csr2011 june14 11_30_winzen_rotated
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Towards a Complexity Theory for
Randomized Search Heuristics: The Ranking-Based Black-Box Model
Benjamin Doerr / Carola Winzen
CSR, June 14, 2011
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B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011
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,
![Page 3: Csr2011 june14 11_30_winzen_rotated](https://reader033.vdocuments.us/reader033/viewer/2022042613/5445e75ab1af9f29098b45b6/html5/thumbnails/3.jpg)
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011
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.
![Page 4: Csr2011 june14 11_30_winzen_rotated](https://reader033.vdocuments.us/reader033/viewer/2022042613/5445e75ab1af9f29098b45b6/html5/thumbnails/4.jpg)
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011
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.
Comparable to the “early days”
of Computer Science
![Page 5: Csr2011 june14 11_30_winzen_rotated](https://reader033.vdocuments.us/reader033/viewer/2022042613/5445e75ab1af9f29098b45b6/html5/thumbnails/5.jpg)
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011
Our Motivation
In classical theoretical computer science:
first results: runtime analysis for
a particular algorithm
on a particular problem.
![Page 6: Csr2011 june14 11_30_winzen_rotated](https://reader033.vdocuments.us/reader033/viewer/2022042613/5445e75ab1af9f29098b45b6/html5/thumbnails/6.jpg)
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011
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
![Page 7: Csr2011 june14 11_30_winzen_rotated](https://reader033.vdocuments.us/reader033/viewer/2022042613/5445e75ab1af9f29098b45b6/html5/thumbnails/7.jpg)
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011
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?
![Page 8: Csr2011 june14 11_30_winzen_rotated](https://reader033.vdocuments.us/reader033/viewer/2022042613/5445e75ab1af9f29098b45b6/html5/thumbnails/8.jpg)
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011
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 problemfor general-purpose (randomized) search heuristics
“Towards a Complexity Theory for Randomized Search Heuristics”
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B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011
A General View on Search Heuristics
Search Heuristic
Black-Box = “Oracle”
ffff
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B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011
A General View on Search Heuristics
Search Heuristic
Black-Box = “Oracle”
xxxx
ffff
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B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011
A General View on Search Heuristics
Search Heuristic
Black-Box = “Oracle”
xxxx
ffff(xxxx)
ffff
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B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011
A General View on Search Heuristics
Search Heuristic
Black-Box = “Oracle”
xxxx
ffff(xxxx)
ffff
Typical cost measure: number of function evaluationsuntil an optimal solution is queried for the first time
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B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011
A General View on Search Heuristics
Search Heuristic
Black-Box = “Oracle”
xxxx
ffff(xxxx)
ffff
Typical cost measure: number of function evaluationsuntil an optimal solution is queried for the first time
Classical Query Complexity Model
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B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011
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”
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B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011
Example: A Mastermind Problem
Carole (=oracle) chooses a binary string of length n:
1 1 0 1 0 0 1 1
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B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011
Example: A Mastermind Problem
Carole (=oracle) chooses a binary string of length n:
Paul (=algo.) tries to find it. He may ask any string of length n:
1 1 0 1 0 0 1 1
![Page 17: Csr2011 june14 11_30_winzen_rotated](https://reader033.vdocuments.us/reader033/viewer/2022042613/5445e75ab1af9f29098b45b6/html5/thumbnails/17.jpg)
