1. Towards a Complexity Theory for
Randomized Search Heuristics:
The Ranking-Based Black-Box Model
Benjamin Doerr / Carola Winzen
CSR, June 14, 2011

2. Our Motivation
General-purpose (randomized) search heuristics are
...easy to implement,
...very flexible,
...need little a priori knowledge about problem at hand,
...can deal with many constraints and nonlinearities,
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011

3. Our Motivation
General-purpose (randomized) search heuristics are
...easy to implement,
...very flexible,
...need little a priori knowledge about problem at hand,
...can deal with many constraints and nonlinearities,
and thus, frequently applied in practice.
Some theoretical results exist.
Typical result: runtime analysis for
a particular algorithm
on a particular problem.
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011

4. Our Motivation
General-purpose (randomized) search heuristics are
...easy to implement,
...very flexible,
...need little a priori knowledge about problem at hand,
...can deal with many constraints and nonlinearities,
and thus, frequently applied in practice.
Some theoretical results exist.
Typical result: runtime analysis for Comparable to
a particular algorithm the “early days”
on a particular problem. of Computer
Science
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011

5. Our Motivation
In classical theoretical computer science:
first results: runtime analysis for
a particular algorithm
on a particular problem.
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011

6. Our Motivation
In classical theoretical computer science:
first results: runtime analysis for
a particular algorithm
on a particular problem.
general lower bounds (“tractability of a problem”)
complexity theory
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011

7. Our Motivation
In classical theoretical computer science:
first results: runtime analysis for
a particular algorithm
on a particular problem.
general lower bounds (“tractability of a problem”)
complexity theory
How to create a
complexity theory for
randomized search
heuristics?
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011

8. Our Motivation
In classical theoretical computer science:
first results: runtime analysis for
a particular algorithm
on a particular problem.
general lower bounds (“tractability of a problem”)
complexity theory
Our aim: to understand the tractability of a problem
for general-purpose (randomized) search heuristics
“Towards a Complexity Theory for
Randomized Search Heuristics”
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011

9. A General View on Search Heuristics
Search
Heuristic f
Black-Box
= “Oracle”
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011

10. A General View on Search Heuristics
x
Search
Heuristic f
Black-Box
= “Oracle”
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011

11. A General View on Search Heuristics
x
Search
Heuristic f
f(x)
x
Black-Box
= “Oracle”
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011

12. A General View on Search Heuristics
x
Search
Heuristic f
f(x)
x
Black-Box
= “Oracle”
Typical cost measure: number of function evaluations
until an optimal solution is queried for the first time
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011

13. A General View on Search Heuristics
Classical Query Complexity Model
x
Search
Heuristic f
f(x)
x
Black-Box
= “Oracle”
Typical cost measure: number of function evaluations
until an optimal solution is queried for the first time
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011

14. A Theory for (Randomized) Search Heuristics
Part 1: Classical query complexity model
Game theoretic view
Example: Mastermind
Part 2: Refinement: ranking-based query complexity
“Towards a Complexity Theory for
Randomized Search Heuristics:
The Ranking-Based Black-Box Model”
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011

15. Example: A Mastermind Problem
Carole (=oracle) chooses a binary string of length n:
1 1 0 1 0 0 1 1
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011

16. Example: A Mastermind Problem
Carole (=oracle) chooses a binary string of length n:
1 1 0 1 0 0 1 1
Paul (=algo.) tries to find it. He may ask any string of length n:
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011

17. Example: A Mastermind Problem
Carole (=oracle) chooses a binary string of length n:
1 1 0 1 0 0 1 1
Paul (=algo.) tries to find it. He may ask any string of length n:
1 0 1 0 1 0 1 0
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011

18. Example: A Mastermind Problem
Carole (=oracle) chooses a binary string of length n:
1 1 0 1 0 0 1 1
Paul (=algo.) tries to find it. He may ask any string of length n:
1 0 1 0 1 0 1 0
Carole computes in how many positions the strings coincide:
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011

19. Example: A Mastermind Problem
Carole (=oracle) chooses a binary string of length n:
1 1 0 1 0 0 1 1
Paul (=algo.) tries to find it. He may ask any string of length n:
1 0 1 0 1 0 1 0
Carole computes in how many positions the strings coincide:
1 0 1 0 1 0 1 0
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011

