and Preliminaries Research Directions Fitness-Level Method Drift Analysis
Offspring Populations and Parallel EAs
Create λ independent offspring: si −→ 1− (1− si )λ
Ai
Ai−3
Ai−1
Ai−2
Ai−1
Ai
p+
Ai
Expected parallel time with µ islands [Lassig and Sudholt, 2010]
Ring:m−1∑i=1
3
(p+si )1/2Torus:
m−1∑i=1
10
p2/3+ s
1/3i
Kµ:4m
p++
4
µ
m−1∑i=1
1
si
Dirk Sudholt Introduction to the Complexity Analysis of Randomized Search Heuristics 17 / 27
Introduction and Preliminaries Research Directions Fitness-Level Method Drift Analysis
Fitness-Levels for Non-Elitist Populations
New population by sampling and mutating λ parents independently:
Pt+1
Ptx
Theorem ([Lehre, GECCO 2011])
If
C1: for one offspring Prob(Ai → Ai+1 ∪ · · · ∪ Am) ≥ si
C2: for one offspring Prob(Ai → Ai ∪ · · · ∪ Am) ≥ p0
C3: selection is sufficiently strong: β(γ,P)/γ ≥ (1 + δ)/p0
C4: population size sufficiently large: λ ≥ 2(1+δ)εδ2 · ln
(m
minisi
)then the expected number of function evaluations is at most
O
(mλ2 +
m−1∑i=1
1
si
).
Dirk Sudholt Introduction to the Complexity Analysis of Randomized Search Heuristics 18 / 27
Introduction and Preliminaries Research Directions Fitness-Level Method Drift Analysis
Fitness-Levels for Non-Elitist Populations
New population by sampling and mutating λ parents independently:
Pt+1
Ptx
Theorem ([Lehre, GECCO 2011])
If
C1: for one offspring Prob(Ai → Ai+1 ∪ · · · ∪ Am) ≥ si
C2: for one offspring Prob(Ai → Ai ∪ · · · ∪ Am) ≥ p0
C3: selection is sufficiently strong: β(γ,P)/γ ≥ (1 + δ)/p0
C4: population size sufficiently large: λ ≥ 2(1+δ)εδ2 · ln
(m
minisi
)then the expected number of function evaluations is at most
O
(mλ2 +
m−1∑i=1
1
si
).
Dirk Sudholt Introduction to the Complexity Analysis of Randomized Search Heuristics 18 / 27
Introduction and Preliminaries Research Directions Fitness-Level Method Drift Analysis
Lower Bounds with Fitness Levels
Upper bounds with fitness levels [Wegener 2002]
Let si be a lower bound on Prob(Ai → Ai+1 ∪ · · · ∪ Am). Then
E(optimization time) ≤m−1∑i=1
Prob(A starts in Ai )m−1∑j=i
1
si.
Lower bounds with fitness levels [Sudholt, 2010]
Let ui · γi,j be an upper bound for Prob(Ai → Aj) and∑m
j=i+1 γi,j = 1.
Assume for all j > i and 0 < χ ≤ 1 that γi,j ≥ χ∑m
k=j γi,k . Then
E(optimization time) ≥m−1∑i=1
Prob(A starts in Ai ) · χm−1∑j=i
1
ui.
Dirk Sudholt Introduction to the Complexity Analysis of Randomized Search Heuristics 19 / 27
Introduction and Preliminaries Research Directions Fitness-Level Method Drift Analysis
Lower Bounds with Fitness Levels
Upper bounds with fitness levels [Wegener 2002]
Let si be a lower bound on Prob(Ai → Ai+1 ∪ · · · ∪ Am). Then
E(optimization time) ≤m−1∑i=1
Prob(A starts in Ai )m−1∑j=i
1
si.
Lower bounds with fitness levels [Sudholt, 2010]
Let ui · γi,j be an upper bound for Prob(Ai → Aj) and∑m
j=i+1 γi,j = 1.
Assume for all j > i and 0 < χ ≤ 1 that γi,j ≥ χ∑m
k=j γi,k . Then
E(optimization time) ≥m−1∑i=1
Prob(A starts in Ai ) · χm−1∑j=i
1
ui.
Dirk Sudholt Introduction to the Complexity Analysis of Randomized Search Heuristics 19 / 27
Introduction and Preliminaries Research Directions Fitness-Level Method Drift Analysis
Overview
1 Introduction and Preliminaries
2 Research Directions
3 Fitness-Level Method
4 Drift Analysis
Dirk Sudholt Introduction to the Complexity Analysis of Randomized Search Heuristics 20 / 27
Introduction and Preliminaries Research Directions Fitness-Level Method Drift Analysis
Additive Drift
0
Drift analysis, drift towards target, [He and Yao, 2004]
1 If E(Xt − Xt+1 | Xt) ≥ δ whenever Xt > 0 then
E(T | X0) ≤ X0
δ.
2 If E(Xt − Xt+1 | Xt) ≤ δ then
E(T | X0) ≥ X0
δ.
