Download Introduction to the Complexity Analysis of Randomized Search

Introduction and Preliminaries
Research Directions
Fitness-Level Method
Drift Analysis
Introduction to the Complexity Analysis of
Randomized 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
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 global
optimum for a given problem.
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 global
optimum 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
...
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
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, 1}n → R
Choose x ∈ {0, 1}n 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 .
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, 1}n → R
Choose x ∈ {0, 1}n 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 . . .
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]
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
fitness
A5
Pr((1+1) EA leaves Ai ) ≥ si
A4
A3
A2
A1
Expected optimization time of (1+1) EA at most
m−1
P
i=1
Dirk Sudholt
1
si .
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) :=
Dirk Sudholt
Pn
i=1 xi
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) :=
Pn
i=1 xi
Ai := {x | OneMax(x) = i}.
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) :=
Pn
i=1 xi
Ai := {x | OneMax(x) = i}.
n−1
1
1
n−i
si ≥ (n − i) · · 1 −
≥
n
n
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) :=
Pn
i=1 xi
Ai := {x | OneMax(x) = i}.
n−1
1
1
n−i
si ≥ (n − i) · · 1 −
≥
n
n
en
Bound on the expected optimization time of (1+1) EA
n−1
n
X
X
en
1
= en
≤ en ln(en)
n−i
i
i=0
Dirk Sudholt
i=1
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, 1}m encodes selection of edges.
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, 1}m encodes selection of edges.
m
m
X
X
f (x) := (#components(x) − 1) · n wmax +nwmax n − 1 −
xi +
wi xi
3
i=1
Dirk Sudholt
Introduction to the Complexity Analysis of Randomized Search Heuristics
i=1
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, 1}m encodes selection of edges.
m
m
X
X
f (x) := (#components(x) − 1) · n wmax +nwmax n − 1 −
xi +
wi xi
3
i=1
i=1
Theorem (Neumann and Wegener, 2007)
The expected time until the (1+1) EA constructs a minimum spanning
tree is O(m2 (log n + log wmax )).
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, 1}m encodes selection of edges.
m
m
X
X
f (x) := (#components(x) − 1) · n wmax +nwmax n − 1 −
xi +
wi xi
3
i=1
i=1
Theorem (Neumann and Wegener, 2007)
The expected time until the (1+1) EA constructs a minimum spanning
tree is O(m2 (log n + log wmax )).
Analysis of Typical Runs
Phase 1: Find some connected graph
Dirk Sudholt
O(m log m).
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, 1}m encodes selection of edges.
m
m
X
X
f (x) := (#components(x) − 1) · n wmax +nwmax n − 1 −
xi +
wi xi
3
i=1
i=1
Theorem (Neumann and Wegener, 2007)
The expected time until the (1+1) EA constructs a minimum spanning
tree 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).
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, 1}m encodes selection of edges.
m
m
X
X
f (x) := (#components(x) − 1) · n wmax +nwmax n − 1 −
xi +
wi xi
3
i=1
i=1
Theorem (Neumann and Wegener, 2007)
The expected time until the (1+1) EA constructs a minimum spanning
tree 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 with
guaranteed 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
A5
A4
A3
A2
A1
Dirk Sudholt
fitness
A7
A6
Introduction to the Complexity Analysis of Randomized Search Heuristics
16 / 27
Introduction and Preliminaries
Research Directions
Fitness-Level Method
Drift Analysis
Populations
A5
A4
A3
A2
A1
fitness
A7
A6
Phase i.1: wait until fraction χ(i) of the population is in Ai .
Phase i.2: try to find improvement from there.