20. Example: A Mastermind Problem
Carole (=oracle) chooses a binary string of length n:
1 1 0 1 0 0 1 1
Paul (=algo.) tries to find it. He may ask any string of length n:
1 0 1 0 1 0 1 0
Carole computes in how many positions the strings coincide:
1 0 1 0 1 0 1 0
“Paul, our strings coincide in 3 bits”
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011

21. Example: A Mastermind Problem
Carole (=oracle) chooses a binary string of length n:
1 1 0 1 0 0 1 1
Paul (=algo.) tries to find it. He may ask any string of length n:
1 0 1 0 1 0 1 0
Carole computes in how many positions the strings coincide:
1 0 1 0 1 0 1 0
We say that the
“Paul, our strings coincide in 3 bits” “fitness” of
Paul’s string is
3
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011

22. Example: A Mastermind Problem
Carole chooses a binary string of length n:
1 1 0 1 0 0 1 1
Paul tries to find it. He may ask any string of length n:
1 0 1 0 1 0 1 0 3
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011

23. Example: A Mastermind Problem
Carole chooses a binary string of length n:
1 1 0 1 0 0 1 1
Paul tries to find it. He may ask any string of length n:
1 0 1 0 1 0 1 0 3
1 0 1 1 0 1 0 1
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011

24. Example: A Mastermind Problem
Carole chooses a binary string of length n:
1 1 0 1 0 0 1 1
Paul tries to find it. He may ask any string of length n:
1 0 1 0 1 0 1 0 3
1 0 1 1 0 1 0 1
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011

25. Example: A Mastermind Problem
Carole chooses a binary string of length n:
1 1 0 1 0 0 1 1
Paul tries to find it. He may ask any string of length n:
1 0 1 0 1 0 1 0 3
1 0 1 1 0 1 0 1 4
...
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011

26. Example: A Mastermind Problem
Carole chooses a binary string of length n:
1 1 0 1 0 0 1 1
Paul tries to find it. He may ask any string of length n:
1 0 1 0 1 0 1 0 3
1 0 1 1 0 1 0 1 4
...
How many queries does Paul need, on average,
until he has identified Carole’s string?
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011

27. Reminder
Our aim: To understand tractability of a problem
for general-purpose (randomized) search heuristics
Measure: number of function evaluations until
an optimal solution is queried for the first time
Our main interest: good lower bounds
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011

28. The Master Mind Problem: What Search Heuristics Do
Paul tries to find Carole’s binary string of length n:
1 1 0 1 0 0 1 1
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011

29. The Master Mind Problem: What Search Heuristics Do
Paul tries to find Carole’s binary string of length n:
1 1 0 1 0 0 1 1
First query is arbitrary:
1 0 1 0 1 0 1 0
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011

30. The Master Mind Problem: What Search Heuristics Do
Paul tries to find Carole’s binary string of length n:
1 1 0 1 0 0 1 1
First query is arbitrary:
1 0 1 0 1 0 1 0 3
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011

31. The Master Mind Problem: What Search Heuristics Do
Paul tries to find Carole’s binary string of length n:
1 1 0 1 0 0 1 1
First query is arbitrary:
1 0 1 0 1 0 1 0 3
Then flip exactly one bit (chosen u.a.r.):
1 1 1 0 1 0 1 0
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011

32. The Master Mind Problem: What Search Heuristics Do
Paul tries to find Carole’s binary string of length n:
1 1 0 1 0 0 1 1
First query is arbitrary:
1 0 1 0 1 0 1 0 3
Then flip exactly one bit (chosen u.a.r.):
1 1 1 0 1 0 1 0 4
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011

33. The Master Mind Problem: What Search Heuristics Do
Paul tries to find Carole’s binary string of length n:
1 1 0 1 0 0 1 1
First query is arbitrary:
1 0 1 0 1 0 1 0 3
Then flip exactly one bit (chosen u.a.r.):
1 1 1 0 1 0 1 0 4
And it continues with the better of the two:
1 1 1 0 1 1 1 0
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011