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011
Example: A Mastermind Problem
Carole (=oracle) chooses a binary string of length n:
Paul (=algo.) tries to find it. He may ask any string of length n:
1 1 0 1 0 0 1 1
1 0 1 0 1 0 1 0
![Page 18: Csr2011 june14 11_30_winzen_rotated](https://reader033.vdocuments.us/reader033/viewer/2022042613/5445e75ab1af9f29098b45b6/html5/thumbnails/18.jpg)
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011
Example: A Mastermind Problem
Carole (=oracle) chooses a binary string of length n:
Paul (=algo.) tries to find it. He may ask any string of length n:
Carole computes in how many positions the strings coincide:
1 1 0 1 0 0 1 1
1 0 1 0 1 0 1 0
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B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011
Example: A Mastermind Problem
Carole (=oracle) chooses a binary string of length n:
Paul (=algo.) tries to find it. He may ask any string of length n:
Carole computes in how many positions the strings coincide:
1 1 0 1 0 0 1 1
1 0 1 0 1 0 1 0
1 0 1 0 1 0 1 0
![Page 20: Csr2011 june14 11_30_winzen_rotated](https://reader033.vdocuments.us/reader033/viewer/2022042613/5445e75ab1af9f29098b45b6/html5/thumbnails/20.jpg)
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011
Example: A Mastermind Problem
Carole (=oracle) chooses a binary string of length n:
Paul (=algo.) tries to find it. He may ask any string of length n:
Carole computes in how many positions the strings coincide:
“Paul, our strings coincide in 3 bits”
1 1 0 1 0 0 1 1
1 0 1 0 1 0 1 0
1 0 1 0 1 0 1 0
![Page 21: Csr2011 june14 11_30_winzen_rotated](https://reader033.vdocuments.us/reader033/viewer/2022042613/5445e75ab1af9f29098b45b6/html5/thumbnails/21.jpg)
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011
Example: A Mastermind Problem
Carole (=oracle) chooses a binary string of length n:
Paul (=algo.) tries to find it. He may ask any string of length n:
Carole computes in how many positions the strings coincide:
“Paul, our strings coincide in 3 bits”
1 1 0 1 0 0 1 1
1 0 1 0 1 0 1 0
1 0 1 0 1 0 1 0
We say that the “fitness” of
Paul’s string is 3
![Page 22: Csr2011 june14 11_30_winzen_rotated](https://reader033.vdocuments.us/reader033/viewer/2022042613/5445e75ab1af9f29098b45b6/html5/thumbnails/22.jpg)
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011
Example: A Mastermind Problem
Carole chooses a binary string of length n:
Paul tries to find it. He may ask any string of length n:
1 1 0 1 0 0 1 1
1 0 1 0 1 0 1 0 3
![Page 23: Csr2011 june14 11_30_winzen_rotated](https://reader033.vdocuments.us/reader033/viewer/2022042613/5445e75ab1af9f29098b45b6/html5/thumbnails/23.jpg)
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011
Example: A Mastermind Problem
Carole chooses a binary string of length n:
Paul tries to find it. He may ask any string of length n:
1 1 0 1 0 0 1 1
1 0 1 0 1 0 1 0 3
1 0 1 1 0 1 0 1
![Page 24: Csr2011 june14 11_30_winzen_rotated](https://reader033.vdocuments.us/reader033/viewer/2022042613/5445e75ab1af9f29098b45b6/html5/thumbnails/24.jpg)
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011
Example: A Mastermind Problem
Carole chooses a binary string of length n:
Paul tries to find it. He may ask any string of length n:
1 1 0 1 0 0 1 1
1 0 1 0 1 0 1 0 3
1 0 1 1 0 1 0 1
![Page 25: Csr2011 june14 11_30_winzen_rotated](https://reader033.vdocuments.us/reader033/viewer/2022042613/5445e75ab1af9f29098b45b6/html5/thumbnails/25.jpg)
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011
Example: A Mastermind Problem
Carole chooses a binary string of length n:
Paul tries to find it. He may ask any string of length n:
1 1 0 1 0 0 1 1
1 0 1 0 1 0 1 0 3
4
...
1 0 1 1 0 1 0 1
![Page 26: Csr2011 june14 11_30_winzen_rotated](https://reader033.vdocuments.us/reader033/viewer/2022042613/5445e75ab1af9f29098b45b6/html5/thumbnails/26.jpg)
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011
Example: A Mastermind Problem
Carole chooses a binary string of length n:
Paul tries to find it. He may ask any string of length n:
1 1 0 1 0 0 1 1
1 0 1 0 1 0 1 0 3
4
...
How many queries does Paul need, on average, until he has identified Carole’s string?
