Rigorous Analyses of Simple Diversity Mechanisms

Tobias Friedrich Nils Hebbinghaus Frank Neumann

Max-Planck-Institut für Informatik Saarbrücken

Tobias Friedrich

Diversity

Important issue for designing successful EAs

Prevents an EA from too large selection pressure

Assumption: The right diversity mechanism may be crucial for the success of an algorithm

Aim of this talk: Show this observed behavior by rigorous runtime analyses

Frank Neumann

Tobias Friedrich

Runtime Analysis Lot of progress in recent years Results for pseudo-Boolean functions Well-known combinatorial optimization

problems Most results are for the (1+1) EA Some examine the choice of the “right”

population size No analyses that consider the impact of

diversity

Question in this talk: What about diversity in populations?

Frank Neumann

Tobias Friedrich

Simple Diversity Mechanisms

Diversify the population with respect to search points

Diversify the population with respect to fitness values

Show situations where the behavior of these strategies differs significantly

Frank Neumann

Tobias Friedrich

Search point diversifying (μ+1)-EA

initial population(fitness = size):

1. select random individual2. mutate this3. if already in population, goto 1.4. add new individual5. delete individual with lowest fitness

current population(fitness = size):

Frank Neumann

Tobias Friedrich

Fitness diversifying (μ+1)-EA

initial population(fitness = size):

1. select random individual2. mutate this3. if individual with same fitness in

population, replace this by new individual and goto 1

current population(fitness = size):

Frank Neumann

Tobias Friedrich

Fitness diversifying (μ+1)-EA

1. select random individual2. mutate this3. if individual with same fitness in population,

replace this by new individual and goto 14. add new individual5. delete individual with lowest fitness

current population(fitness = size):

Frank Neumann

Tobias Friedrich

Plateaus Examine the choice of diversity on plateau

functions

Plateaus are regions in the search space where all search points have the same fitness

Size and structure determines difficulty for evolutionary search

Investigations for the (1+1) EA on pseudo-Boolean functions, maximum matchings, Eulerian cycles

Frank Neumann

Tobias Friedrich

Investigations Search point vs. Fitness diversifying (μ+1)-EA Constant population size Search space {0,1}n, mutate each bit with 1/n

Compare them on different plateau functions

Runtime:= Number of fitness evaluations to reach an optimal search point

Show advantage/disadvantage of the different diversity mechanisms

Frank Neumann

Tobias Friedrich

Theorem 1

Theorem 1: On f (x) :=

8<

:

jxj0 : x 62 f1i 0n¡ i ;0< i · ngn + 1 : x 2 f1i 0n¡ i ;0< i < ngn + 2 : x = 1n

search point diversifying (¹ +1)-EA hasexpected runtime O(n3),

¯tness diversifying (¹ +1)-EA hasexponential runtime with overwhelming probability.

Tobias Friedrich

Formal Proof of Theorem 1

Tonto Plateau (Grand Canyon)© by Prof. Ian Parker, Univ. of California

f (x) :=

8<

:

jxj0 : x 62 f1i 0n¡ i ;0< i · ngn +1 : x 2 f1i 0n¡ i ;0< i < ngn +2 : x = 1n

Plateau function in theory:

Plateau in the real world:

Tobias Friedrich

Proof of Theorem 1

Tonto Plateau (Grand Canyon)© by Prof. Ian Parker, Univ. of California

0n1n

otherwise

f 1i 0n ¡ i ; 0 < i < ng

Plateau with ¯tness n + 1

Plateau function

f (x) :=

8<

:

jxj0 : x 62 f1i 0n¡ i ;0< i · ngn +1 : x 2 f1i 0n¡ i ;0< i < ngn +2 : x = 1n

Optimum with¯tness n + 2

Tobias Friedrich

Proof of Theorem 1

0n1n

otherwise

f 1i 0n ¡ i ; 0 < i < ng

Fitness diversifying (μ+1)-EA

Tobias Friedrich

Proof of Theorem 1

0n1n

otherwise

f 1i 0n ¡ i ; 0 < i < ng

Mutation with probability Selection kills individual on plateau

MutationSelection

? ?

