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Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio A Gentle Introduction to the Time Complexity Analysis of Evolutionary Algorithms Pietro S. Oliveto and Xin Yao CERCIA, School of Computer Science, University of Birmingham, UK WCCI 2012 Brisbane, Australia, 10 June 2012 P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 1 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Aims and Goals of this Tutorial This tutorial will provide an overview of the goals of time complexity analysis of Evolutionary Algorithms (EAs) the most common and effective techniques You should attend if you wish to theoretically understand the behaviour and performance of the search algorithms you design familiarise with the techniques used in the time complexity analysis of EAs pursue research in the area enable you or enhance your ability to 1 2 3 4 5 P. S. Oliveto & X. Yao understand theoretically the behaviour of EAs on different problems perform time complexity analysis of simple EAs on common toy problems read and understand research papers on the computational complexity of EAs have the basic skills to start independent research in the area follow the time complexity of EAs for combinatorial optimization Tutorial later on today (University of Birmingham) Runtime Analysis of EAs WCCI 2012 2 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Outline I 1 Motivation Introduction to the theory of EAs Convergence analysis of EAs Computational complexity of EAs 2 Basic Probability Theory Probability space Union bound Random variables and expectations Law of total probability 3 Evolutionary Algorithms General EAs (1+1)-EA and RLS General properties General upper bound 4 Tail Inequalities Markov’s inequality Chernoff bounds 5 Artificial Fitness Levels Coupon collector’s problem AFL method for upper bounds AFL method for parent populations AFL for non-elitist EAs P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 3 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Outline II AFL for lower bounds 6 Drift Analysis Additive Drift Theorem Multiplicative Drift Theorem Simplified Negative Drift Theorem 7 Typical Run Investigations 8 Conclusions Overview State-of-the-art Further reading P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 4 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Introduction to the theory of EAs Evolutionary Algorithms and Computer Science Goals of design and analysis of algorithms 1 correctness “does the algorithm always output the correct solution?” 2 computational complexity “how many computational resources are required?” For Evolutionary Algorithms (General purpose) 1 convergence “Does the EA find the solution in finite time?” 2 time complexity “how long does it take to find the optimum?” (time = n. of fitness function evaluations) P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 5 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Introduction to the theory of EAs Brief history Theoretical studies of Evolutionary Algorithms (EAs), albeit few, have always existed since the seventies [Goldberg, 1989]; Early studies were concerned with explaining the behaviour rather than analysing their performance. Schema Theory was considered fundamental; First proposed to understand the behaviour of the simple GA [Holland, 1992]; It cannot explain the performance or limit behaviour of EAs; Building Block Hypothesis was wrong [Reeves and Rowe, 2002]; Convergence results appeared in the nineties [Rudolph, 1998]; Related to the time limit behaviour of EAs. P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 6 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Convergence analysis of EAs Convergence Definition Ideally the EA should find the solution in finite steps with probability 1 (visit the global optimum in finite time); If the solution is held forever after, then the algorithm converges to the optimum! P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 7 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Convergence analysis of EAs Convergence Definition Ideally the EA should find the solution in finite steps with probability 1 (visit the global optimum in finite time); If the solution is held forever after, then the algorithm converges to the optimum! Conditions for Convergence ([Rudolph, 1998]) 1 There is a positive probability to reach any point in the search space from any other point 2 The best found solution is never removed from the population (elitism) P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 7 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Convergence analysis of EAs Convergence Definition Ideally the EA should find the solution in finite steps with probability 1 (visit the global optimum in finite time); If the solution is held forever after, then the algorithm converges to the optimum! Conditions for Convergence ([Rudolph, 1998]) 1 There is a positive probability to reach any point in the search space from any other point 2 The best found solution is never removed from the population (elitism) Canonical GAs using mutation, crossover and proportional selection Do Not converge! Elitist variants Do converge! In practice, is it interesting that an algorithm converges to the optimum? Most EAs visit the global optimum in finite time (RLS does not!) How much time? P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 7 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Computational complexity of EAs Computational Complexity of EAs P. K. Lehre, 2011 P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 8 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Computational complexity of EAs Computational Complexity of EAs P. K. Lehre, 2011 Generally means predicting the resources the algorithm requires: Usually the computational time: the number of primitive steps; Usually grows with size of the input; Usually expressed in asymptotic notation; Exponential runtime: Inefficient algorithm Polynomial runtime: “Efficient” algorithm P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 8 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Computational complexity of EAs Computational Complexity of EAs P. K. Lehre, 2011 However (EAs): 1 2 In practice the time for a fitness function evaluation is much higher than the rest; EAs are randomised algorithms They do not perform the same operations even if the input is the same! They do not output the same result if run twice! Hence, the runtime of an EA is a random variable Tf . We are interested in: 1 Estimating E(Tf ), the expected runtime of the EA for f ; 2 Estimating p(Tf ≤ t), the success probability of the EA in t steps for f . P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 8 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Computational complexity of EAs Asymptotic notation f (n) ∈ O(g(n)) ⇐⇒ ∃ constants c, n0 > 0 st. 0 ≤ f (n)≤cg(n) ∀n ≥ n0 f (n) ∈ Ω(g(n)) ⇐⇒ ∃ constants c, n0 > 0 st. 0 ≤ cg(n)≤f (n) ∀n ≥ n0 f (n) ∈ Θ(g(n)) ⇐⇒ f (n) ∈ O(g(n)) and f (n) ∈ Ω(g(n)) f (n) f (n) ∈ o(g(n)) ⇐⇒ lim =0 n→∞ g(n) P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 9 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Computational complexity of EAs Exercise 1: Asymptotic Notation [Lehre, Tutorial] P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 10 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Computational complexity of EAs Motivation Overview Overview Goal: Analyze the correctness and performance of EAs; Difficulties: General purpose, randomised; EAs find the solution in finite time; (convergence analysis) How much time? → Derive the expected runtime and the success probability; Next Basic Probability Theory: probability space, random variables, expectations (expected runtime) Randomised Algorithm Tools: Tail inequalities (success probabilities) Along the way Understand that the analysis cannot be done over all functions Understand why the success probability is important (expected runtime not always sufficient) P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 11 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Probability space Probability Axioms Probability space Sample space Ω (eg. 1 die: {1, 2, 3, 4, 5, 6} ∈ Ω) Allowable events: = ⊆ Ω (eg. E ∈ = := 2 or 3 is the outcome of 1 die) Probability function: P r : = → R (eg. P r(E) = 2/6 = 1/3) Probability Axioms For any event E, 0 ≤ P r(E) ≤ 1 P r(Ω) = 1 For any countably finite sequence of pairwise mutually disjoint events E1 , E2 , E3 , . . . ´ X `[ P r(Ei ) Ei = Pr i≥1 P. S. Oliveto & X. Yao (University of Birmingham) i≥1 Runtime Analysis of EAs WCCI 2012 12 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Probability space