1 0 1 1 0 1 0 1
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B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011
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
![Page 28: Csr2011 june14 11_30_winzen_rotated](https://reader033.vdocuments.us/reader033/viewer/2022042613/5445e75ab1af9f29098b45b6/html5/thumbnails/28.jpg)
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011
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
![Page 29: Csr2011 june14 11_30_winzen_rotated](https://reader033.vdocuments.us/reader033/viewer/2022042613/5445e75ab1af9f29098b45b6/html5/thumbnails/29.jpg)
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011
The Master Mind Problem: What Search Heuristics Do
Paul tries to find Carole’s binary string of length n:
First query is arbitrary:
1 1 0 1 0 0 1 1
1 0 1 0 1 0 1 0
![Page 30: Csr2011 june14 11_30_winzen_rotated](https://reader033.vdocuments.us/reader033/viewer/2022042613/5445e75ab1af9f29098b45b6/html5/thumbnails/30.jpg)
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011
The Master Mind Problem: What Search Heuristics Do
Paul tries to find Carole’s binary string of length n:
First query is arbitrary:
1 1 0 1 0 0 1 1
1 0 1 0 1 0 1 0 3
![Page 31: Csr2011 june14 11_30_winzen_rotated](https://reader033.vdocuments.us/reader033/viewer/2022042613/5445e75ab1af9f29098b45b6/html5/thumbnails/31.jpg)
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011
The Master Mind Problem: What Search Heuristics Do
Paul tries to find Carole’s binary string of length n:
First query is arbitrary:
Then flip exactly one bit (chosen u.a.r.):
1 1 0 1 0 0 1 1
1 0 1 0 1 0 1 0 3
1 1 1 0 1 0 1 0
![Page 32: Csr2011 june14 11_30_winzen_rotated](https://reader033.vdocuments.us/reader033/viewer/2022042613/5445e75ab1af9f29098b45b6/html5/thumbnails/32.jpg)
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011
The Master Mind Problem: What Search Heuristics Do
Paul tries to find Carole’s binary string of length n:
First query is arbitrary:
Then flip exactly one bit (chosen u.a.r.):
1 1 0 1 0 0 1 1
1 0 1 0 1 0 1 0 3
1 1 1 0 1 0 1 0 4
![Page 33: Csr2011 june14 11_30_winzen_rotated](https://reader033.vdocuments.us/reader033/viewer/2022042613/5445e75ab1af9f29098b45b6/html5/thumbnails/33.jpg)
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011
The Master Mind Problem: What Search Heuristics Do
Paul tries to find Carole’s binary string of length n:
First query is arbitrary:
Then flip exactly one bit (chosen u.a.r.):
And it continues with the better of the two:
1 1 0 1 0 0 1 1
1 0 1 0 1 0 1 0 3
1 1 1 0 1 0 1 0 4
1 1 1 0 1 1 1 0
![Page 34: Csr2011 june14 11_30_winzen_rotated](https://reader033.vdocuments.us/reader033/viewer/2022042613/5445e75ab1af9f29098b45b6/html5/thumbnails/34.jpg)
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011
The Master Mind Problem: What Search Heuristics Do
Paul tries to find Carole’s binary string of length n:
First query is arbitrary:
Then flip exactly one bit (chosen u.a.r.):
1 1 0 1 0 0 1 1
1 0 1 0 1 0 1 0 3
1 1 1 0 1 0 1 0 4
Random Local Search algorithm:
Θ(n logn)Coupon Collector [Folklore]
![Page 35: Csr2011 june14 11_30_winzen_rotated](https://reader033.vdocuments.us/reader033/viewer/2022042613/5445e75ab1af9f29098b45b6/html5/thumbnails/35.jpg)
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011
The Master Mind Problem: What Search Heuristics Do
Paul tries to find Carole’s binary string of length n:
First query is arbitrary:
1 1 0 1 0 0 1 1
1 0 1 0 1 0 1 0
![Page 36: Csr2011 june14 11_30_winzen_rotated](https://reader033.vdocuments.us/reader033/viewer/2022042613/5445e75ab1af9f29098b45b6/html5/thumbnails/36.jpg)
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011
The Master Mind Problem: What Search Heuristics Do
Paul tries to find Carole’s binary string of length n:
First query is arbitrary:
Flip each bit with probability 1/n:
1 1 0 1 0 0 1 1
1 0 1 0 1 0 1 0 3
![Page 37: Csr2011 june14 11_30_winzen_rotated](https://reader033.vdocuments.us/reader033/viewer/2022042613/5445e75ab1af9f29098b45b6/html5/thumbnails/37.jpg)