1n

Tobias Friedrich

Proof of Theorem 1

0n1n

otherwise

f 1i 0n ¡ i ; 0 < i < ng

Individual on plateau cannot perform random walk

exponential runtime with overwhelming probability

Tobias Friedrich

Proof of Theorem 1

0n1n

otherwise

f 1i 0n ¡ i ; 0 < i < ng

Search point diversifying (μ+1)-EA expected polynomial runtime

Optimumfound!

Tobias Friedrich

Theorem 2

Theorem 2: On f (x) :=

8>>>><

>>>>:

n + 1 : x 2 f0i 1n¡ i ;1< i < n ¡ 1gnf03n=41n=4g

n + 2 : x 2 f1i 0n¡ i ;1< i < n ¡ 1gn + 3 : x = 03n=41n=4

jxj0 : otherwise.

search point diversifying (¹ +1)-EA hasexponential runtime with probability 1=2¡ o(1),

¯tness diversifying (¹ +1)-EA hasexpected runtime O(n3).

Tobias Friedrich

Proof of Theorem 2

Double-plateau function:

f (x) :=

8>>>><

>>>>:

n +1 : x 2 f0i 1n¡ i ;1< i < n ¡ 1gnf03n=41n=4g

n +2 : x 2 f1i 0n¡ i ;1< i < n ¡ 1gn +3 : x = 03n=41n=4

jxj0 : otherwise.

Tobias Friedrich

Proof of Theorem 2

f (x) :=

8>><

>>:

n + 1 : x 2 Plateau 1 (without Optimum)n + 2 : x 2 Plateau 2n + 3 : x = Optimum(on Plateau 1)jxj0 : otherwise.

Double-plateau function:

Double-plateau in the real world:

Tobias Friedrich

Proof of Theorem 2

f (x) :=

8>><

>>:

n + 1 : x 2 Plateau 1 (without Optimum)n + 2 : x 2 Plateau 2n + 3 : x = Optimum(on Plateau 1)jxj0 : otherwise.

Double-plateau function:

Double-plateau \ close to the real world":

Tobias Friedrich

Proof of Theorem 2

Plateau 2 Plateau 1

Optimum

f (x) :=

8>><

>>:

n + 1 : x 2 Plateau 1 (without Optimum)n + 2 : x 2 Plateau 2n + 3 : x = Optimum(on Plateau 1)jxj0 : otherwise.

Double-plateau function:

Tobias Friedrich

Proof of Theorem 2

Mutation12

Search point diversifying (μ+1)-EA(only avoiding duplicates)

12

Tobias Friedrich

Proof of Theorem 2

12

Optimumfound!

Search point diversifying (μ+1)-EA reaches Optimum with prob.

12

Tobias Friedrich

Proof of Theorem 2

12

Search point diversifying (μ+1)-EA expected exponential runtime

Tobias Friedrich

Proof of Theorem 2

Mutation12 1

2

Fitness diversifying (μ+1)-EA expected polynomial runtime

Optimumfound!

Tobias Friedrich

Larger Populations

The expected optimization time of the¯tness diversifying (¹ +1)-EA with ¹ = n ¡ k(0 · k < n) on the ¯rst plateau functionis O(¹ nk+2).

The expected optimization time of thesearch point diversifying (¹ +1)-EA with¹ = 2n ¡ k (6 · k < n) on the double plateaufunction is O(¹ nmax(2;dk=2e¡ 3)).

Tobias Friedrich

Conclusions

Ensuring diversity is important for successful EAs

First rigorous runtime analysis on this topic Using the “right” strategy may have a great

impact on the runtime Proven for some basic plateau functions Same effect can be observed in multi-

objective optimization (upcoming CEC paper)

Future work: Other measures for diversity, classical combinatorial optimization problems

Thanks!