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):  select random individual  mutate this  if already in population, goto 1.  add new individual  delete individual with lowest fitness  current population (fitness = size): Frank Neumann

Tobias Friedrich Fitness diversifying (μ+1)-EA  initial population (fitness = size):  select random individual  mutate this  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  select random individual  mutate this  if individual with same fitness in population, replace this by new individual and goto 1  add new individual  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   Th eorem 1 : O n f ( x ) : = 8 < : j x j 0 : x 6 2 f 1 i 0 n ¡ i ; 0 < i · n g n + 1 : x 2 f 1 i 0 n ¡ i ; 0 < i < n g n + 2 : x = 1 n searc h po i n t d i vers i f y i ng ( ¹ + 1 ) - EAh as expec t e d run t i me O ( n 3 ), ¯ t ness d i vers i f y i ng ( ¹ + 1 ) - EAh as exponen t i a l run t i mew i t h overw h e l m i ngpro b a b i l i t y.

Tobias Friedrich Formal Proof of Theorem 1 Tonto Plateau (Grand Canyon)© by Prof. Ian Parker, Univ. of California f ( x ) : = 8 < : j x j 0 : x 6 2 f 1 i 0 n ¡ i ; 0 < i · n g n + 1 : x 2 f 1 i 0 n ¡ i ; 0 < i < n g n + 2 : x = 1 n Pl a t eau f unc t i on i n t h eory: Pl a t eau i n t h erea l wor ld :

Tobias Friedrich Proof of Theorem 1 Tonto Plateau (Grand Canyon)© by Prof. Ian Parker, Univ. of California 0 n 1 n o t h erw i se f 1 i 0 n ¡ i ; 0 < i < n g Pl a t eauw i t h¯ t ness n + 1 Pl a t eau f unc t i on f ( x ) : = 8 < : j x j 0 : x 6 2 f 1 i 0 n ¡ i ; 0 < i · n g n + 1 : x 2 f 1 i 0 n ¡ i ; 0 < i < n g n + 2 : x = 1 n O p t i mumw i t h ¯ t ness n + 2

Tobias Friedrich Proof of Theorem 1 0 n 1 n o t h erw i se f 1 i 0 n ¡ i ; 0 < i < n g  Fitness diversifying ( μ +1)-EA

Tobias Friedrich Proof of Theorem 1 0 n 1 n o t h erw i se f 1 i 0 n ¡ i ; 0 < i < n g  Mutation with probability  Selection kills individual on plateau Mutation Selection ? ? 1 n

Tobias Friedrich Proof of Theorem 1 0 n 1 n o t h erw i se f 1 i 0 n ¡ i ; 0 < i < n g  Individual on plateau cannot perform random walk  exponential runtime with overwhelming probability

Tobias Friedrich Proof of Theorem 1 0 n 1 n o t h erw i se f 1 i 0 n ¡ i ; 0 < i < n g  Search point diversifying ( μ +1)-EA  expected polynomial runtime Optimum found!

Tobias Friedrich Theorem 2   searc h po i n t d i vers i f y i ng ( ¹ + 1 ) - EAh as exponen t i a l run t i mew i t h pro b a b i l i t y 1 = 2 ¡ o ( 1 ), ¯ t ness d i vers i f y i ng ( ¹ + 1 ) - EAh as expec t e d run t i me O ( n 3 ).

Tobias Friedrich Proof of Theorem 2 D ou bl e-p l a t eau f unc t i on:

Tobias Friedrich Proof of Theorem 2 f ( x ) : = 8 > > < > > : n + 1 : x 2 Pl a t eau 1 ( w i t h ou t O p t i mum ) n + 2 : x 2 Pl a t eau 2 n + 3 : x = O p t i mum ( on Pl a t eau 1 ) j x j 0 : o t h erw i se. D ou bl e-p l a t eau f unc t i on: D ou bl e-p l a t eau i n t h erea l wor ld :

Tobias Friedrich Proof of Theorem 2 f ( x ) : = 8 > > < > > : n + 1 : x 2 Pl a t eau 1 ( w i t h ou t O p t i mum ) n + 2 : x 2 Pl a t eau 2 n + 3 : x = O p t i mum ( on Pl a t eau 1 ) j x j 0 : o t h erw i se. D ou bl e-p l a t eau f unc t i on: D ou bl e-p l a t eau \ c l ose t o t h erea l wor ld" :

Tobias Friedrich Proof of Theorem 2 Plateau 2 Plateau 1 Optimum f ( x ) : = 8 > > < > > : n + 1 : x 2 Pl a t eau 1 ( w i t h ou t O p t i mum ) n + 2 : x 2 Pl a t eau 2 n + 3 : x = O p t i mum ( on Pl a t eau 1 ) j x j 0 : o t h erw i se. D ou bl e-p l a t eau f unc t i on:

Tobias Friedrich Proof of Theorem 2 Mutation 1 2  Search point diversifying ( μ +1)-EA (only avoiding duplicates) 1 2

Tobias Friedrich Proof of Theorem Optimum found!  Search point diversifying ( μ +1)-EA  reaches Optimum with prob. 1 2

Tobias Friedrich Proof of Theorem  Search point diversifying ( μ +1)-EA  expected exponential runtime

Tobias Friedrich Proof of Theorem 2 Mutation  Fitness diversifying ( μ +1)-EA  expected polynomial runtime Optimum found!

Tobias Friedrich Larger Populations  

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!