34. The Master Mind Problem: What Search Heuristics Do
Paul tries to find Carole’s binary string of length n:
1 1 0 1 0 0 1 1
First query is arbitrary:
1 0 1 0 1 0 1 0 3
Then flip exactly one bit (chosen u.a.r.):
1 1 1 0 1 0 1 0 4
Random Local Search algorithm:
Θ(n log n) Coupon Collector [Folklore]
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011

35. The Master Mind Problem: What Search Heuristics Do
Paul tries to find Carole’s binary string of length n:
1 1 0 1 0 0 1 1
First query is arbitrary:
1 0 1 0 1 0 1 0
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011

36. The Master Mind Problem: What Search Heuristics Do
Paul tries to find Carole’s binary string of length n:
1 1 0 1 0 0 1 1
First query is arbitrary:
1 0 1 0 1 0 1 0 3
Flip each bit with probability 1/n:
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011

37. The Master Mind Problem: What Search Heuristics Do
Paul tries to find Carole’s binary string of length n:
1 1 0 1 0 0 1 1
First query is arbitrary:
1 0 1 0 1 0 1 0 3
Flip each bit with probability 1/n:
0 0 1 0 1 1 1 0 1
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011

38. The Master Mind Problem: What Search Heuristics Do
Paul tries to find Carole’s binary string of length n:
1 1 0 1 0 0 1 1
First query is arbitrary:
1 0 1 0 1 0 1 0 3
Flip each bit with probability 1/n:
0 0 1 0 1 1 1 0 1
And it continues with the better of the two
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011

39. The Master Mind Problem: What Search Heuristics Do
Paul tries to find Carole’s binary string of length n:
1 1 0 1 0 0 1 1
First query is arbitrary:
1 0 1 0 1 0 1 0 3
Flip each bit with probability 1/n:
0 0 1 0 1 1 1 0 1
(1+1) Evolutionary Algorithm:
Θ(n log n)[Droste/Jansen/Wegener 92]
[Mühlenbein
02]
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011

40. The Master Mind Problem: Optimal Strategies (1/2)
Paul tries to find Carole’s binary string of length n:
1 0 0 1 0 0 1 1
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011

41. The Master Mind Problem: Optimal Strategies (1/2)
Paul tries to find Carole’s binary string of length n:
1 0 0 1 0 0 1 1
He can go through the string bit by bit:
0 0 0 0 0 0 0 0 4
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011

42. The Master Mind Problem: Optimal Strategies (1/2)
Paul tries to find Carole’s binary string of length n:
1 0 0 1 0 0 1 1
He can go through the string bit by bit:
0 0 0 0 0 0 0 0 4
1 0 0 0 0 0 0 0 5
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011

43. The Master Mind Problem: Optimal Strategies (1/2)
Paul tries to find Carole’s binary string of length n:
1 0 0 1 0 0 1 1
He can go through the string bit by bit:
0 0 0 0 0 0 0 0 4
1 0 0 0 0 0 0 0 5
1 1 0 0 0 0 0 0 4
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011

44. The Master Mind Problem: Optimal Strategies (1/2)
Paul tries to find Carole’s binary string of length n:
1 0 0 1 0 0 1 1
He can go through the string bit by bit:
0 0 0 0 0 0 0 0 4
1 0 0 0 0 0 0 0 5
1 1 0 0 0 0 0 0 4
...
1 0 0 1 0 0 1 1 8
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011

45. The Master Mind Problem: Optimal Strategies (1/2)
Paul tries to find Carole’s binary string of length n:
1 0 0 1 0 0 1 1
He can go through the string bit by bit:
0 0 0 0 0 0 0 0 4
1 0 0 0 0 0 0 0 5
1 1 0 0 0 0 0 0 4 O(n)
...
1 0 0 1 0 0 1 1 8
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011

46. The Master Mind Problem: Optimal Strategies (1/2)
Paul tries to find Carole’s binary string of length n:
1 0 0 1 0 0 1 1
He can go through the string bit by bit:
0 0 0 0 0 0 0 0 4
1 0 0 0 0 0 0 0 5
1 1 0 0 0 0 0 0 4 O(n)
...
Can we do
1 0 0 1 0 0 1 1 even8
better?
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011