Probability Axioms Probability space Sample space Ω (eg. 1 die: {1, 2, 3, 4, 5, 6} ∈ Ω) Allowable events: = ⊆ Ω (eg. E ∈ = := 2 or 3 is the outcome of 1 die) Probability function: P r : = → R (eg. P r(E) = 2/6 = 1/3) Probability Axioms For any event E, 0 ≤ P r(E) ≤ 1 P r(Ω) = 1 For any countably finite sequence of pairwise mutually disjoint events E1 , E2 , E3 , . . . ´ X `[ P r(Ei ) Ei = Pr i≥1 i≥1 1 Die (Ei the event that i shows up) Ω 1 2 3 4 5 6 P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 12 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Probability space Probability Axioms Probability space Sample space Ω (eg. 1 die: {1, 2, 3, 4, 5, 6} ∈ Ω) Allowable events: = ⊆ Ω (eg. E ∈ = := 2 or 3 is the outcome of 1 die) Probability function: P r : = → R (eg. P r(E) = 2/6 = 1/3) Probability Axioms For any event E, 0 ≤ P r(E) ≤ 1 P r(Ω) = 1 For any countably finite sequence of pairwise mutually disjoint events E1 , E2 , E3 , . . . ´ X `[ P r(Ei ) Ei = Pr i≥1 i≥1 1 Die (Ei the event that i shows up) Ω P r(E1 ) = P (E2 ) = · · · = P (E6 ) = 1/6 1 2 3 P r(E1 ∪ E2 ) = P r(E1 ) + P r(E2 ) = 2/6 = 1/3 P r(E1 ∪ E2 ∪ E3 ) = P r(E1 ) + P r(E2 ) + P r(E3 ) = 3/6 = 1/2 4 5 P. S. Oliveto & X. Yao 6 P r(E1 ∪ E2 ∪ · · · ∪ E6 ) = P r(E1 ) + . . . P r(E6 ) = 6/6 = 1 = P r(Ω) (University of Birmingham) Runtime Analysis of EAs WCCI 2012 12 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Probability space Independent Events and Conditional Probabilities Two dice E1 : Event that the first die is a 1; E2 : Event that the second die is a 1; What is the probability that the first die is 1? P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 13 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Probability space Independent Events and Conditional Probabilities Two dice E1 : Event that the first die is a 1; E2 : Event that the second die is a 1; What is the probability that the first die is 1? P r(E1 ) = P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs 6 36 WCCI 2012 13 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Probability space Independent Events and Conditional Probabilities Two dice E1 : Event that the first die is a 1; E2 : Event that the second die is a 1; What is the probability that the first die is 1? P r(E1 ) = What is the probability that the second die is 1? P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs 6 36 WCCI 2012 13 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Probability space Independent Events and Conditional Probabilities Two dice E1 : Event that the first die is a 1; E2 : Event that the second die is a 1; 6 What is the probability that the first die is 1? P r(E1 ) = 36 What is the probability that the second die is 1? P r(E2 ) = P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs 6 36 WCCI 2012 13 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Probability space Independent Events and Conditional Probabilities Two dice E1 : Event that the first die is a 1; E2 : Event that the second die is a 1; 6 What is the probability that the first die is 1? P r(E1 ) = 36 What is the probability that the second die is 1? P r(E2 ) = What is the probability that both dice are 1? P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs 6 36 WCCI 2012 13 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Probability space Independent Events and Conditional Probabilities Two dice E1 : Event that the first die is a 1; E2 : Event that the second die is a 1; 6 What is the probability that the first die is 1? P r(E1 ) = 36 6 What is the probability that the second die is 1? P r(E2 ) = 36 What is the probability that both dice are 1? P r(E3 ) = P r(E1 ∩ E2 ) = (independent events) P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs 1 36 WCCI 2012 13 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Probability space Independent Events and Conditional Probabilities Two dice E1 : Event that the first die is a 1; E2 : Event that the second die is a 1; 6 What is the probability that the first die is 1? P r(E1 ) = 36 6 What is the probability that the second die is 1? P r(E2 ) = 36 What is the probability that both dice are 1? P r(E3 ) = P r(E1 ∩ E2 ) = (independent events) 1 36 Definition Two events E and F are independent if and only if P r(E ∩ F ) = P r(E) · P r(F ) What is the probability that second die is 1 given that the first die is 1? P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 13 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Probability space Independent Events and Conditional Probabilities Two dice E1 : Event that the first die is a 1; E2 : Event that the second die is a 1; 6 What is the probability that the first die is 1? P r(E1 ) = 36 6 What is the probability that the second die is 1? P r(E2 ) = 36 What is the probability that both dice are 1? P r(E3 ) = P r(E1 ∩ E2 ) = (independent events) 1 36 Definition Two events E and F are independent if and only if P r(E ∩ F ) = P r(E) · P r(F ) What is the probability that second die is 1 given that the first die is 1? P r(E2 ∩E1 ) 1/36 P r(E2 |E1 ) = P r(E ) = 1/6 = 1/6 (conditional probabilities) ) 1 P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 13 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Probability space Independent Events and Conditional Probabilities Two dice E1 : Event that the first die is a 1; E2 : Event that the second die is a 1; 6 What is the probability that the first die is 1? P r(E1 ) = 36 6 What is the probability that the second die is 1? P r(E2 ) = 36 What is the probability that both dice are 1? P r(E3 ) = P r(E1 ∩ E2 ) = (independent events) 1 36 Definition Two events E and F are independent if and only if P r(E ∩ F ) = P r(E) · P r(F ) What is the probability that second die is 1 given that the first die is 1? P r(E2 ∩E1 ) 1/36 P r(E2 |E1 ) = P r(E ) = 1/6 = 1/6 (conditional probabilities) ) 1 What about at least one of the two dice is a 1? P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 13 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Probability space Independent Events and Conditional Probabilities Two dice E1 : Event that the first die is a 1; E2 : Event that the second die is a 1; 6 What is the probability that the first die is 1? P r(E1 ) = 36 6 What is the probability that the second die is 1? P r(E2 ) = 36 What is the probability that both dice are 1? P r(E3 ) = P r(E1 ∩ E2 ) = (independent events) 1 36 Definition Two events E and F are independent if and only if P r(E ∩ F ) = P r(E) · P r(F ) What is the probability that second die is 1 given that the first die is 1? P r(E2 ∩E1 ) 1/36 P r(E2 |E1 ) = P r(E ) = 1/6 = 1/6 (conditional probabilities) ) 1 What about at least one of the two dice is a 1?P r(E1 ∪ E2 ) = P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs 11 36 WCCI 2012 13 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Probability space Independent Events and Conditional Probabilities Two dice E1 : Event that the first die is a 1; E2 : Event that the second die is a 1; 6 What is the probability that the first die is 1? P r(E1 ) = 36 6 What is the probability that the second die is 1? P r(E2 ) = 36 What is the probability that both dice are 1? P r(E3 ) = P r(E1 ∩ E2 ) = (independent events) 1 36 Definition Two events E and F are independent if and only if P r(E ∩ F ) = P r(E) · P r(F ) What is the probability that second die is 1 given that the first die is 1? P r(E2 ∩E1 ) 1/36 P r(E2 |E1 ) = P r(E ) = 1/6 = 1/6 (conditional probabilities) ) 1 What about at least one of the two dice is a 1?P r(E1 ∪ E2 ) = 11 36 E1 and E2 are not mutually disjoint events! P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 13 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Union bound Union bound Ω 1,1 1,2 1,3 1,4 1,5 1,6 2,1 2,2 2,3 2,4 2,5 2,6 3,1 3,2 3,3 3,4 3,5 3,6 4,1 4,2 4,3 4,4 4,5 4,6 5,1 5,2 5,3 5,4 5,5 5,6 6,1 6,2 6,3 6,4 6,5 6,6 P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 14 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Union bound Union bound Ω 1,1 1,2 1,3 1,4 1,5 1,6 2,1 2,2 2,3 2,4 2,5 2,6 3,1 3,2 3,3 3,4 3,5 3,6 4,1 4,2 4,3 4,4 4,5 4,6 5,1 5,2 5,3 5,4 5,5 5,6 6,1 6,2 6,3 6,4 6,5 6,6 P r(E1 ) = 6/36 = 1/6 P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 14 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Union bound Union bound Ω 1,1 1,2 1,3 1,4 1,5 1,6 2,1 2,2 2,3 2,4 2,5 2,6 3,1 3,2 3,3 3,4 3,5 3,6 4,1 4,2 4,3 4,4 4,5 4,6 5,1 5,2 5,3 5,4 5,5 5,6 6,1 6,2 6,3 6,4 6,5 6,6 P r(E1 ) = 6/36 = 1/6 P r(E2 ) = 6/36 = 1/6 P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 14 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Union bound Union bound Ω 1,1 1,2 1,3 1,4 1,5 1,6 2,1 2,2 2,3 2,4 2,5 2,6 3,1 3,2 3,3 3,4 3,5 3,6 4,1 4,2 4,3 4,4 