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011
The Master Mind Problem: What Search Heuristics Do
Paul tries to find Carole’s binary string of length n:
First query is arbitrary:
Flip each bit with probability 1/n:
1 1 0 1 0 0 1 1
1 0 1 0 1 0 1 0 3
0 0 1 0 1 1 1 0 1
![Page 38: Csr2011 june14 11_30_winzen_rotated](https://reader033.vdocuments.us/reader033/viewer/2022042613/5445e75ab1af9f29098b45b6/html5/thumbnails/38.jpg)
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011
The Master Mind Problem: What Search Heuristics Do
Paul tries to find Carole’s binary string of length n:
First query is arbitrary:
Flip each bit with probability 1/n:
And it continues with the better of the two
1 1 0 1 0 0 1 1
1 0 1 0 1 0 1 0 3
0 0 1 0 1 1 1 0 1
![Page 39: Csr2011 june14 11_30_winzen_rotated](https://reader033.vdocuments.us/reader033/viewer/2022042613/5445e75ab1af9f29098b45b6/html5/thumbnails/39.jpg)
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011
The Master Mind Problem: What Search Heuristics Do
Paul tries to find Carole’s binary string of length n:
First query is arbitrary:
Flip each bit with probability 1/n:
1 1 0 1 0 0 1 1
1 0 1 0 1 0 1 0 3
0 0 1 0 1 1 1 0 1
(1+1) Evolutionary Algorithm:
Θ(n logn) [Mühlenbein 92] [Droste/Jansen/Wegener 02]
![Page 40: Csr2011 june14 11_30_winzen_rotated](https://reader033.vdocuments.us/reader033/viewer/2022042613/5445e75ab1af9f29098b45b6/html5/thumbnails/40.jpg)
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011
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
![Page 41: Csr2011 june14 11_30_winzen_rotated](https://reader033.vdocuments.us/reader033/viewer/2022042613/5445e75ab1af9f29098b45b6/html5/thumbnails/41.jpg)
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011
The Master Mind Problem: Optimal Strategies (1/2)
Paul tries to find Carole’s binary string of length n:
He can go through the string bit by bit:
1 0 0 1 0 0 1 1
0 0 0 0 0 0 0 0 4
![Page 42: Csr2011 june14 11_30_winzen_rotated](https://reader033.vdocuments.us/reader033/viewer/2022042613/5445e75ab1af9f29098b45b6/html5/thumbnails/42.jpg)
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011
The Master Mind Problem: Optimal Strategies (1/2)
Paul tries to find Carole’s binary string of length n:
He can go through the string bit by bit:
1 0 0 1 0 0 1 1
0 0 0 0 0 0 0 0 4
1 0 0 0 0 0 0 0 5
![Page 43: Csr2011 june14 11_30_winzen_rotated](https://reader033.vdocuments.us/reader033/viewer/2022042613/5445e75ab1af9f29098b45b6/html5/thumbnails/43.jpg)
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011
The Master Mind Problem: Optimal Strategies (1/2)
Paul tries to find Carole’s binary string of length n:
He can go through the string bit by bit:
1 0 0 1 0 0 1 1
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
![Page 44: Csr2011 june14 11_30_winzen_rotated](https://reader033.vdocuments.us/reader033/viewer/2022042613/5445e75ab1af9f29098b45b6/html5/thumbnails/44.jpg)
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011
The Master Mind Problem: Optimal Strategies (1/2)
Paul tries to find Carole’s binary string of length n:
He can go through the string bit by bit:
1 0 0 1 0 0 1 1
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
![Page 45: Csr2011 june14 11_30_winzen_rotated](https://reader033.vdocuments.us/reader033/viewer/2022042613/5445e75ab1af9f29098b45b6/html5/thumbnails/45.jpg)
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011
The Master Mind Problem: Optimal Strategies (1/2)
Paul tries to find Carole’s binary string of length n:
He can go through the string bit by bit:
1 0 0 1 0 0 1 1
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
O(n)
![Page 46: Csr2011 june14 11_30_winzen_rotated](https://reader033.vdocuments.us/reader033/viewer/2022042613/5445e75ab1af9f29098b45b6/html5/thumbnails/46.jpg)
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011
The Master Mind Problem: Optimal Strategies (1/2)
Paul tries to find Carole’s binary string of length n:
He can go through the string bit by bit:
1 0 0 1 0 0 1 1
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
O(n)
Can we do even
better?