47. The Master Mind Problem: Optimal Strategies (2/2)
1 1 0 1 0 0 1 1
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011

48. The Master Mind Problem: Optimal Strategies (2/2)
1 1 0 1 0 0 1 1
1 1 1 1 0 1 0 0 1 4
2 0 1 0 1 0 0 0 0 5
cn
0 0 0 1 1 1 1 1 4
log n
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011

49. Optimal Strategies (2/2)
0
1
1
1
0
0
1
1
0
0
0
1
0
0
1
0
1 1 0 1 0 0 1 1
0
0
1
1
0
0
0
1
0
0
1
1
1 1 1 1 0 1 0 0 1 4
0
0
1
1
2 0 1 0 1 0 0 0 0 5 1 1 ... 1 3 4 . . 4 . 5
0 1 ... 0 6 5 . . 5 . 2
cn 0 0 ... 1 3 4 . . 4 . 5
0 0 0 1 1 1 1 1 4
log n
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011

50. Optimal Strategies (2/2)
0
1
1
1
0
0
1
1
0
0
0
1
0
0
1
0
1 1 0 1 0 0 1 1
0
0
1
1
0
0
0
1
0
0
1
1
1 1 1 1 0 1 0 0 1 4
0
0
1
1
2 0 1 0 1 0 0 0 0 5 1 1 ... 1 3 4 . . 4 . 5
0 1 ... 0 6 5 . . 5 . 2
cn 0 0 ... 1 3 4 . . 4 . 5
0 0 0 1 1 1 1 1 4
log n
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011

51. Optimal Strategies (2/2)
0
1
1
1
0
0
1
1
0
0
0
1
0
0
1
0
1 1 0 1 0 0 1 1
0
0
1
1
0
0
0
1
0
0
1
1
1 1 1 1 0 1 0 0 1 4
0
0
1
1
2 0 1 0 1 0 0 0 0 5 1 1 ... 1 3 4 . . 4 . 5
0 1 ... 0 6 5 . . 5 . 2
cn 0 0 ... 1 3 4 . . 4 . 5
0 0 0 1 1 1 1 1 4
log n
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011

52. Optimal Strategies (2/2)
0
1
1
1
0
0
1
1
0
0
0
1
0
0
1
0
1 1 0 1 0 0 1 1
0
0
1
1
0
0
0
1
0
0
1
1
1 1 1 1 0 1 0 0 1 4
0
0
1
1
2 0 1 0 1 0 0 0 0 5 1 1 ... 1 3 4 . . 4 . 5
0 1 ... 0 6 5 . . 5 . 2
cn 0 0 ... 1 3 4 . . 4 . 5
0 0 0 1 1 1 1 1 4
log n
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011

53. Optimal Strategies (2/2)
0
1
1
1
0
0
1
1
0
0
0
1
0
0
1
0
1 1 0 1 0 0 1 1
0
0
1
1
0
0
0
1
0
0
1
1
1 1 1 1 0 1 0 0 1 4
0
0
1
1
2 0 1 0 1 0 0 0 0 5 1 1 ... 1 3 4 . . 4 . 5
0 1 ... 0 6 5 . . 5 . 2
Fitness elimination
technique
n
O( log n )
cn 0 0 ... 1 3 4 . . 4 . 5
0 0 0 1 1 1 1 1 4
log n
[Anil/Wiegand 09], see also [D./Johannsen/Kötzing/Lehre/Wagner/W. 11]
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011

54. Optimal Strategies (2/2)
0
1
1
1
0
0
1
1
0
0
0
1
0
0
1
0
1 1 0 1 0 0 1 1
0
0
1
1
0
0
0
1
0
0
1
1
1 1 1 1 0 1 0 0 1 4
0
0
1
1
2 0 1 0 1 0 0 0 0 5 1 1 ... 1 3 4 . . 4 . 5
0 1 ... 0 6 5 . . 5 . 2
Fitness elimination
technique
n
Θ( log n )
cn 0 0 ... 1 3 4 . . 4 . 5
0 0 0 1 1 1 1 1 4
log n
[Anil/Wiegand 09], see also [D./Johannsen/Kötzing/Lehre/Wagner/W. 11]
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011