4,5 4,6 5,1 5,2 5,3 5,4 5,5 5,6 6,1 6,2 6,3 6,4 6,5 6,6 P r(E1 ) = 6/36 = 1/6 P r(E2 ) = 6/36 = 1/6 P r(E2 |E1 ) = 1/6 P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 14 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Union bound Union bound Ω 1,1 1,2 1,3 1,4 1,5 1,6 2,1 2,2 2,3 2,4 2,5 2,6 3,1 3,2 3,3 3,4 3,5 3,6 4,1 4,2 4,3 4,4 4,5 4,6 5,1 5,2 5,3 5,4 5,5 5,6 6,1 6,2 6,3 6,4 6,5 6,6 P r(E1 ) = 6/36 = 1/6 P r(E2 ) = 6/36 = 1/6 P r(E2 |E1 ) = 1/6 P r(E1 ∪ E2 ) = 6/36 + 5/36 = 11/36 ≤ 6/36 + 6/36 = P r(E1 ) + P r(E2 ) P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 14 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Union bound Union bound Ω 1,1 1,2 1,3 1,4 1,5 1,6 2,1 2,2 2,3 2,4 2,5 2,6 3,1 3,2 3,3 3,4 3,5 3,6 4,1 4,2 4,3 4,4 4,5 4,6 5,1 5,2 5,3 5,4 5,5 5,6 6,1 6,2 6,3 6,4 6,5 6,6 P r(E1 ) = 6/36 = 1/6 P r(E2 ) = 6/36 = 1/6 P r(E2 |E1 ) = 1/6 P r(E1 ∪ E2 ) = 6/36 + 5/36 = 11/36 ≤ 6/36 + 6/36 = P r(E1 ) + P r(E2 ) E1 and E2 are not mutually disjoint events! P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 14 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Union bound Union Bound Theorem (Union bound) For any finite or countably finite sequence of events E1 , E2 , . . . `[ ´ X Pr Ei ≤ P r(Ei ) i≥1 P. S. Oliveto & X. Yao (University of Birmingham) i≥1 Runtime Analysis of EAs WCCI 2012 15 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Random variables and expectations Random Variables Definition (Random Variable) A random variable X on a sample space Ω is a real-valued function X : Ω → R. A discrete random variable takes only a finite or countably finite number of values. Probability: P r(X = a) = P s∈Ω;X(s)=a P r(s) Expectation: E(X) = P. S. Oliveto & X. Yao (University of Birmingham) P i i · P r(X = i) Runtime Analysis of EAs WCCI 2012 16 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Random variables and expectations Random Variables Definition (Random Variable) A random variable X on a sample space Ω is a real-valued function X : Ω → R. A discrete random variable takes only a finite or countably finite number of values. Probability: P r(X = a) = P s∈Ω;X(s)=a P r(s) Expectation: E(X) = P i i · P r(X = i) Example 1 (one die):Let X be the value of one die; 1 6 1 1 1 1 7 E(X) = · 1 + · 2 + · 3 + . . . · 6 = 6 6 6 6 2 P r(X = 6) = P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 16 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Random variables and expectations Random Variables Definition (Random Variable) A random variable X on a sample space Ω is a real-valued function X : Ω → R. A discrete random variable takes only a finite or countably finite number of values. Probability: P r(X = a) = P s∈Ω;X(s)=a P r(s) Expectation: E(X) = P i i · P r(X = i) Example 1 (two dice): Let X be the value of the sum of two dice; P r(X = 4) = P r(1, 3) + P r(3, 1) + P r(2, 2) = E(X) = P. S. Oliveto & X. Yao (University of Birmingham) 3 1 = 36 12 1 2 3 1 ·2+ ·3+ · 4 + ... · 12 = 7 36 36 36 36 Runtime Analysis of EAs WCCI 2012 16 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Random variables and expectations Linearity of Expectation Definition For any collection of discrete random variables X1 , X2 , . . . , Xn with finite expectations, n n ˆX ˜ X E Xi = E(Xi ) i=1 i=1 Example 1 (two dice): Let X be the value of the sum of two dice, X1 the value of die 1 and X2 the value of die 2 7 2 E(X) = E(X1 ) + E(X2 ) = 7 E(Xi ) = P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 17 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Random variables and expectations Binomial Random Variables We run an experiment that succeeds with probability p and fails 1 − p (Eg. coin flips) Consider n trials. Definition A binomial random variable X ∼ B(n, p) with parameters n and p represents the number of successes in n independent experiments each of which succeds with probability p. Probability: P r(X = j) = `n´ j p (1 − p)n−j j Expectation: Let Xi = 1 if the ith trial is successful; (linearity of expectation) ˆ Pn ˜ Pn E(X) = E i Xi = i=1 E(Xi ) = np P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 18 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Random variables and expectations Binomial Random Variables We run an experiment that succeeds with probability p and fails 1 − p (Eg. coin flips) Consider n trials. Definition A binomial random variable X ∼ B(n, p) with parameters n and p represents the number of successes in n independent experiments each of which succeds with probability p. Probability: P r(X = j) = `n´ j p (1 − p)n−j j Expectation: Let Xi = 1 if the ith trial is successful; (linearity of expectation) ˆ Pn ˜ Pn E(X) = E i Xi = i=1 E(Xi ) = np Example 1 (Initialisation of EA): Let X be the number of ones in the initial bit-string (p= 1/2, bit-string length =n) E(X) = np = n/2 P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 18 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Random variables and expectations Geometric Random Variables We run an experiment that succeeds with probability p and fails 1 − p (Eg. coin flips) What is the number of trials until we get a success? (eg. heads?) Definition A geometric random variable X with parameter p represents the number of trials until the first success. Probability: P r(X = n) = (1 − p)n−1 p Expectation: E(X) = 1/p (waiting time argument) P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 19 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Random variables and expectations Geometric Random Variables We run an experiment that succeeds with probability p and fails 1 − p (Eg. coin flips) What is the number of trials until we get a success? (eg. heads?) Definition A geometric random variable X with parameter p represents the number of trials until the first success. Probability: P r(X = n) = (1 − p)n−1 p Expectation: E(X) = 1/p (waiting time argument) Example (expected time for bit i to flip): Let X be the number of steps until bit i flips with mutation rate pm = 1/n P r(X = n · P. S. Oliveto & X. Yao √ n + 1) = (1 − (University of Birmingham) p)n E(X) = 1/p = n „ n+1−1 p = 1 − √ Runtime Analysis of EAs 1 n «n√n 1 n „ «√n ≤ 1 e 1 n √ ≤ e− n WCCI 2012 19 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Law of total probability Law of Total Probability Let E, F be mutually disjoint events and E ∪ F = Ω. Theorem P r(E) = P r(E|F ) · P r(F ) + P r(E|F̄ ) · P r(F̄ ) E(X) = E(X|F ) · P r(F ) + E(X|F̄ ) · P r(F̄ ) Immediate Consequence: P r(E) ≥ P r(E|F ) · P r(F ) E(X) ≥ E(X|F ) · P r(F ) Often used to derive lower bounds on the expected time! We will use this to show that expected values may not be sufficient → success probabilities! P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 20 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio General EAs Evolutionary Algorithms Algorithm ((µ+λ)-EA) 1 Let t = 0; 2 3 Initialize P0 with µ individuals chosen uniformly at random; Repeat Create λ new individuals: 1 2 choose x ∈ Pt uniformly at random; flip each bit in x with probability p; 4 Create the new population Pt+1 by choosing the best µ individuals out of µ + λ; 5 Let t = t + 1. Until a stopping condition is fulfilled. P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 21 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio General EAs Evolutionary Algorithms Algorithm ((µ+λ)-EA) 1 Let t = 0; 2 3 Initialize P0 with µ individuals chosen uniformly at random; Repeat Create λ new individuals: 1 2 choose x ∈ Pt uniformly at random; flip each bit in x with probability p; 4 Create the new population Pt+1 by choosing the best µ individuals out of µ + λ; 5 Let t = t + 1. Until a stopping condition is fulfilled. if µ = λ = 1 we get a (1+1)-EA; P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 21 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio General EAs Evolutionary Algorithms Algorithm ((µ+λ)-EA) 1 Let t = 0; 2 3 Initialize P0 with µ individuals chosen uniformly at random; Repeat Create λ new individuals: 1 2 choose x ∈ Pt uniformly at random; flip each bit in x with probability p; 4 Create the new population Pt+1 by choosing the best µ individuals out of µ + λ; 5 Let t = t + 1. Until a stopping condition is fulfilled. if µ = λ = 1 we get a (1+1)-EA; p = 1/n is generally considered as best choice [Bäck, 1993, Droste et al., 1998]; P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 21 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio General EAs Evolutionary Algorithms Algorithm ((µ+λ)-EA) 1 Let t = 0; 2 3 Initialize P0 with µ individuals chosen uniformly at random; Repeat Create λ new individuals: 1 2 choose x ∈ Pt uniformly at random; flip each bit in x with probability p; 4 Create the new population Pt+1 by choosing the