![Page 47: Csr2011 june14 11_30_winzen_rotated](https://reader033.vdocuments.us/reader033/viewer/2022042613/5445e75ab1af9f29098b45b6/html5/thumbnails/47.jpg)
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011
The Master Mind Problem: Optimal Strategies (2/2)
1 1 0 1 0 0 1 1
![Page 48: Csr2011 june14 11_30_winzen_rotated](https://reader033.vdocuments.us/reader033/viewer/2022042613/5445e75ab1af9f29098b45b6/html5/thumbnails/48.jpg)
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011
The Master Mind Problem: Optimal Strategies (2/2)
1 1 0 1 0 0 1 1
1 1 1 0 1 0 0 1
0 1 0 1 0 0 0 0
0 0 0 1 1 1 1 1
1
2
c n
log n
4
5
4
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B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011
Optimal Strategies (2/2)
1 1 0 1 0 0 1 1
1 1 1 0 1 0 0 1
0 1 0 1 0 0 0 0
0 0 0 1 1 1 1 1
1
2
c n
log n
4
5
4
3 4 . . 4 . 5
6 5 . . 5 . 2
3 4 . . 4 . 5
00
00
00
00
00
00
00
01
11
01
00
11
0 0 ... 1
1 1 ... 1
0 1 ... 0
11
11
11
11
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B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011
Optimal Strategies (2/2)
1 1 0 1 0 0 1 1
1 1 1 0 1 0 0 1
0 1 0 1 0 0 0 0
0 0 0 1 1 1 1 1
1
2
c n
log n
4
5
4
3 4 . . 4 . 5
6 5 . . 5 . 2
3 4 . . 4 . 5
00
00
00
00
00
00
00
01
11
01
00
11
0 0 ... 1
1 1 ... 1
0 1 ... 0
11
11
11
11
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B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011
Optimal Strategies (2/2)
1 1 0 1 0 0 1 1
1 1 1 0 1 0 0 1
0 1 0 1 0 0 0 0
0 0 0 1 1 1 1 1
1
2
c n
log n
4
5
4
3 4 . . 4 . 5
6 5 . . 5 . 2
3 4 . . 4 . 5
00
00
00
00
00
00
00
01
11
01
00
11
0 0 ... 1
1 1 ... 1
0 1 ... 0
11
11
11
11
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B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011
3 4 . . 4
00
00
00
00Optimal Strategies (2/2)
1 1 0 1 0 0 1 1
1 1 1 0 1 0 0 1
0 1 0 1 0 0 0 0
0 0 0 1 1 1 1 1
1
2
c n
log n
4
5
4
. 5
6 5 . . 5 . 2
3 4 . . 4 . 5
00
00
00
01
11
01
00
11
0 0 ... 1
1 1 ... 1
0 1 ... 0
11
11
11
11
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B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011
3 4 . . 4
00
00
00
00Optimal Strategies (2/2)
1 1 0 1 0 0 1 1
1 1 1 0 1 0 0 1
0 1 0 1 0 0 0 0
0 0 0 1 1 1 1 1
1
2
c n
log n
4
5
4
. 5
6 5 . . 5 . 2
3 4 . . 4 . 5
00
00
00
01
11
01
00
11
0 0 ... 1
1 1 ... 1
0 1 ... 0
11
11
11
11
Fitness elimination
technique
[Anil/Wiegand 09], see also [D./Johannsen/Kötzing/Lehre/Wagner/W. 11]
O( n
logn)
![Page 54: Csr2011 june14 11_30_winzen_rotated](https://reader033.vdocuments.us/reader033/viewer/2022042613/5445e75ab1af9f29098b45b6/html5/thumbnails/54.jpg)
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011
3 4 . . 4
00
00
00
00Optimal Strategies (2/2)
1 1 0 1 0 0 1 1
1 1 1 0 1 0 0 1
0 1 0 1 0 0 0 0
0 0 0 1 1 1 1 1
1
2
c n
log n
4
5
4
. 5
6 5 . . 5 . 2
3 4 . . 4 . 5
00
00
00
01
11
01
00
11
0 0 ... 1
1 1 ... 1
0 1 ... 0
11
11
11
11
Fitness elimination
technique
[Anil/Wiegand 09], see also [D./Johannsen/Kötzing/Lehre/Wagner/W. 11]
Θ( n
logn)
![Page 55: Csr2011 june14 11_30_winzen_rotated](https://reader033.vdocuments.us/reader033/viewer/2022042613/5445e75ab1af9f29098b45b6/html5/thumbnails/55.jpg)
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011
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
Query Complexity
Θ( n
logn)
Search Heuristics
Θ(n logn)
![Page 56: Csr2011 june14 11_30_winzen_rotated](https://reader033.vdocuments.us/reader033/viewer/2022042613/5445e75ab1af9f29098b45b6/html5/thumbnails/56.jpg)
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011
The Ranking-Based Black-Box Model
Observation: many randomized search heuristics use fitness
values only to compare
![Page 57: Csr2011 june14 11_30_winzen_rotated](https://reader033.vdocuments.us/reader033/viewer/2022042613/5445e75ab1af9f29098b45b6/html5/thumbnails/57.jpg)
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011
...
