2 two typical geometries of fitness landscapes
play

2. Two typical geometries of fitness landscapes Fitness landscape - PowerPoint PPT Presentation

Multimodal fitness landscape Rugged fitness landscapes Neutral fitness landscapes Neutral Networks 2. Two typical geometries of fitness landscapes Fitness landscape analysis for understanding and designing local search heuristics S


  1. Multimodal fitness landscape Rugged fitness landscapes Neutral fitness landscapes Neutral Networks 2. Two typical geometries of fitness landscapes Fitness landscape analysis for understanding and designing local search heuristics S´ ebastien Verel LISIC - Universit´ e du Littoral Cˆ ote d’Opale, Calais, France http://www-lisic.univ-littoral.fr/~verel/ The 51st CREST Open Workshop Tutorial on Landscape Analysis University College London 27th, February, 2017

  2. Multimodal fitness landscape Rugged fitness landscapes Neutral fitness landscapes Neutral Networks Outline of this part Basis of fitness landscape : introductory example (Done) brief history and background of fitness landscape (Done) fundamental definitions (Done) Two typical geometries of fitness landscapes : multimodality ruggedness neutrality neutral networks

  3. Multimodal fitness landscape Rugged fitness landscapes Neutral fitness landscapes Neutral Networks Multimodal Fitness landscapes Local optima s ∗ no neighbor solution with strictly higher fitness value (maximization) ∀ s ∈ N ( s ∗ ) , f ( s ) � f ( s ∗ ) Fitness Search space

  4. Multimodal fitness landscape Rugged fitness landscapes Neutral fitness landscapes Neutral Networks Typical example : bit strings Search space : X = { 0 , 1 } N N ( x ) = { y ∈ X | d Hamming ( x , y ) = 1 } Example : x = 01101 and f 1 ( x ) = f 2 ( x ) = f 3 ( x ) = 5 11101 00101 01001 01111 01100 f 1 4 2 3 0 3 f 2 2 3 6 2 3 f 3 1 5 2 2 4 Question Is x is a local maximum for f 1 , f 2 , and/or f 3 ?

  5. Multimodal fitness landscape Rugged fitness landscapes Neutral fitness landscapes Neutral Networks Not so typical example : continuous optimization Still an open question... x 1 Search space : X = [0 , 1] d x N α ( x ) = { y ∈ X | � y − x � � α } with α > 0 x 2 Classical definition of local optimum x is local maximum iff ∃ ε > 0 , ∀ y such that � y − x � � ε, f ( y ) � f ( x ) Questions Local search definition with N α ⇒ classical definition ? Classical definition ⇒ local search definition with N α ?

  6. Multimodal fitness landscape Rugged fitness landscapes Neutral fitness landscapes Neutral Networks Not so typical example : continuous optimization Still an open question... x 1 Search space : X = [0 , 1] d x N α ( x ) = { y ∈ X | � y − x � � α } with α > 0 x 2 Classical definition of local optimum x is local maximum iff ∃ ε > 0 , ∀ y such that � y − x � � ε, f ( y ) � f ( x ) Questions Local search definition with N α ⇒ classical definition ? Classical definition ⇒ local search definition with N α ? Still some works to do...

  7. Multimodal fitness landscape Rugged fitness landscapes Neutral fitness landscapes Neutral Networks Sampling local optima Basic estimator (Alyahya, K., & Rowe, J. E. 2016 [AR16]) Expected proportion of local optima : Proportion of local optima in a sample of random solutions Complexity : n × |N| Pros : unbiased estimator Cons : poor estimation when expected proportion is lower than 1 / n

  8. Multimodal fitness landscape Rugged fitness landscapes Neutral fitness landscapes Neutral Networks Sampling local optima by adaptive walks Adaptive walk ( x 1 , x 2 , . . . , x ℓ ) such that x i +1 ∈ N ( x i ) and f ( x i ) < f ( x i +1 )