best µ individuals out of µ + λ; 5 Let t = t + 1. Until a stopping condition is fulfilled. if µ = λ = 1 we get a (1+1)-EA; p = 1/n is generally considered as best choice [Bäck, 1993, Droste et al., 1998]; By introducing stochastic selection and crossover we obtain a Genetic Algorithm(GA) P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 21 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio (1+1)-EA and RLS 1+1-EA Algorithm ((1+1)-EA) Initialise P0 with x ∈ {1, 0}n by flipping each bit with p = 1/2 ; Repeat Create x0 by flipping each bit in x with p = 1/n; If f (x0 ) ≥ f (x) Then x0 ∈ Pt+1 Else x ∈ Pt+1 ; Let t = t + 1; Until stopping condition. If only one bit is flipped per iteration: Random Local Search (RLS). How does it work? P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 22 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio (1+1)-EA and RLS 1+1-EA Algorithm ((1+1)-EA) Initialise P0 with x ∈ {1, 0}n by flipping each bit with p = 1/2 ; Repeat Create x0 by flipping each bit in x with p = 1/n; If f (x0 ) ≥ f (x) Then x0 ∈ Pt+1 Else x ∈ Pt+1 ; Let t = t + 1; Until stopping condition. If only one bit is flipped per iteration: Random Local Search (RLS). How does it work? Given x, how many bits will flip in expectation? P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 22 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio (1+1)-EA and RLS 1+1-EA Algorithm ((1+1)-EA) Initialise P0 with x ∈ {1, 0}n by flipping each bit with p = 1/2 ; Repeat Create x0 by flipping each bit in x with p = 1/n; If f (x0 ) ≥ f (x) Then x0 ∈ Pt+1 Else x ∈ Pt+1 ; Let t = t + 1; Until stopping condition. If only one bit is flipped per iteration: Random Local Search (RLS). How does it work? Given x, how many bits will flip in expectation? E[X] = E[X1 + X2 + · · · + Xn ] = E[X1 ] + E[X2 ] + · · · + E[Xn ] = P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 22 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio (1+1)-EA and RLS 1+1-EA Algorithm ((1+1)-EA) Initialise P0 with x ∈ {1, 0}n by flipping each bit with p = 1/2 ; Repeat Create x0 by flipping each bit in x with p = 1/n; If f (x0 ) ≥ f (x) Then x0 ∈ Pt+1 Else x ∈ Pt+1 ; Let t = t + 1; Until stopping condition. If only one bit is flipped per iteration: Random Local Search (RLS). How does it work? Given x, how many bits will flip in expectation? E[X] = E[X1 + X2 + · · · + Xn ] = E[X1 ] + E[X2 ] + · · · + E[Xn ] = „ E[Xi ] = 1 · 1/n + 0 · (1 − 1/n) = 1 · 1/n = 1/n P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs « E(X) = np WCCI 2012 22 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio (1+1)-EA and RLS 1+1-EA Algorithm ((1+1)-EA) Initialise P0 with x ∈ {1, 0}n by flipping each bit with p = 1/2 ; Repeat Create x0 by flipping each bit in x with p = 1/n; If f (x0 ) ≥ f (x) Then x0 ∈ Pt+1 Else x ∈ Pt+1 ; Let t = t + 1; Until stopping condition. If only one bit is flipped per iteration: Random Local Search (RLS). How does it work? Given x, how many bits will flip in expectation? E[X] = E[X1 + X2 + · · · + Xn ] = E[X1 ] + E[X2 ] + · · · + E[Xn ] = „ E[Xi ] = 1 · 1/n + 0 · (1 − 1/n) = 1 · 1/n = 1/n = n X « E(X) = np 1 · 1/n = n/n = 1 i=1 P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 22 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio General properties 1+1-EA: 2 How likely is it that exactly one bit flips? P. S. Oliveto & X. Yao (University of Birmingham) „ « ` ´ j n−j P r(X = j) = n p (1 − p) j Runtime Analysis of EAs WCCI 2012 23 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio General properties 1+1-EA: 2 How likely is it that exactly one bit flips? „ « ` ´ j n−j P r(X = j) = n p (1 − p) j What is the probability of exactly one bit flipping? P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 23 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio General properties 1+1-EA: 2 How likely is it that exactly one bit flips? „ « ` ´ j n−j P r(X = j) = n p (1 − p) j What is the probability of exactly one bit flipping? “n” P r(X = 1) = · 1/n · (1 − 1/n)n−1 = (1 − 1/n)n−1 ≥ 1/e ≈ 0.37 1 P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 23 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio General properties 1+1-EA: 2 How likely is it that exactly one bit flips? „ « ` ´ j n−j P r(X = j) = n p (1 − p) j What is the probability of exactly one bit flipping? “n” P r(X = 1) = · 1/n · (1 − 1/n)n−1 = (1 − 1/n)n−1 ≥ 1/e ≈ 0.37 1 Is it more likely that 2 bits flip or none? P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 23 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio General properties 1+1-EA: 2 How likely is it that exactly one bit flips? „ « ` ´ j n−j P r(X = j) = n p (1 − p) j What is the probability of exactly one bit flipping? “n” P r(X = 1) = · 1/n · (1 − 1/n)n−1 = (1 − 1/n)n−1 ≥ 1/e ≈ 0.37 1 Is it more likely that 2 bits flip or none? “n” P r(X = 2) = · 1/n2 · (1 − 1/n)n−2 = 2 = n · (n − 1) 1/n2 · (1 − 1/n)n−2 = 2 = 1/2 · (1 − 1/n)n−1 ≈ 1/(2e) P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 23 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio General properties 1+1-EA: 2 How likely is it that exactly one bit flips? „ « ` ´ j n−j P r(X = j) = n p (1 − p) j What is the probability of exactly one bit flipping? “n” P r(X = 1) = · 1/n · (1 − 1/n)n−1 = (1 − 1/n)n−1 ≥ 1/e ≈ 0.37 1 Is it more likely that 2 bits flip or none? “n” P r(X = 2) = · 1/n2 · (1 − 1/n)n−2 = 2 = n · (n − 1) 1/n2 · (1 − 1/n)n−2 = 2 = 1/2 · (1 − 1/n)n−1 ≈ 1/(2e) While P r(X = 0) = P. S. Oliveto & X. Yao (University of Birmingham) “n” (1/n)0 · (1 − 1/n)n ≈ 1/e 0 Runtime Analysis of EAs WCCI 2012 23 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio General upper bound 1+1-EA: General Upper bound Theorem ([Droste et al., 2002]) The expected runtime of the (1+1)-EA for an arbitrary function defined in {0, 1}n is O(nn ) P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 24 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio General upper bound 1+1-EA: General Upper bound Theorem ([Droste et al., 2002]) The expected runtime of the (1+1)-EA for an arbitrary function defined in {0, 1}n is O(nn ) Proof P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 24 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio General upper bound 1+1-EA: General Upper bound Theorem ([Droste et al., 2002]) The expected runtime of the (1+1)-EA for an arbitrary function defined in {0, 1}n is O(nn ) Proof 1 Let i be the number of bit positions in which the current solution x and the global optimum x∗ differ; P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 24 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio General upper bound 1+1-EA: General Upper bound Theorem ([Droste et al., 2002]) The expected runtime of the (1+1)-EA for an arbitrary function defined in {0, 1}n is O(nn ) Proof 1 Let i be the number of bit positions in which the current solution x and the global optimum x∗ differ; 2 Each bit flips with probability 1/n, hence does not flip with probability (1 − 1/n); P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 24 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio General upper bound 1+1-EA: General Upper bound Theorem ([Droste et al., 2002]) The expected runtime of the (1+1)-EA for an arbitrary function defined in {0, 1}n is O(nn ) Proof 1 Let i be the number of bit positions in which the current solution x and the global optimum x∗ differ; 2 Each bit flips with probability 1/n, hence does not flip with probability (1 − 1/n); 3 In order to reach the global optimum the algorithm has to mutate the i bits and leave the n − i bits unchanged; P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 24 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio General upper bound 1+1-EA: General Upper bound Theorem ([Droste et al., 2002]) The expected runtime of the (1+1)-EA for an arbitrary function defined in {0, 1}n is O(nn ) Proof 1 Let i be the number of bit positions in which the current solution x and the global optimum x∗ differ; 2 Each bit flips with probability 1/n, hence does not flip with probability (1 − 1/n); 3 In order to reach the global optimum the algorithm has to mutate the i bits and leave the n − i bits unchanged; 4 Then: p(x∗ |x) = P. S. Oliveto & X. Yao (University of Birmingham) „ «i „ « „ «n ` ´ 1 1 n−i 1 1− ≥ = n−n p = n−n n n n Runtime Analysis of EAs WCCI 2012 24 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio General upper bound 1+1-EA: General Upper bound Theorem ([Droste et al., 2002]) The expected runtime of the (1+1)-EA for an arbitrary function defined in {0, 1}n is O(nn ) Proof 1 Let i be the number of bit positions in which the current solution x and the global optimum x∗ differ; 2 Each bit flips with probability 