The Ranking-Based Black-Box Model
Observation: many randomized search heuristics use fitness
values only to compare
(1+1) EARLS
![Page 58: Csr2011 june14 11_30_winzen_rotated](https://reader033.vdocuments.us/reader033/viewer/2022042613/5445e75ab1af9f29098b45b6/html5/thumbnails/58.jpg)
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011
...
The Ranking-Based Black-Box Model
Observation: many randomized search heuristics use fitness
values only to compare
RLS (1+1) EA
![Page 59: Csr2011 june14 11_30_winzen_rotated](https://reader033.vdocuments.us/reader033/viewer/2022042613/5445e75ab1af9f29098b45b6/html5/thumbnails/59.jpg)
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011
The Ranking-Based Black-Box Model
Does not reveal absolute fitness values:
Algorithm
Black-Box = “Oracle”
ffff
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B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011
The Ranking-Based Black-Box Model
Does not reveal absolute fitness values:
Algorithm
Black-Box = “Oracle”
xxxx
ffff
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B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011
The Ranking-Based Black-Box Model
Does not reveal absolute fitness values:
Algorithm
Black-Box = “Oracle”
Rank 1
Rank 2
Rank 2
xxxx
xxxx
ffff
xxxxxxxx
Rank 4 xxxx
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B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011
The Ranking-Based Black-Box Model
Algorithm
Black-Box = “Oracle”
xxxx
gggg(ffff(xxxx))
ffff
Equivalent formulation: Let be a strictly monotone functiong : R→ R
ggggg(f(x))=127 Rank 1
g(f(x))=125 Rank 2
g(f(x))=125 Rank 2
g(f(x))=27 Rank 4
![Page 63: Csr2011 june14 11_30_winzen_rotated](https://reader033.vdocuments.us/reader033/viewer/2022042613/5445e75ab1af9f29098b45b6/html5/thumbnails/63.jpg)
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011
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
Intermediate Summary
Mastermind problem
1 1 0 1 0 0 1 1Does it
help?
![Page 64: Csr2011 june14 11_30_winzen_rotated](https://reader033.vdocuments.us/reader033/viewer/2022042613/5445e75ab1af9f29098b45b6/html5/thumbnails/64.jpg)
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011
The Ranking-Based BBC of Mastermind is ΘΘΘΘ(nnnn / log nnnn)
Mastermind problem
1 1 0 1 0 0 1 1
Ranking-BasedQuery Complexity
Θ( n
logn)
Search Heuristics
Θ(n logn)
ClassicalQuery Complexity
Θ( n
logn)
![Page 65: Csr2011 june14 11_30_winzen_rotated](https://reader033.vdocuments.us/reader033/viewer/2022042613/5445e75ab1af9f29098b45b6/html5/thumbnails/65.jpg)
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011
Carole chooses a binary string of length n and a permutation σ
Paul wants to find it. He may ask any string of length n:
Carole computes the weighted fitness value:
“Paul, your string has a score of 336 (=24+28+26)”
Example: BinaryValue //Weighted Mastermind
1 1 0 1 0 0 1 1
1 0 1 0 1 0 1 0
24 22 21 23 25 28 26 27 σ=(4 2 1 3 5 8 6 7)
1 0 1 0 1 0 1 0