55. Intermediate Summary
Want to understand tractability of a problem for general-
purpose (randomized) search heuristics
Query complexity as such is not a sufficient measure:
Mastermind problem
1 1 0 1 0 0 1 1
Search Heuristics Query Complexity
n
Θ(n log n) Θ( log n )
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011

56. The Ranking-Based Black-Box Model
Observation: many randomized search heuristics use fitness
values only to compare
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011

57. The Ranking-Based Black-Box Model
Observation: many randomized search heuristics use fitness
values only to compare
RLS (1+1) EA ...
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011

58. The Ranking-Based Black-Box Model
Observation: many randomized search heuristics use fitness
values only to compare
RLS (1+1) EA ...
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011

59. The Ranking-Based Black-Box Model
Does not reveal absolute fitness values:
Algorithm
f
Black-Box
= “Oracle”
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011

60. The Ranking-Based Black-Box Model
Does not reveal absolute fitness values:
x
Algorithm
f
Black-Box
= “Oracle”
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011

61. The Ranking-Based Black-Box Model
Does not reveal absolute fitness values:
x
Algorithm
f
Rank 1 x
Rank 2 x Black-Box
Rank 2 x = “Oracle”
Rank 4 x
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011

62. The Ranking-Based Black-Box Model
Equivalent formulation: Let g : R → R be a strictly monotone function
x
Algorithm f
g
g(f(x))=127 Rank 1 g(f(x))
fx
g(f(x))=125 Rank 2 Black-Box
g(f(x))=125 Rank 2
g(f(x))=27 Rank 4
= “Oracle”
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011

63. Intermediate Summary
Want to understand tractability of a problem
for general-purpose (randomized) search heuristics
Query complexity as such is not a sufficient measure
(Many) Randomized search heuristics do selection based on
relative fitness values only, not on absolute values:
Ranking-Based Black-Box Model
Does it Mastermind problem
help? 1 1 0 1 0 0 1 1
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011

64. The Ranking-Based BBC of Mastermind is Θ(n / log n)
n
Mastermind problem
1 1 0 1 0 0 1 1
Classical
Query Complexity
n
Θ( log n )
Search Heuristics
Θ(n log n)
Ranking-Based
Query Complexity
n
Θ( log n )
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011

65. Example: BinaryValue //Weighted Mastermind
Carole chooses a binary string of length n and a permutation σ
1 1 0 1 0 0 1 1
24 22 21 23 25 28 26 27 σ=(4 2 1 3 5 8 6 7)
Paul wants to find it. He may ask any string of length n:
1 0 1 0 1 0 1 0
Carole computes the weighted fitness value:
1 0 1 0 1 0 1 0
“Paul, your string has a score of 336 (=24+28+26)”
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011

66. The Query Complexity of BinaryValue is O(log n)
Paul can do a binary search (parallel for each i≤n):
1 1 0 1 0 0 1 1
24 22 21 23 25 28 26 27
1 1 1 1 1 1 1 1 24+22+23+26+27
1 1 1 1 0 0 0 0 24+22+23+25+28
1 1 0 0 1 1 0 0 24+22+21
1 0 1 0 1 0 1 0 24+28+26
1 1 0 1 0 0 1 1
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011

67. Binary Search not Possible in Ranking-Based Model
Paul can do a binary search (parallel for each i≤n):
1 1 0 1 0 0 1 1
24 22 21 23 25 28 26 27
1 1 1 1 1 1 1 1 28
1 1 1 1 0 0 0 0 317
1 1 0 0 1 1 0 0 -29
1 0 1 0 1 0 1 0 29
? ? ? ? ? ? ? ?
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011

68. Binary Search not Possible in Ranking-Based Model
Paul can do a binary search (parallel for each i≤n):
1 1 0 1 0 0 1 1
24 22 21 23 25 28 26 27
1 1 1 1 1 1 1 1 28
1 1 1 1 0 0 0 0 317
In fact,
1 1 0 RBBBC(BinaryValue)=Θ(n)
0 1 1 0 0 -29
1 0 1 0 1 0 1 0 29
? ? ? ? ? ? ? ?
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011