1/n, hence does not flip with probability (1 − 1/n); 3 In order to reach the global optimum the algorithm has to mutate the i bits and leave the n − i bits unchanged; 4 Then: p(x∗ |x) = 5 „ «i „ « „ «n ` ´ 1 1 n−i 1 1− ≥ = n−n p = n−n n n n it implies an upper bound on the expected runtime of O(nn ) (E(X) = 1/p = nn ) (waiting time argument). P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 24 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio General upper bound General Upper bound Exercises Theorem The expected runtime of the (1+1)-EA with mutation probability p = 1/2 for an arbitrary function defined in {0, 1}n is O(2n ) P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 25 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio General upper bound General Upper bound Exercises Theorem The expected runtime of the (1+1)-EA with mutation probability p = 1/2 for an arbitrary function defined in {0, 1}n is O(2n ) Proof Left as Exercise. Theorem The expected runtime of the (1+1)-EA with mutation probability p = χ/n for an arbitrary function defined in {0, 1}n is O((n/χ)n ) P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 25 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio General upper bound General Upper bound Exercises Theorem The expected runtime of the (1+1)-EA with mutation probability p = 1/2 for an arbitrary function defined in {0, 1}n is O(2n ) Proof Left as Exercise. Theorem The expected runtime of the (1+1)-EA with mutation probability p = χ/n for an arbitrary function defined in {0, 1}n is O((n/χ)n ) Proof Left as Exercise. Theorem The expected runtime of RLS for an arbitrary function defined in {0, 1}n is infinite. P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 25 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio General upper bound General Upper bound Exercises Theorem The expected runtime of the (1+1)-EA with mutation probability p = 1/2 for an arbitrary function defined in {0, 1}n is O(2n ) Proof Left as Exercise. Theorem The expected runtime of the (1+1)-EA with mutation probability p = χ/n for an arbitrary function defined in {0, 1}n is O((n/χ)n ) Proof Left as Exercise. Theorem The expected runtime of RLS for an arbitrary function defined in {0, 1}n is infinite. Proof Left as Exercise. P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 25 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio General upper bound 1+1-EA: Conclusions & Exercises In general: P (i − bitf lip) = P. S. Oliveto & X. Yao (University of Birmingham) « „ « “n” 1 „ 1 1 1 n−i 1 n−i ≤ ≈ 1 − 1 − i ni n i! n i!e Runtime Analysis of EAs WCCI 2012 26 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio General upper bound 1+1-EA: Conclusions & Exercises In general: P (i − bitf lip) = « „ « “n” 1 „ 1 1 1 n−i 1 n−i ≤ ≈ 1 − 1 − i ni n i! n i!e What about RLS? P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 26 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio General upper bound 1+1-EA: Conclusions & Exercises In general: P (i − bitf lip) = « „ « “n” 1 „ 1 1 1 n−i 1 n−i ≤ ≈ 1 − 1 − i ni n i! n i!e What about RLS? Expectation: E[X] = 1 P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 26 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio General upper bound 1+1-EA: Conclusions & Exercises In general: P (i − bitf lip) = « „ « “n” 1 „ 1 1 1 n−i 1 n−i ≤ ≈ 1 − 1 − i ni n i! n i!e What about RLS? Expectation: E[X] = 1 P(1-bitflip) = 1 P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 26 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio General upper bound 1+1-EA: Conclusions & Exercises In general: P (i − bitf lip) = « „ « “n” 1 „ 1 1 1 n−i 1 n−i ≤ ≈ 1 − 1 − i ni n i! n i!e What about RLS? Expectation: E[X] = 1 P(1-bitflip) = 1 What about initialisation? P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 26 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio General upper bound 1+1-EA: Conclusions & Exercises In general: P (i − bitf lip) = « „ « “n” 1 „ 1 1 1 n−i 1 n−i ≤ ≈ 1 − 1 − i ni n i! n i!e What about RLS? Expectation: E[X] = 1 P(1-bitflip) = 1 What about initialisation? How many one-bits in expectation after initialisation? P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 26 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio General upper bound 1+1-EA: Conclusions & Exercises In general: P (i − bitf lip) = « „ « “n” 1 „ 1 1 1 n−i 1 n−i ≤ ≈ 1 − 1 − i ni n i! n i!e What about RLS? Expectation: E[X] = 1 P(1-bitflip) = 1 What about initialisation? How many one-bits in expectation after initialisation? E[X] = n · 1/2 = n/2 How likely is it that we get exactly n/2 one-bits? P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 26 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio General upper bound 1+1-EA: Conclusions & Exercises In general: P (i − bitf lip) = « „ « “n” 1 „ 1 1 1 n−i 1 n−i ≤ ≈ 1 − 1 − i ni n i! n i!e What about RLS? Expectation: E[X] = 1 P(1-bitflip) = 1 What about initialisation? How many one-bits in expectation after initialisation? E[X] = n · 1/2 = n/2 How likely is it that we get exactly n/2 one-bits? P r(X = n/2) = « „ « “ n ” 1 „ 1 n/2 1 − n = 100, P r(X = 50) ≈ 0.0796 n/2 nn/2 n Tail Inequalities help us to deal with these kind of problems. P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 26 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Tail Inequalities Given a random variable X it may assume values that are considerably larger or lower than its expectation; Tail inequalities: Estimate the probability that X deviates from the expectation by a defined amount δ; For many intermediate results, expected values are useless; May turn expected runtimes into bounds that hold with overwhelming probability. P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 27 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Markov’s inequality Markov Inequality The fundamental inequality from which many others are derived. P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 28 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Markov’s inequality Markov Inequality The fundamental inequality from which many others are derived. Definition (Markov’s Inequality) Let X be a random variable assuming only non-negative values, and E[X] its expectation. Then for all t ∈ R+ , P r[X ≥ t] ≤ P. S. Oliveto & X. Yao (University of Birmingham) E[X] . t Runtime Analysis of EAs WCCI 2012 28 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Markov’s inequality Markov Inequality The fundamental inequality from which many others are derived. Definition (Markov’s Inequality) Let X be a random variable assuming only non-negative values, and E[X] its expectation. Then for all t ∈ R+ , P r[X ≥ t] ≤ E[X] = 1; then: P r[X ≥ 2] ≤ P. S. Oliveto & X. Yao (University of Birmingham) E[X] 2 ≤ 1 2 E[X] . t (Number of bits that flip) Runtime Analysis of EAs WCCI 2012 28 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Markov’s inequality Markov Inequality The fundamental inequality from which many others are derived. Definition (Markov’s Inequality) Let X be a random variable assuming only non-negative values, and E[X] its expectation. Then for all t ∈ R+ , P r[X ≥ t] ≤ E[X] = 1; then: P r[X ≥ 2] ≤ E[X] 2 E[X] . t 1 (Number of bits that 2 E[X] n/2 = (2/3)n ≤ 43 (2/3)n ≤ flip) E[X] = n/2; then P r[X ≥ (2/3)n] ≤ (Number of one-bits after initialisation) P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 28 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Markov’s inequality Markov Inequality The fundamental inequality from which many others are derived. Definition (Markov’s Inequality) Let X be a random variable assuming only non-negative values, and E[X] its expectation. Then for all t ∈ R+ , P r[X ≥ t] ≤ E[X] = 1; then: P r[X ≥ 2] ≤ E[X] 2 E[X] . t 1 (Number of bits that 2 E[X] n/2 = (2/3)n ≤ 43 (2/3)n ≤ flip) E[X] = n/2; then P r[X ≥ (2/3)n] ≤ (Number of one-bits after initialisation) Usually Markov’s inequality is used iteratively to obtain stronger bounds! P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 28 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Chernoff bounds Chernoff Bounds Let X1 , XP 2 , . . . Xn be independent Poisson trials Pneach with probability pi ; For X = n i=1 pi . i=1 Xi the expectation is E(X) = Definition (Chernoff Bounds) −E[X]δ 2 1 2 2 . for 0 ≤ δ ≤ 1, P r(X ≤ (1 − δ)E[X]) ≤ e –E[X] » δ e . for δ > 0, P r(X > (1 + δ)E[X]) ≤ (1+δ) 1+δ P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 29 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Chernoff bounds Chernoff Bounds Let X1 , XP 2 , . . . Xn be independent Poisson trials Pneach with probability pi ; For X = n i=1 pi . i=1 Xi the expectation is E(X) = Definition (Chernoff Bounds) −E[X]δ 