  9. Multimodal fitness landscape Rugged fitness landscapes Neutral fitness landscapes Neutral Networks Sampling local optima by adaptive walks Adaptive walk ( x 1 , x 2 , . . . , x ℓ ) such that x i +1 ∈ N ( x i ) and f ( x i ) < f ( x i +1 ) Hill-Climbing algorithm (first-improvement) Choose initial solution x ∈ X repeat ′ ∈ { y ∈ N ( x ) | f ( y ) > f ( x ) } choose x ′ ) then if f ( x ) < f ( x ′ x ← x end if until x is a Local Optimum

  10. Multimodal fitness landscape Rugged fitness landscapes Neutral fitness landscapes Neutral Networks Sampling local optima by adaptive walks Adaptive walk ( x 1 , x 2 , . . . , x ℓ ) such that x i +1 ∈ N ( x i ) and f ( x i ) < f ( x i +1 ) Hill-Climbing algorithm (first-improvement) Choose initial solution x ∈ X repeat ′ ∈ { y ∈ N ( x ) | f ( y ) > f ( x ) } choose x ′ ) then if f ( x ) < f ( x ′ x ← x end if until x is a Local Optimum Basin of attraction of x ∗ { x ∈ X | HillClimbing ( x ) = x ∗ } .

  11. Multimodal fitness landscape Rugged fitness landscapes Neutral fitness landscapes Neutral Networks Multimodal Fitness landscapes and difficulty The idea : if the size of attractive basin of global optimum is ”small”, then, the ”time” to find the Fitness global optimum is ”long” Optimisation difficulty : Number and size of attractive basins (Garnier et al. [GK02]) Search space Feature to estimate basin size : Length of adaptive walks complexity : sample size × ℓ × |N|

  12. Multimodal fitness landscape Rugged fitness landscapes Neutral fitness landscapes Neutral Networks Multimodal Fitness landscapes and difficulty 200 The idea : 180 Length of adaptive walk 160 if the size of attractive basin of 140 120 global optimum is ”small”, 100 80 then, the ”time” to find the 60 40 global optimum is ”long” 20 64 128 256 512 N Optimisation difficulty : 260 240 Length of adaptive walk Number and size of attractive 220 basins (Garnier et al. [GK02]) 200 180 160 Feature to estimate basin size : 140 120 Length of adaptive walks 100 0 1 2 3 4 5 6 7 8 K complexity : sample size × ℓ × |N| ex. nk-landscapes with n = 512

  13. Multimodal fitness landscape Rugged fitness landscapes Neutral fitness landscapes Neutral Networks Practice : the Squares Problem a program design problem ? Squares Problem (SP) Find the position of 5 squares Candidate solutions in order to maximize inside X = ([0 , 1000] × [0 , 1000]) 5 squares the number of brown x 1 x 2 points without blue points 1 577 701 2 609 709 3 366 134 1000 ● ● ● ● 4 261 408 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 5 583 792 ● ● ● ● ● ● ● ● ● ● ● ● ● ● 750 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● Fitness function ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● x2 ● ● 500 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● f ( x ) = number of brown points ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● − number of blue points ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 250 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● inside squares ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0 ● 0 250 500 750 1000 x1

  14. Multimodal fitness landscape Rugged fitness landscapes Neutral fitness landscapes Neutral Networks Source code in R : ex01.R Source code : http://www-lisic.univ-littoral.fr/~verel/ Different functions are already defined : main : example to execute the following functions draw and draw solution : draw a problem and the squares of a solution fitness create : create a fitness function from a data frame of points pb1 create and pb2 create : create two particular SP problems init : create a random solution with n squares hc ngh : hill-climbing local search based on neighborhood

  15. Multimodal fitness landscape Rugged fitness landscapes Neutral fitness landscapes Neutral Networks Neighborhood Questions Execute line by line the main function Define the neighborhood create which creates a neighborhood : a neighbor move one square

Recommend


More recommend