69. The Ranking-Based Black-Box Complexity of BinaryValue is Θ(n)
n
Limited
If Algorithm queries two strings x and y,
Learning
it can learn at most 1 bit of the target string z.
t=2
Example:
g(BVz,σ (x)) > g(BVz,σ (y))
x=100000 ⇔
y=000000 z1 = x1 = 1
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011

70. The Ranking-Based Black-Box Complexity of BinaryValue is Θ(n)
n
Limited
If Algorithm queries two strings x and y,
Learning
it can learn at most 1 bit of the target string z.
t=2
Limited
If Algorithm queries t strings x1,...,xt,
x
Learning
it can learn at most t-1 bit of the target string z.
general t
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011

71. The Ranking-Based Black-Box Complexity of BinaryValue is Θ(n)
n
Limited
If Algorithm queries two strings x and y,
Learning
it can learn at most 1 bit of the target string z.
t=2
Limited
If Algorithm queries t strings x1,...,xt,
x
Learning
it can learn at most t-1 bit of the target string z.
general t
Ω(n)
There is no deterministic algorithm which
deterministic
optimizes BinaryValue in sublinear time.
lower bound
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011

72. The Ranking-Based Black-Box Complexity of BinaryValue is Θ(n)
n
Limited
If Algorithm queries two strings x and y,
Learning
it can learn at most 1 bit of the target string z.
t=2
Limited
If Algorithm queries t strings x1,...,xt,
x
Learning
it can learn at most t-1 bit of the target string z.
general t
Ω(n)
There is no deterministic algorithm which
deterministic
optimizes BinaryValue in sublinear time.
lower bound
Ω(n)
Follows from deterministic lower bound and
randomized
Yao’s minimax principle
lower bound
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011

73. Summary
Want to understand tractability of different problems
for (randomized) search heuristics
Measure: number of function evaluations
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011

74. Summary
Want to understand tractability of different problems
for (randomized) search heuristics
Measure: number of function evaluations
Our main interest are good lower bounds
Classical query complexity: often too weak lower bounds
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011

75. Summary
Want to understand tractability of different problems
for (randomized) search heuristics
Measure: number of function evaluations
Our main interest are good lower bounds
Classical query complexity: often too weak lower bounds
Ranking-Based Black-Box Model:
query complexity model
only relative, not absolute fitness values are given
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011

76. Summary
Want to understand tractability of different problems
for (randomized) search heuristics
Measure: number of function evaluations
Our main interest are good lower bounds
Classical query complexity: often too weak lower bounds
Ranking-Based Black-Box Model:
only relative, not absolute fitness values are given
BinaryValue problem
11010011
Classical
Query Complexity
Θ(log n)
Search Heuristics
Θ(n log n)
Ranking-Based
Query Complexity
Θ(n)
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011

77. Summary
Want to understand tractability of different problems
for (randomized) search heuristics
Measure: number of function evaluations
Our main interest are good lower bounds
Classical query complexity: often too weak lower bounds
Ranking-Based Black-Box Model:
only relative, not absolute fitness values are given
BinaryValue problem Mastermind problem
11010011 11010011
Classical Classical
Query Complexity Query Complexity
n
Θ(log n) Θ( log n )
Search Heuristics Search Heuristics
Θ(n log n) Θ(n log n)
Ranking-Based Ranking-Based
Query Complexity Query Complexity
n
Θ(n) Θ( log n )
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011

78. Future Work
Plenty!
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011

79. Future Work
Plenty!
Other black-box models:
restricted memory models
unbiased sampling strategies
combinations thereof
...
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011

80. Future Work
Plenty!
Other black-box models:
restricted memory models
unbiased sampling strategies
combinations thereof
...
Combinatorial problems:
MST, SSSP problems
partition problem
...
...
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011

81. Future Work
Plenty!
Other black-box models:
restricted memory models
unbiased sampling strategies
Almost equivalent problem:
combinations thereof n distinguishable balls of unknown weight
...
10
Combinatorial problems: 8 9
MST, SSSP problems 5 6 7
1 2 3 4
partition problem
... How often do you need to use the
balance to find a perfect partition
... of the balls?
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011