2 1 2 2 . for 0 ≤ δ ≤ 1, P r(X ≤ (1 − δ)E[X]) ≤ e –E[X] » δ e . for δ > 0, P r(X > (1 + δ)E[X]) ≤ (1+δ) 1+δ What is the probability that we have more than (2/3)n one-bits at initialisation? P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 29 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Chernoff bounds Chernoff Bounds Let X1 , XP 2 , . . . Xn be independent Poisson trials Pneach with probability pi ; For X = n i=1 pi . i=1 Xi the expectation is E(X) = Definition (Chernoff Bounds) −E[X]δ 2 1 2 2 . for 0 ≤ δ ≤ 1, P r(X ≤ (1 − δ)E[X]) ≤ e –E[X] » δ e . for δ > 0, P r(X > (1 + δ)E[X]) ≤ (1+δ) 1+δ What is the probability that we have more than (2/3)n one-bits at initialisation? pi = 1/2, E[X] = n · 1/2 = n/2, P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 29 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Chernoff bounds Chernoff Bounds Let X1 , XP 2 , . . . Xn be independent Poisson trials Pneach with probability pi ; For X = n i=1 pi . i=1 Xi the expectation is E(X) = Definition (Chernoff Bounds) −E[X]δ 2 1 2 2 . for 0 ≤ δ ≤ 1, P r(X ≤ (1 − δ)E[X]) ≤ e –E[X] » δ e . for δ > 0, P r(X > (1 + δ)E[X]) ≤ (1+δ) 1+δ What is the probability that we have more than (2/3)n one-bits at initialisation? pi = 1/2, E[X] = n · 1/2 = n/2, (we fix δ = 1/3 → (1 + δ)E[X] = (2/3)n); then: „ «n/2 e1/3 P r[X > (2/3)n] ≤ = c−n/2 4/3 (4/3) P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 29 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Chernoff bounds Chernoff Bound Simple Application Bitstring of length n = 100 P r(Xi ) = 1/2 and E(X) = np = 100/2 = 50. P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 30 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Chernoff bounds Chernoff Bound Simple Application Bitstring of length n = 100 P r(Xi ) = 1/2 and E(X) = np = 100/2 = 50. What is the probability to have at least 75 1-bits? P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 30 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Chernoff bounds Chernoff Bound Simple Application Bitstring of length n = 100 P r(Xi ) = 1/2 and E(X) = np = 100/2 = 50. What is the probability to have at least 75 1-bits? Markov: P r(X ≥ 75) ≤ 50 75 = 2 3 „ Chernoff: P r(X ≥ (1 + 1/2)50) ≤ Truth: P r(X ≥ 75) = P. S. Oliveto & X. Yao (University of Birmingham) P100 `100´ i=75 i √ e (3/2)3/2 «50 < 0.0045 2−100 < 0.000000282 Runtime Analysis of EAs WCCI 2012 30 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Chernoff bounds OneMax P OneMax (x)= n i=1 x[i]) f(x) n 2 1 n 1 2 P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs ones(x) WCCI 2012 31 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio RLS for OneMax ( OneMax (x)= P. S. Oliveto & X. Yao 0 0 0 0 0 0 0 1 2 3 4 5 (University of Birmingham) Pn i=1 x[i]) p0 = Runtime Analysis of EAs 6 6 E(T0 ) = 6 6 WCCI 2012 32 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio RLS for OneMax ( OneMax (x)= P. S. Oliveto & X. Yao 0 0 0 0 0 0 0 1 2 3 4 5 (University of Birmingham) Pn i=1 x[i]) p0 = Runtime Analysis of EAs 6 6 E(T0 ) = 6 6 WCCI 2012 32 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio RLS for OneMax ( OneMax (x)= P. S. Oliveto & X. Yao 0 0 0 0 0 1 0 1 2 3 4 5 (University of Birmingham) Pn i=1 x[i]) p0 = Runtime Analysis of EAs 6 6 E(T0 ) = 6 6 WCCI 2012 32 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio RLS for OneMax ( OneMax (x)= P. S. Oliveto & X. Yao 0 0 0 0 0 1 0 1 2 3 4 5 0 0 0 0 0 1 0 1 2 3 4 5 (University of Birmingham) Pn i=1 x[i]) Runtime Analysis of EAs p0 = 6 6 E(T0 ) = 6 6 p0 = 6 6 E(T0 ) = 6 6 WCCI 2012 32 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio RLS for OneMax ( OneMax (x)= P. S. Oliveto & X. Yao 0 0 0 0 0 1 0 1 2 3 4 5 0 0 0 0 0 1 0 1 2 3 4 5 (University of Birmingham) Pn i=1 x[i]) Runtime Analysis of EAs p0 = 6 6 E(T0 ) = 6 6 p1 = 5 6 E(T1 ) = 6 5 WCCI 2012 32 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio RLS for OneMax ( OneMax (x)= P. S. Oliveto & X. Yao 0 0 0 0 0 1 0 1 2 3 4 5 0 0 1 0 0 1 0 1 2 3 4 5 (University of Birmingham) Pn i=1 x[i]) Runtime Analysis of EAs p0 = 6 6 E(T0 ) = 6 6 p1 = 5 6 E(T1 ) = 6 5 WCCI 2012 32 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio RLS for OneMax ( OneMax (x)= P. S. Oliveto & X. Yao 0 0 0 0 0 1 0 1 2 3 4 5 0 0 1 0 0 1 0 1 2 3 4 5 0 0 1 0 0 1 0 1 2 3 4 5 (University of Birmingham) Pn i=1 x[i]) Runtime Analysis of EAs p0 = 6 6 E(T0 ) = 6 6 p1 = 5 6 E(T1 ) = 6 5 p1 = 5 6 E(T1 ) = 6 5 WCCI 2012 32 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio RLS for OneMax ( OneMax (x)= P. S. Oliveto & X. Yao 0 0 0 0 0 1 0 1 2 3 4 5 0 0 1 0 0 1 0 1 2 3 4 5 0 0 1 0 0 1 0 1 2 3 4 5 (University of Birmingham) Pn i=1 x[i]) Runtime Analysis of EAs p0 = 6 6 E(T0 ) = 6 6 p1 = 5 6 E(T1 ) = 6 5 p2 = 4 6 E(T2 ) = 6 4 WCCI 2012 32 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio RLS for OneMax ( OneMax (x)= P. S. Oliveto & X. Yao 0 0 0 0 0 1 0 1 2 3 4 5 0 0 1 0 0 1 0 1 2 3 4 5 1 0 1 0 0 1 0 1 2 3 4 5 (University of Birmingham) Pn i=1 x[i]) Runtime Analysis of EAs p0 = 6 6 E(T0 ) = 6 6 p1 = 5 6 E(T1 ) = 6 5 p2 = 4 6 E(T2 ) = 6 4 WCCI 2012 32 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio RLS for OneMax ( OneMax (x)= P. S. Oliveto & X. Yao 0 0 0 0 0 1 0 1 2 3 4 5 0 0 1 0 0 1 0 1 2 3 4 5 1 0 1 0 0 1 0 1 2 3 4 5 1 0 1 0 0 1 0 1 2 3 4 5 (University of Birmingham) Pn i=1 x[i]) Runtime Analysis of EAs p0 = 6 6 E(T0 ) = 6 6 p1 = 5 6 E(T1 ) = 6 5 p2 = 4 6 E(T2 ) = 6 4 p2 = 4 6 E(T2 ) = 6 4 WCCI 2012 32 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio RLS for OneMax ( OneMax (x)= P. S. Oliveto & X. Yao 0 0 0 0 0 1 0 1 2 3 4 5 0 0 1 0 0 1 0 1 2 3 4 5 1 0 1 0 0 1 0 1 2 3 4 5 1 0 1 0 0 1 0 1 2 3 4 5 (University of Birmingham) Pn i=1 x[i]) Runtime Analysis of EAs p0 = 6 6 E(T0 ) = 6 6 p1 = 5 6 E(T1 ) = 6 5 p2 = 4 6 E(T2 ) = 6 4 p3 = 3 6 E(T0 ) = 6 3 WCCI 2012 32 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio RLS for OneMax ( OneMax (x)= P. S. Oliveto & X. Yao 0 0 0 0 0 1 0 1 2 3 4 5 0 0 1 0 0 1 0 1 2 3 4 5 1 0 1 0 0 1 0 1 2 3 4 5 1 0 1 0 0 0 0 1 2 3 4 5 (University of Birmingham) Pn i=1 x[i]) Runtime Analysis of EAs p0 = 6 6 E(T0 ) = 6 6 p1 = 5 6 E(T1 ) = 6 5 p2 = 4 6 E(T2 ) = 6 4 p3 = 3 6 E(T3 ) = 6 3 WCCI 2012 32 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio RLS for OneMax ( OneMax (x)= P. S. Oliveto & X. Yao 0 0 0 0 0 1 0 1 2 3 4 5 0 0 1 0 0 1 0 1 2 3 4 5 1 0 1 0 0 1 0 1 2 3 4 5 1 0 1 0 0 1 0 1 2 3 4 5 (University of Birmingham) Pn i=1 x[i]) Runtime Analysis of EAs p0 = 6 6 E(T0 ) = 6 6 p1 = 5 6 E(T1 ) = 6 5 p2 = 4 6 E(T2 ) = 6 4 p2 = 4 6 E(T2 ) = 6 4 WCCI 2012 32 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio RLS for OneMax ( OneMax (x)= P. S. Oliveto & X. Yao 0 0 0 0 0 1 0 1 2 3 4 5 0 0 1 0 0 1 0 1 2 3 4 5 1 0 1 0 0 1 0 1 2 3 4 5 1 0 1 0 0 1 0 1 2 3 4 5 (University of Birmingham) Pn i=1 x[i]) Runtime Analysis of EAs p0 = 6 6 E(T0 ) = 6 6 p1 = 5 6 E(T1 ) = 6 5 p2 = 4 6 E(T2 ) = 6 4 p3 = 3 6 E(T3 ) = 6 3 WCCI 2012 32 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio RLS for OneMax ( OneMax (x)= P. S. Oliveto & X. Yao 0 0 0 0 0 1 0 1 2 3 4 5 0 0 1 0 0 1 0 1 2 3 4 5 1 0 1 0 0 1 0 1 2 3 4 5 1 0 1 0 1 1 0 1 2 3 4 5 (University of Birmingham) Pn i=1 x[i]) Runtime Analysis of EAs p0 = 6 6 E(T0 ) = 6 6 p1 = 5 6 E(T1 ) = 6 5 p2 = 4 6 E(T2 ) = 6 4 p3 = 3 6 E(T3 ) = 6 3 WCCI 2012 32 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio RLS for OneMax ( OneMax (x)= P. S. Oliveto & X. Yao 0 0 0 0 0 1 0 1 2 3 4 5 0 0 1 0 0 1 0 1 2 3 4 5 1 0 1 0 0 1 0 1 2 3 4 5 1 0 1 0 1 1 0 1 2 3 4 5 1 0 1 0 1 1 0 1 2 3 4 5 (University of Birmingham) Pn i=1 x[i]) Runtime Analysis of EAs p0 = 6 6 E(T0 ) = 6 6 p1 = 5 6 E(T1 ) = 6 5 p2 = 4 6 E(T2 ) = 6 4 p3 = 3 6 E(T3 ) = 6 3 p3 = 3 6 E(T3 ) = 6 3 WCCI 2012 32 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio RLS for OneMax ( OneMax (x)= P. S. Oliveto & X. Yao 0 0 0 0 0 1 0 1 2 3 4 5 0 0 1 0 0 1 0 1 2 3 4 5 1 0 1 0 0 1 0 1 2 3 4 5 1 0 1 0 1 1 0 1 2 3 4 5 1 0 1 0 1 1 0 1 2 3 4 5 (University of Birmingham) Pn i=1 x[i]) Runtime Analysis of EAs p0 = 6 6 E(T0 ) = 6 6 p1 = 5 6 E(T1 ) = 6 5 p2 = 4 6 E(T2 ) = 6 4 p3 = 3 6 E(T3 ) = 6 3 p4 = 2 6 E(T4 ) = 6 2 WCCI 2012 32 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio RLS for OneMax ( OneMax (x)= P. S. Oliveto & X. Yao 0 0 0 0 0 1 0 1 2 3 4 5 0 0 1 0 0 1 0 1 2 3 4 5 1 0 1 0 0 1 0 1 2 3 4 5 1 0 1 0 1 1 0 1 2 3 4 5 1 1 1 0 1 1 0 1 2 3 4 5 (University of Birmingham) Pn i=1 x[i]) Runtime Analysis of EAs p0 = 6 6 E(T0 ) = 6 6 p1 = 5 6 E(T1 ) = 6 5 p2 = 4 6 E(T2 ) = 6 4 p3 = 3 6 E(T3 ) = 6 3 p4 = 2 6 E(T4 ) = 6 2 WCCI 2012 32 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio RLS for OneMax ( OneMax (x)= P. S. Oliveto & X. Yao 0 0 0 0 0 1 0 1 2 3 4 5 0 0 1 0 0 1 0 1 2 3 4 5 1 0 1 0 0 1 