Dirk Sudholt Introduction to the Complexity Analysis of Randomized Search Heuristics 21 / 27
Introduction and Preliminaries Research Directions Fitness-Level Method Drift Analysis
Variable Drift
0
Variable Drift results
1 [Mitavskiy, Rowe, and Cannings 2009] If E(Xt − Xt+1 | Xt) ≥ δiwhenever Xt > i then
E(T | X0) ≤dX0e∑i=1
1
δi.
2 [Johannsen, 2010] If h : R+0 → R+ is continuous and monotone
increasing and E(Xt − Xt+1 | Xt) ≤ h(Xt) then
E(T | X0) ≤ xmin
h(xmin)+
∫ X0
xmin
1
h(x)dx .
Dirk Sudholt Introduction to the Complexity Analysis of Randomized Search Heuristics 22 / 27
Introduction and Preliminaries Research Directions Fitness-Level Method Drift Analysis
Multiplicative Drift
Multiplicative Drift Theorem [Doerr, Johannsen, Winzen, 2010]
If there exists a constant δ > 0 such that E(Xt − Xt+1 | Xt) ≥ δXt then
E(T | X0) ≤ 1 + ln(X0/smin)
δ
Bounds also hold with high probability [Doerr and Goldberg, 2010].
Dirk Sudholt Introduction to the Complexity Analysis of Randomized Search Heuristics 23 / 27
Introduction and Preliminaries Research Directions Fitness-Level Method Drift Analysis
Exponential Lower Bounds with Drift
0 a b
Theorem (Simplified Drift Theorem [Oliveto and Witt, 2008])
Assume
1 E (Xt − Xt+1 | Xt) ≤ −ε for a < Xt < b,
2 Prob(Xt − Xt+1 ≥ j) ≤ r(1+δ)j for i > a and j ∈ N0.
If X0 ≥ b it holds Prob(T ≤ 2c∗`
)= 2−Ω(`) for a constant c∗.
Dirk Sudholt Introduction to the Complexity Analysis of Randomized Search Heuristics 24 / 27
Introduction and Preliminaries Research Directions Fitness-Level Method Drift Analysis
Example: (1,λ) EA
How to choose the offspring population size λ for the (1,λ) EA?
Theorem ([Jagerskupper and Storch, 2007])
Exponential gaps on OneMax for λ ≤ 1/14 · ln n vs. λ ≥ 3 ln n.
Refined result: phase transition at log ee−1
n ≈ 2.18 ln n.
Theorem ([Rowe and Sudholt, in preparation])
If λ ≥ log ee−1
n the expected number of function evaluations on
OneMax is O(n log n + nλ).
If λ ≤ (1− ε) log ee−1
n it is at least 2cnε/2
with probability 1− 2−Ω(nε/2).
Dirk Sudholt Introduction to the Complexity Analysis of Randomized Search Heuristics 25 / 27
Introduction and Preliminaries Research Directions Fitness-Level Method Drift Analysis
Example: (1,λ) EA
How to choose the offspring population size λ for the (1,λ) EA?
Theorem ([Jagerskupper and Storch, 2007])
Exponential gaps on OneMax for λ ≤ 1/14 · ln n vs. λ ≥ 3 ln n.
Refined result: phase transition at log ee−1
n ≈ 2.18 ln n.
Theorem ([Rowe and Sudholt, in preparation])
If λ ≥ log ee−1
n the expected number of function evaluations on
OneMax is O(n log n + nλ).
If λ ≤ (1− ε) log ee−1
n it is at least 2cnε/2
with probability 1− 2−Ω(nε/2).
Dirk Sudholt Introduction to the Complexity Analysis of Randomized Search Heuristics 25 / 27
Introduction and Preliminaries Research Directions Fitness-Level Method Drift Analysis
Example: (1,λ) EA
How to choose the offspring population size λ for the (1,λ) EA?
Theorem ([Jagerskupper and Storch, 2007])
Exponential gaps on OneMax for λ ≤ 1/14 · ln n vs. λ ≥ 3 ln n.
Refined result: phase transition at log ee−1
n ≈ 2.18 ln n.
Theorem ([Rowe and Sudholt, in preparation])
If λ ≥ log ee−1
n the expected number of function evaluations on
OneMax is O(n log n + nλ).
If λ ≤ (1− ε) log ee−1
n it is at least 2cnε/2
with probability 1− 2−Ω(nε/2).
Dirk Sudholt Introduction to the Complexity Analysis of Randomized Search Heuristics 25 / 27
Introduction and Preliminaries Research Directions Fitness-Level Method Drift Analysis
Further Reading
Dirk Sudholt Introduction to the Complexity Analysis of Randomized Search Heuristics 26 / 27
Introduction and Preliminaries Research Directions Fitness-Level Method Drift Analysis
Further Learning: Tutorials at GECCO 2011
Runtime Analysis Tutorials on Wednesday
8:30 Thomas Jansen and Frank Neumann: Computational Complexityand Evolutionary Computation
10:40 Carsten Witt: Theory of Randomized Search Heuristics
14:00 Dirk Sudholt: Theory of Swarm Intelligence
14:00 Tobias Friedrich and Frank Neumann: Foundations of EvolutionaryMulti-Objective Optimization
16:10 Benjamin Doerr: Drift Analysis
Slides available from the ACM digital library.
Dirk Sudholt Introduction to the Complexity Analysis of Randomized Search Heuristics 27 / 27