introduction to the complexity analysis of randomized search...

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

Overview

1 Introduction and Preliminaries

2 Research Directions

3 Fitness-Level Method

4 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

Scheme of an Evolutionary Algorithm

Select parents for reproduction

Mutation/Recombination

Selection for new population

Motivation

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

Motivation

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

Approach

Markov chain theory

asymptotic notation

amortized analysis

Challenge

Perspective

Approach

Markov chain theory

asymptotic notation

amortized analysis

Challenge

Perspective

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

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

Overview

4 Drift Analysis

Runtime Analyses

Design aspects

parent populations

offspring populations

crossover vs. mutation

population diversity

operator bias

coping with obstacles: paths, plateaus, multimodality, . . .

multiobjective optimization

hybridization

parallelization

stochastic optimization

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]

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]

Overview

4 Drift Analysis

Fitness-level Method for the (1+1) EA

essPr((1+1) EA leaves Ai ) ≥ si

Expected optimization time of (1+1) EA at mostm−1∑i=1

Fitness-Level Method: Example

OneMax (x) :=∑n

i=1 xi

Ai := x | OneMax(x) = i.

si ≥ (n − i) · 1

1− 1

)n−1

≥ n − i

Bound on the expected optimization time of (1+1) EA

n−1∑i=0

n − i= en

n∑i=1

i≤ en ln(en)

OneMax (x) :=∑n

i=1 xi

si ≥ (n − i) · 1

1− 1

)n−1

≥ n − i

n−1∑i=0

n − i= en

n∑i=1

i≤ en ln(en)

OneMax (x) :=∑n

i=1 xi

si ≥ (n − i) · 1

1− 1

)n−1

≥ n − i

n−1∑i=0

n − i= en

n∑i=1

i≤ en ln(en)

OneMax (x) :=∑n

i=1 xi

si ≥ (n − i) · 1

1− 1

)n−1

≥ n − i

n−1∑i=0

n − i= en

n∑i=1

i≤ en ln(en)

(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

∣∣∣∣∣+m∑i=1

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.

∣∣∣∣∣n − 1−m∑i=1

∣∣∣∣∣+m∑i=1

∣∣∣∣∣n − 1−m∑i=1

∣∣∣∣∣+m∑i=1

∣∣∣∣∣n − 1−m∑i=1

∣∣∣∣∣+m∑i=1

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.

∣∣∣∣∣n − 1−m∑i=1

∣∣∣∣∣+m∑i=1

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.

∣∣∣∣∣n − 1−m∑i=1

∣∣∣∣∣+m∑i=1

Populations

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

si+ time for population takeover to fraction χ(i)

Populations

m−1∑i=1

Populations

m−1∑i=1

)Dirk Sudholt Introduction to the Complexity Analysis of Randomized Search Heuristics 16 / 27

Offspring Populations and Parallel EAs

Create λ independent offspring: si −→ 1− (1− si )λ

Ai−3

Ai−1

Ai−2

Ai−1

Expected parallel time with µ islands [Lassig and Sudholt, 2010]

Ring:m−1∑i=1

(p+si )1/2Torus:

m−1∑i=1

p2/3+ s

Kµ:4m

m−1∑i=1

Ai−3

Ai−1

Ai−2

Ai−1

Ring:m−1∑i=1

(p+si )1/2Torus:

m−1∑i=1

p2/3+ s

Kµ:4m

m−1∑i=1

Ai−3

Ai−1

Ai−2

Ai−1

Ring:m−1∑i=1

(p+si )1/2Torus:

m−1∑i=1

p2/3+ s

Kµ:4m

m−1∑i=1

Ai−3

Ai−1

Ai−2

Ai−1

Ring:m−1∑i=1

(p+si )1/2Torus:

m−1∑i=1

p2/3+ s

Kµ:4m

m−1∑i=1

Ai−3

Ai−1

Ai−2

Ai−1

Ring:m−1∑i=1

(p+si )1/2Torus:

m−1∑i=1

p2/3+ s

Kµ:4m

m−1∑i=1

Ai−3

Ai−1

Ai−2

Ai−1

Ring:m−1∑i=1

(p+si )1/2

Torus:m−1∑i=1

p2/3+ s

Kµ:4m

m−1∑i=1

Ai−3

Ai−1

Ai−2

Ai−1

Ring:m−1∑i=1

(p+si )1/2Torus:

m−1∑i=1

p2/3+ s

Kµ:4m

m−1∑i=1

Ai−3

Ai−1

Ai−2

Ai−1

Ring:m−1∑i=1

(p+si )1/2Torus:

m−1∑i=1

p2/3+ s

Kµ:4m

m−1∑i=1

Fitness-Levels for Non-Elitist Populations

New population by sampling and mutating λ parents independently:

Theorem ([Lehre, GECCO 2011])

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

minisi

)then the expected number of function evaluations is at most

(mλ2 +

m−1∑i=1

Fitness-Levels for Non-Elitist Populations

New population by sampling and mutating λ parents independently:

Theorem ([Lehre, GECCO 2011])

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

minisi

)then the expected number of function evaluations is at most

(mλ2 +

m−1∑i=1

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

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

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

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

Overview

4 Drift Analysis

Additive Drift

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

2 If E(Xt − Xt+1 | Xt) ≤ δ then

E(T | X0) ≥ X0

Variable Drift

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

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

h(x)dx .

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)

Bounds also hold with high probability [Doerr and Goldberg, 2010].

Exponential Lower Bounds with Drift

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∗.

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).

Example: (1,λ) EA

n ≈ 2.18 ln n.

Example: (1,λ) EA

n ≈ 2.18 ln n.

introduction to the complexity analysis of randomized search...

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