0 1 2 3 4 5 1 0 1 0 1 1 0 1 2 3 4 5 1 1 1 0 1 1 0 1 2 3 4 5 1 1 1 0 1 1 0 1 2 3 4 5 (University of Birmingham) Pn i=1 x[i]) Runtime Analysis of EAs p0 = 6 6 E(T0 ) = 6 6 p1 = 5 6 E(T1 ) = 6 5 p2 = 4 6 E(T2 ) = 6 4 p3 = 3 6 E(T3 ) = 6 3 p4 = 2 6 E(T4 ) = 6 2 p4 = 2 6 E(T4 ) = 6 2 WCCI 2012 32 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio RLS for OneMax ( OneMax (x)= P. S. Oliveto & X. Yao 0 0 0 0 0 1 0 1 2 3 4 5 0 0 1 0 0 1 0 1 2 3 4 5 1 0 1 0 0 1 0 1 2 3 4 5 1 0 1 0 1 1 0 1 2 3 4 5 1 1 1 0 1 1 0 1 2 3 4 5 1 1 1 0 1 1 0 1 2 3 4 5 (University of Birmingham) Pn i=1 x[i]) Runtime Analysis of EAs p0 = 6 6 E(T0 ) = 6 6 p1 = 5 6 E(T1 ) = 6 5 p2 = 4 6 E(T2 ) = 6 4 p3 = 3 6 E(T3 ) = 6 3 p4 = 2 6 E(T4 ) = 6 2 p5 = 1 6 E(T5 ) = 6 1 WCCI 2012 32 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio RLS for OneMax ( OneMax (x)= P. S. Oliveto & X. Yao 0 0 0 0 0 1 0 1 2 3 4 5 0 0 1 0 0 1 0 1 2 3 4 5 1 0 1 0 0 1 0 1 2 3 4 5 1 0 1 0 1 1 0 1 2 3 4 5 1 1 1 0 1 1 0 1 2 3 4 5 1 1 1 1 1 1 0 1 2 3 4 5 (University of Birmingham) Pn i=1 x[i]) Runtime Analysis of EAs p0 = 6 6 E(T0 ) = 6 6 p1 = 5 6 E(T1 ) = 6 5 p2 = 4 6 E(T2 ) = 6 4 p3 = 3 6 E(T3 ) = 6 3 p4 = 2 6 E(T4 ) = 6 2 p5 = 1 6 E(T5 ) = 6 1 WCCI 2012 32 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio RLS for OneMax ( OneMax (x)= 0 0 0 0 0 1 0 1 2 3 4 5 0 0 1 0 0 1 0 1 2 3 4 5 1 0 1 0 0 1 0 1 2 3 4 5 1 0 1 0 1 1 0 1 2 3 4 5 1 1 1 0 1 1 0 1 2 3 4 5 1 1 1 1 1 1 0 1 2 3 4 5 Pn i=1 x[i]) p0 = 6 6 E(T0 ) = 6 6 p1 = 5 6 E(T1 ) = 6 5 p2 = 4 6 E(T2 ) = 6 4 p3 = 3 6 E(T3 ) = 6 3 p4 = 2 6 E(T4 ) = 6 2 p5 = 1 6 E(T5 ) = 6 1 E(T ) = E(T0 ) + E(T1 ) + · · · + E(T5 ) = 1/p0 + 1/p1 + · · · + 1/p5 = = P. S. Oliveto & X. Yao 5 5 6 X X X 1 6 1 = =6 = 6 · 2.45 = 14.7 p i i i i=0 i=0 i=1 (University of Birmingham) Runtime Analysis of EAs WCCI 2012 32 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio RLS for OneMax ( OneMax (x)= 0 0 0 0 0 1 2 3 P. S. Oliveto & X. Yao 0 (University of Birmingham) 0 0 0 Pn i=1 x[i]) 0 0 n Runtime Analysis of EAs : Generalisation p0 = n n E(T0 ) = n n WCCI 2012 33 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio RLS for OneMax ( OneMax (x)= 0 0 0 0 0 1 2 3 P. S. Oliveto & X. Yao 0 (University of Birmingham) 0 5 0 0 Pn i=1 x[i]) 0 0 n Runtime Analysis of EAs : Generalisation p0 = n n E(T0 ) = n n WCCI 2012 33 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio RLS for OneMax ( OneMax (x)= 0 0 0 0 0 1 2 3 P. S. Oliveto & X. Yao 0 (University of Birmingham) 1 0 0 Pn i=1 x[i]) 0 0 n Runtime Analysis of EAs : Generalisation p0 = n n E(T0 ) = n n WCCI 2012 33 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio RLS for OneMax ( OneMax (x)= 0 0 0 0 0 1 2 3 0 0 0 0 0 1 2 3 P. S. Oliveto & X. Yao Pn i=1 x[i]) : Generalisation 0 1 0 0 0 0 n p0 = n n E(T0 ) = n n 0 1 0 0 0 0 n p0 = n n E(T0 ) = n n (University of Birmingham) Runtime Analysis of EAs WCCI 2012 33 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio RLS for OneMax ( OneMax (x)= 0 0 0 0 0 1 2 3 0 0 0 0 0 1 2 3 P. S. Oliveto & X. Yao Pn i=1 x[i]) : Generalisation 0 1 0 0 0 0 n p0 = n n 0 1 0 0 0 0 n p1 = n−1 n (University of Birmingham) Runtime Analysis of EAs E(T0 ) = E(T1 ) = n n n n−1 WCCI 2012 33 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio RLS for OneMax ( OneMax (x)= 0 0 0 0 0 1 2 3 0 0 1 0 0 1 2 3 P. S. Oliveto & X. Yao Pn i=1 x[i]) : Generalisation 0 1 0 0 0 0 n p0 = n n 0 1 0 0 0 0 n p1 = n−1 n (University of Birmingham) Runtime Analysis of EAs E(T0 ) = E(T1 ) = n n n n−1 WCCI 2012 33 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio RLS for OneMax ( OneMax (x)= 0 0 0 0 0 1 2 3 0 0 1 0 0 1 2 3 0 0 1 0 0 1 2 3 P. S. Oliveto & X. Yao Pn i=1 x[i]) : Generalisation 0 1 0 0 0 0 n p0 = n n 0 1 0 0 0 0 n p1 = n−1 n E(T1 ) = n n−1 0 1 0 0 0 0 n p1 = n−1 n E(T1 ) = n n−1 (University of Birmingham) Runtime Analysis of EAs E(T0 ) = n n WCCI 2012 33 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio RLS for OneMax ( OneMax (x)= 0 0 0 0 0 1 2 3 0 0 1 0 0 1 2 3 0 0 1 0 0 1 2 3 P. S. Oliveto & X. Yao Pn i=1 x[i]) : Generalisation 0 1 0 0 0 0 n p0 = n n 0 1 0 0 0 0 n p1 = n−1 n E(T1 ) = n n−1 0 1 0 0 0 0 n p2 = n−2 n E(T2 ) = n n−2 (University of Birmingham) Runtime Analysis of EAs E(T0 ) = n n WCCI 2012 33 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio RLS for OneMax ( OneMax (x)= 0 0 0 0 0 1 2 3 0 0 1 0 0 1 2 3 0 0 1 0 0 1 2 3 P. S. Oliveto & X. Yao Pn i=1 x[i]) : Generalisation 0 1 0 0 0 0 n p0 = n n 0 1 0 0 0 0 n p1 = n−1 n E(T1 ) = n n−1 0 1 0 0 0 1 n p2 = n−2 n E(T2 ) = n n−2 (University of Birmingham) Runtime Analysis of EAs E(T0 ) = n n WCCI 2012 33 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio RLS for OneMax ( OneMax (x)= 0 0 0 0 0 1 2 3 0 0 1 0 0 1 2 3 0 0 1 0 0 1 2 3 1 1 1 1 0 1 2 3 P. S. Oliveto & X. Yao Pn i=1 x[i]) : Generalisation 0 1 0 0 0 0 n p0 = n n 0 1 0 0 0 0 n p1 = n−1 n E(T1 ) = n n−1 0 1 0 0 0 1 n p2 = n−2 n E(T2 ) = n n−2 0 (University of Birmingham) 1 1 1 1 1 n Runtime Analysis of EAs pn−1 = 1 n E(T0 ) = n n E(Tn−1 ) = n 1 WCCI 2012 33 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio RLS for OneMax ( OneMax (x)= 0 0 0 0 0 1 2 3 0 0 1 0 0 1 2 3 0 0 1 0 0 1 2 3 1 1 1 1 0 0 1 2 3 4 P. S. Oliveto & X. Yao Pn i=1 x[i]) : Generalisation 0 1 0 0 0 0 n p0 = n n 0 1 0 0 0 0 n p1 = n−1 n E(T1 ) = n n−1 0 1 0 0 0 1 n p2 = n−2 n E(T2 ) = n n−2 (University of Birmingham) 1 1 1 1 1 n Runtime Analysis of EAs pn−1 = 1 n E(T0 ) = n n E(Tn−1 ) = n 1 WCCI 2012 33 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio RLS for OneMax ( OneMax (x)= 0 0 0 0 0 1 2 3 0 0 1 0 0 1 2 3 0 0 1 0 0 1 2 3 1 1 1 1 0 1 2 3 P. S. Oliveto & X. Yao Pn i=1 x[i]) : Generalisation 0 1 0 0 0 0 n p0 = n n 0 1 0 0 0 0 n p1 = n−1 n E(T1 ) = n n−1 0 1 0 0 0 1 n p2 = n−2 n E(T2 ) = n n−2 1 (University of Birmingham) 1 1 1 1 1 n Runtime Analysis of EAs pn−1 = 1 n E(T0 ) = n n E(Tn−1 ) = n 1 WCCI 2012 33 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio RLS for OneMax ( OneMax (x)= 1 0 0 0 0 0 0 1 2 3 0 0 1 0 0 1 2 3 0 0 1 0 0 1 2 3 1 1 1 1 2 3 Pn i=1 x[i]) : Generalisation 0 1 0 0 0 0 n p0 = n n 0 1 0 0 0 0 n p1 = n−1 n E(T1 ) = n n−1 0 1 0 0 0 1 n p2 = n−2 n E(T2 ) = n n−2 1 1 1 1 1 1 n pn−1 = 1 n E(T0 ) = n n E(Tn−1 ) = n 1 E(T ) = E(T0 ) + E(T1 ) + · · · + E(Tn−1 ) = 1/p1 + 1/p2 + · · · + 1/pn−1 = = n−1 X i=0 P. S. Oliveto & X. Yao n n X X 1 n 1 = =n = n · H(n) = n log n + Θ(n) = O(n log n) pi i i i=1 i=1 (University of Birmingham) Runtime Analysis of EAs WCCI 2012 33 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Coupon collector’s problem Coupon collector’s problem The Coupon collector’s problem There are n types of coupons and at each trial one coupon is chosen at random. Each coupon has the same probability of being extracted. The goal is to find the exact number of trials before the collector has obtained all the n coupons. Theorem (The coupon collector’s Theorem) Let T be the time for all the n coupons to be collected. Then E(T ) = n−1 X 1 i=0 pi+1 = n−1 X i=0 n−1 X 1 n =n = n−i i i=0 = n(log n + Θ(1)) = n log n + O(n). P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 34 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Coupon collector’s problem Coupon collector’s problem: Upper bound on time What is the probability that the time to collect n coupons is greater than n ln n + O(n)? Theorem (Coupon collector upper bound on time) Let T be the time for all the n coupons to be collected. Then P r(T ≥ (1 + )n ln n) ≤ n− Proof P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 35 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Coupon collector’s problem Coupon collector’s problem: Upper bound on time What is the probability that the time to collect n coupons is greater than n ln n + O(n)? Theorem (Coupon collector upper bound on time) Let T be the time for all the n coupons to be collected. Then P r(T ≥ (1 + )n ln n) ≤ n− Proof 1 n 1 1− n „ «t 1 1− n Probability of choosing a given coupon Probability of not choosing a given coupon Probability of not choosing a given coupon for t rounds The probability that one of the n coupons is not chosen in t rounds is less than „ «t 1 n· 1− n (Union Bound) Hence, for t = cn ln n ` P r(T ≥ cn ln n) ≤ n 1 − 1/n)cn ln n ≤ n · e−c ln n = n · n−c = n−c+1 P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 35 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Coupon collector’s problem Coupon collector’s problem: lower bound on time What is the probability that the time to collect n coupons is less than n ln n + O(n)? Theorem (Coupon collector lower bound on time (Doerr, 2011)) Let T be the time for all the n coupons to be collected. Then for all > 0 P r(T < (1 − )(n − 1) ln n) ≤ exp(−n ) P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 