![Page 66: Csr2011 june14 11_30_winzen_rotated](https://reader033.vdocuments.us/reader033/viewer/2022042613/5445e75ab1af9f29098b45b6/html5/thumbnails/66.jpg)
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011
Paul can do a binary search (parallel for each i≤n):
The Query Complexity of BinaryValue is O(log n)
1 1 1 1 1 1 1 1
1 1 0 1 0 0 1 1
1 1 1 1 0 0 0 0
1 1 0 0 1 1 0 0
1 0 1 0 1 0 1 0
1 1 0 1 0 0 1 1
24 22 21 23 25 28 26 27
24+22+23+26+27
24+22+23+25+28
24+22+21
24+28+26
![Page 67: Csr2011 june14 11_30_winzen_rotated](https://reader033.vdocuments.us/reader033/viewer/2022042613/5445e75ab1af9f29098b45b6/html5/thumbnails/67.jpg)
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011
Paul can do a binary search (parallel for each i≤n):
Binary Search not Possible in Ranking-Based Model
1 1 1 1 1 1 1 1
1 1 1 1 0 0 0 0
1 1 0 0 1 1 0 0
1 0 1 0 1 0 1 0
1 1 0 1 0 0 1 1
24 22 21 23 25 28 26 27
28
317
-29
29
? ? ? ? ? ? ? ?
![Page 68: Csr2011 june14 11_30_winzen_rotated](https://reader033.vdocuments.us/reader033/viewer/2022042613/5445e75ab1af9f29098b45b6/html5/thumbnails/68.jpg)
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011
Paul can do a binary search (parallel for each i≤n):
Binary Search not Possible in Ranking-Based Model
1 1 1 1 1 1 1 1
1 1 1 1 0 0 0 0
1 1 0 0 1 1 0 0
1 0 1 0 1 0 1 0
1 1 0 1 0 0 1 1
24 22 21 23 25 28 26 27
28
317
-29
29
In fact, RBBBC(BinaryValue)=Θ(n)
? ? ? ? ? ? ? ?
![Page 69: Csr2011 june14 11_30_winzen_rotated](https://reader033.vdocuments.us/reader033/viewer/2022042613/5445e75ab1af9f29098b45b6/html5/thumbnails/69.jpg)
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011
The Ranking-Based Black-Box Complexity of BinaryValue is ΘΘΘΘ(nnnn)
Limited Learning
t=2
If Algorithm queries two strings xxxx and yyyy,
it can learn at most 1 bit of the target string zzzz.
Example:
x =100000
y =000000
g(BVz,σ(x)) > g(BVz,σ(y))⇔
z1 = x1 = 1
![Page 70: Csr2011 june14 11_30_winzen_rotated](https://reader033.vdocuments.us/reader033/viewer/2022042613/5445e75ab1af9f29098b45b6/html5/thumbnails/70.jpg)
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011
The Ranking-Based Black-Box Complexity of BinaryValue is ΘΘΘΘ(nnnn)
Limited Learning tttt=2
If Algorithm queries two strings xxxx and yyyy,
it can learn at most 1 bit of the target string zzzz.
Limited
Learning general tttt
If Algorithm queries tttt strings xxxx1,...,xxxxtttt, it can learn at most tttt-1 bit of the target string zzzz.
![Page 71: Csr2011 june14 11_30_winzen_rotated](https://reader033.vdocuments.us/reader033/viewer/2022042613/5445e75ab1af9f29098b45b6/html5/thumbnails/71.jpg)
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011
ΩΩΩΩ(n)
deterministic
lower bound
The Ranking-Based Black-Box Complexity of BinaryValue is ΘΘΘΘ(nnnn)
Limited Learning tttt=2
If Algorithm queries two strings xxxx and yyyy,
it can learn at most 1 bit of the target string zzzz.
Limited
Learning general tttt
If Algorithm queries tttt strings xxxx1,...,xxxxtttt, it can learn at most tttt-1 bit of the target string zzzz.
There is no deterministic algorithm which
optimizes BinaryValue in sublinear time.