Dirk Sudholt
Introduction to the Complexity Analysis of Randomized Search Heuristics
16 / 27
Introduction and Preliminaries
Research Directions
Fitness-Level Method
Drift Analysis
Populations
A5
A4
A3
A2
A1
fitness
A7
A6
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
X
i=1
Dirk Sudholt
1
+ time for population takeover to fraction χ(i)
si
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
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
Ai−3
Ai−1
Ai−1
Ai−2
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
p+
Ai
Ai−3
Ai−1
Ai−1
Ai−2
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
p+
Ai
Ai
Ai−1
Ai−1
Ai−2
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
p+
Ai
Ai
Ai−1
Ai−1
Ai−2
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
p+
Ai
Ai
Ai−1
Ai−1
Ai−2
Expected parallel time with µ islands [Lässig and Sudholt, 2010]
Ring:
m−1
X
i=1
Dirk Sudholt
3
(p+ si )1/2
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
p+
Ai
Ai
Ai−1
Ai−1
Ai−2
Expected parallel time with µ islands [Lässig and Sudholt, 2010]
Ring:
m−1
X
i=1
Dirk Sudholt
3
(p+ si )1/2
Torus:
m−1
X
10
2/3 1/3
i=1 p+ si
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
p+
Ai
Ai
Ai−1
Ai−1
Ai−2
Expected parallel time with µ islands [Lässig and Sudholt, 2010]
Ring:
m−1
X
i=1
Dirk Sudholt
3
(p+ si )1/2
Torus:
m−1
X
10
i=1
p+ si
2/3 1/3
Kµ :
m−1
4m
4 X 1
+
p+
µ
si
Introduction to the Complexity Analysis of Randomized Search Heuristics
i=1
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:
x
Pt
Pt+1
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:
x
Pt
Pt+1
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
m
C4: population size sufficiently large: λ ≥ 2(1+δ)
·
ln
2
εδ
mini {si }
then the expected number of function evaluations is at most
!
m−1
X 1
2
O mλ +
.
si
i=1
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
X
i=1
Dirk Sudholt
Prob(A starts in Ai )
m−1
X
j=i
Introduction to the Complexity Analysis of Randomized Search Heuristics
1
.
si
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
X
Prob(A starts in Ai )
i=1
m−1
X
j=i
1
.
si
Lower bounds with fitness levels [Sudholt, 2010]
Pm
Let ui · γi,j be an upper bound for Prob(Ai → Aj ) and j=i+1 γi,j = 1.
Pm
Assume for all j > i and 0 < χ ≤ 1 that γi,j ≥ χ k=j γi,k . Then
E(optimization time) ≥
m−1
X
i=1
Dirk Sudholt
Prob(A starts in Ai ) · χ
m−1
X
j=i
Introduction to the Complexity Analysis of Randomized Search Heuristics
1
.
ui
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 ) ≤
2
X0
.
δ
If E(Xt − Xt+1 | Xt ) ≤ δ then
E(T | X0 ) ≥
Dirk Sudholt
X0
.
δ
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 ) ≥ δi
whenever Xt > i then
dX0 e
E(T | X0 ) ≤
X 1
.
δi
i=1
2
+
[Johannsen, 2010] If h : R+
0 → R is continuous and monotone
increasing and E(Xt − Xt+1 | Xt ) ≤ h(Xt ) then
xmin
E(T | X0 ) ≤
+
h(xmin )
Dirk Sudholt
Z
X0
xmin
1
dx.
h(x)
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
a
0
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
c∗`
If X0 ≥ b it holds Prob T ≤ 2
Dirk Sudholt
for i > a and j ∈ N0 .
= 2−Ω(`) for a constant c ∗ .
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?
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 ([Jägersküpper and Storch, 2007])
Exponential gaps on OneMax for λ ≤ 1/14 · ln n vs. λ ≥ 3 ln n.
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 ([Jägersküpper and Storch, 2007])
Exponential gaps on OneMax for λ ≤ 1/14 · ln n vs. λ ≥ 3 ln n.
e
Refined result: phase transition at log e−1
n ≈ 2.18 ln n.
Theorem ([Rowe and Sudholt, in preparation])
e
If λ ≥ log e−1
n the expected number of function evaluations on
OneMax is O(n log n + nλ).
e
If λ ≤ (1 − ε) log e−1
n it is at least 2cn
Dirk Sudholt
ε/2
with probability 1 − 2−Ω(n
Introduction to the Complexity Analysis of Randomized Search Heuristics
ε/2
)
.
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 Complexity
and 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 Evolutionary
Multi-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
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