![Page 72: Csr2011 june14 11_30_winzen_rotated](https://reader033.vdocuments.us/reader033/viewer/2022042613/5445e75ab1af9f29098b45b6/html5/thumbnails/72.jpg)
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011
ΩΩΩΩ(n)
deterministic
lower bound
The Ranking-Based Black-Box Complexity of BinaryValue is ΘΘΘΘ(nnnn)
Limited Learning tttt=2
If Algorithm queries two strings xxxx and yyyy,
it can learn at most 1 bit of the target string zzzz.
Limited
Learning general tttt
If Algorithm queries tttt strings xxxx1,...,xxxxtttt, it can learn at most tttt-1 bit of the target string zzzz.
There is no deterministic algorithm which
optimizes BinaryValue in sublinear time.
ΩΩΩΩ(n)
randomized lower bound
Follows from deterministic lower bound and Yao’s minimax principle
![Page 73: Csr2011 june14 11_30_winzen_rotated](https://reader033.vdocuments.us/reader033/viewer/2022042613/5445e75ab1af9f29098b45b6/html5/thumbnails/73.jpg)
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011
Want to understand tractability of different problems for (randomized) search heuristics
Measure: number of function evaluations
Summary
![Page 74: Csr2011 june14 11_30_winzen_rotated](https://reader033.vdocuments.us/reader033/viewer/2022042613/5445e75ab1af9f29098b45b6/html5/thumbnails/74.jpg)
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011
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
Summary
![Page 75: Csr2011 june14 11_30_winzen_rotated](https://reader033.vdocuments.us/reader033/viewer/2022042613/5445e75ab1af9f29098b45b6/html5/thumbnails/75.jpg)
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011
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
Summary
![Page 76: Csr2011 june14 11_30_winzen_rotated](https://reader033.vdocuments.us/reader033/viewer/2022042613/5445e75ab1af9f29098b45b6/html5/thumbnails/76.jpg)
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011
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
Summary
BinaryValue problem
1 1 0 1 0 0 1 1
Ranking-BasedQuery Complexity
Search Heuristics
ClassicalQuery Complexity
Θ(n logn)
Θ(log n)
Θ(n)
![Page 77: Csr2011 june14 11_30_winzen_rotated](https://reader033.vdocuments.us/reader033/viewer/2022042613/5445e75ab1af9f29098b45b6/html5/thumbnails/77.jpg)
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011
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
Summary
BinaryValue problem
1 1 0 1 0 0 1 1
Ranking-BasedQuery Complexity
Search Heuristics
ClassicalQuery Complexity
Θ(n logn)
Mastermind problem
1 1 0 1 0 0 1 1
Ranking-BasedQuery Complexity
Search Heuristics
ClassicalQuery Complexity
Θ(n logn)
Θ( n
logn)
Θ( n
log n)
Θ(log n)
Θ(n)
![Page 78: Csr2011 june14 11_30_winzen_rotated](https://reader033.vdocuments.us/reader033/viewer/2022042613/5445e75ab1af9f29098b45b6/html5/thumbnails/78.jpg)
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011
Plenty!
Future Work
![Page 79: Csr2011 june14 11_30_winzen_rotated](https://reader033.vdocuments.us/reader033/viewer/2022042613/5445e75ab1af9f29098b45b6/html5/thumbnails/79.jpg)
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011
Plenty!
Other black-box models:
restricted memory models
unbiased sampling strategies
combinations thereof
...
Future Work
![Page 80: Csr2011 june14 11_30_winzen_rotated](https://reader033.vdocuments.us/reader033/viewer/2022042613/5445e75ab1af9f29098b45b6/html5/thumbnails/80.jpg)
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011
Plenty!
Other black-box models:
restricted memory models
unbiased sampling strategies
combinations thereof
...
Combinatorial problems:
MST, SSSP problems
partition problem
...
...
Future Work
![Page 81: Csr2011 june14 11_30_winzen_rotated](https://reader033.vdocuments.us/reader033/viewer/2022042613/5445e75ab1af9f29098b45b6/html5/thumbnails/81.jpg)
B. Doerr/C. Winzen: Ranking-Based Black-Box Model CSR, June 14, 2011
Plenty!
Other black-box models:
restricted memory models
unbiased sampling strategies
combinations thereof
...
Combinatorial problems:
MST, SSSP problems
partition problem
...
...
Future Work
10
47
36
25
1
98
How often do you need to use the
balance to find a perfect partition of the balls?
Almost equivalent problem:
n distinguishable balls of unknown weight