36 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Coupon collector’s problem Coupon collector’s problem: lower bound on time What is the probability that the time to collect n coupons is less than n ln n + O(n)? Theorem (Coupon collector lower bound on time (Doerr, 2011)) Let T be the time for all the n coupons to be collected. Then for all > 0 P r(T < (1 − )(n − 1) ln n) ≤ exp(−n ) Corollary The expected time for RLS to optimise OneMax is Θ(n ln n). Furthermore, P r(T ≥ (1 + )n ln n) ≤ n− and P r(T < (1 − )(n − 1) ln n) ≤ exp(−n ) What about the (1+1)-EA? P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 36 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio AFL method for upper bounds Artificial Fitness Levels [Droste et al., 2002] Observation Divide the search space |S| = 2n into m < 2n sets A1 , . . . Am such that: Idea 1 2 3 Due to elitism, fitness is monotone increasing ∀i 6= j : Ai ∩ Aj = ∅ Sm n i=0 Ai = {0, 1} for all points a ∈ Ai and b ∈ Aj it happens that f (a) < f (b) if i < j. requirement P. S. Oliveto & X. Yao Am contains only optimal search points. (University of Birmingham) Runtime Analysis of EAs WCCI 2012 37 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio AFL method for upper bounds Artificial Fitness Levels [Droste et al., 2002] Observation Divide the search space |S| = 2n into m < 2n sets A1 , . . . Am such that: Idea 1 2 3 Due to elitism, fitness is monotone increasing ∀i 6= j : Ai ∩ Aj = ∅ Sm n i=0 Ai = {0, 1} for all points a ∈ Ai and b ∈ Aj it happens that f (a) < f (b) if i < j. requirement Am contains only optimal search points. Then: pi probability that point in Ai is mutated to a point in Aj with j > i. P Expected time: E(T ) ≤ i p1 i Very simple, yet often powerful method for upper bounds P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 37 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio AFL method for upper bounds Artificial Fitness Levels D. Sudholt, Tutorial 2011 Let: p(Ai ) be the probability that a random chosen point belongs to level Ai si be the probability to leave level Ai for Aj with j > i Then: „ « „ « m−1 X X 1 1 1 1 1 E(T ) ≤ p(Ai ) · + ··· + ≤ + ··· + = s s s s s m−1 1 m−1 i i=1 i 1≤i≤m−1 ` ´ P Inequality 1: Law of total probability E(T ) = i P r(F ) · E(T |F ) Inequality 2: Trivial! P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 38 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio AFL method for upper bounds (1+1)-EA for OneMax Theorem The expected runtime of the (1+1)-EA for OneMax is O(n ln n). Proof P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 39 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio AFL method for upper bounds (1+1)-EA for OneMax Theorem The expected runtime of the (1+1)-EA for OneMax is O(n ln n). Proof The current solution is in level Ai if it has i zeroes (hence n − i ones) P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 39 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio AFL method for upper bounds (1+1)-EA for OneMax Theorem The expected runtime of the (1+1)-EA for OneMax is O(n ln n). Proof The current solution is in level Ai if it has i zeroes (hence n − i ones) To reach a higher fitness level it is sufficient to flip a zero into a one and leave the other bits unchanged P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 39 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio AFL method for upper bounds (1+1)-EA for OneMax Theorem The expected runtime of the (1+1)-EA for OneMax is O(n ln n). Proof The current solution is in level Ai if it has i zeroes (hence n − i ones) To reach a higher fitness level it is sufficient to flip a zero into a one and leave the other bits unchanged „ «n−1 1 1 i The probability is si ≥ i · n 1− n ≥ en P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 39 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio AFL method for upper bounds (1+1)-EA for OneMax Theorem The expected runtime of the (1+1)-EA for OneMax is O(n ln n). Proof The current solution is in level Ai if it has i zeroes (hence n − i ones) To reach a higher fitness level it is sufficient to flip a zero into a one and leave the other bits unchanged „ «n−1 1 1 i The probability is si ≥ i · n 1− n ≥ en Hence, P. S. Oliveto & X. Yao 1 si ≤ en i (University of Birmingham) Runtime Analysis of EAs WCCI 2012 39 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio AFL method for upper bounds (1+1)-EA for OneMax Theorem The expected runtime of the (1+1)-EA for OneMax is O(n ln n). Proof The current solution is in level Ai if it has i zeroes (hence n − i ones) To reach a higher fitness level it is sufficient to flip a zero into a one and leave the other bits unchanged „ «n−1 1 1 i The probability is si ≥ i · n 1− n ≥ en Hence, s1 ≤ en i i ` ´ Then Artificial Fitness Levels : E(T ) ≤ m−1 X s−1 ≤ i i=1 m−1 n X 1 X en ≤e·n ≤ e · n · (ln n + 1) = O(n ln n) i i i=1 i=1 Is the (1+1)-EA quicker than n ln n? P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 39 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio AFL method for upper bounds (1+1)-EA lower bound for OneMax Theorem (Droste,Jansen,Wegener, 2002) The expected runtime of the (1+1)-EA for OneMax is Ω(n ln n). Proof Idea P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 40 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio AFL method for upper bounds (1+1)-EA lower bound for OneMax Theorem (Droste,Jansen,Wegener, 2002) The expected runtime of the (1+1)-EA for OneMax is Ω(n ln n). Proof Idea 1 At most n/2 one-bits are created during initialisation with probability at least 1/2 (By symmetry of the binomial distribution). P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 40 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio AFL method for upper bounds (1+1)-EA lower bound for OneMax Theorem (Droste,Jansen,Wegener, 2002) The expected runtime of the (1+1)-EA for OneMax is Ω(n ln n). Proof Idea 1 At most n/2 one-bits are created during initialisation with probability at least 1/2 (By symmetry of the binomial distribution). 2 There is a constant probability that in cn ln n steps one of the n/2 remaining zero-bits does not flip. P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 40 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio AFL method for upper bounds Lower bound for OneMax Theorem (Droste,Jansen,Wegener, 2002) The expected runtime of the (1+1)-EA for OneMax is Ω(n log n). Proof of 2. 1 − 1/n P. S. Oliveto & X. Yao (University of Birmingham) a given bit does not flip Runtime Analysis of EAs WCCI 2012 41 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio AFL method for upper bounds Lower bound for OneMax Theorem (Droste,Jansen,Wegener, 2002) The expected runtime of the (1+1)-EA for OneMax is Ω(n log n). Proof of 2. 1 − 1/n (1 − 1/n)t P. S. Oliveto & X. Yao (University of Birmingham) a given bit does not flip a given bit does not flip in t steps Runtime Analysis of EAs WCCI 2012 41 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio AFL method for upper bounds Lower bound for OneMax Theorem (Droste,Jansen,Wegener, 2002) The expected runtime of the (1+1)-EA for OneMax is Ω(n log n). Proof of 2. 1 − 1/n (1 − 1/n)t 1 − (1 − 1/n)t P. S. Oliveto & X. Yao (University of Birmingham) a given bit does not flip a given bit does not flip in t steps it flips at least once in t steps Runtime Analysis of EAs WCCI 2012 41 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio AFL method for upper bounds Lower bound for OneMax Theorem (Droste,Jansen,Wegener, 2002) The expected runtime of the (1+1)-EA for OneMax is Ω(n log n). Proof of 2. 1 − 1/n (1 − 1/n)t 1 − (1 − 1/n)t (1 − (1 − 1/n)t )n/2 P. S. Oliveto & X. Yao (University of Birmingham) a given bit does not flip a given bit does not flip in t steps it flips at least once in t steps n/2 bits flip at least once in t steps Runtime Analysis of EAs WCCI 2012 41 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio AFL method for upper bounds Lower bound for OneMax Theorem (Droste,Jansen,Wegener, 2002) The expected runtime of the (1+1)-EA for OneMax is Ω(n log n). Proof of 2. 1 − 1/n (1 − 1/n)t 1 − (1 − 1/n)t (1 − (1 − 1/n)t )n/2 1 − [1 − (1 − 1/n)t ]n/2 P. S. Oliveto & X. Yao (University of Birmingham) a given bit does not flip a given bit does not flip in t steps it flips at least once in t steps n/2 bits flip at least once in t steps at least one of the n/2 bits does not flip in t steps Runtime Analysis of EAs WCCI 2012 41 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio AFL method for upper bounds Lower bound for OneMax Theorem (Droste,Jansen,Wegener, 2002) The expected runtime of the (1+1)-EA for OneMax is Ω(n log n). Proof of 2. 1 − 1/n (1 − 1/n)t 1 − (1 − 1/n)t (1 − (1 − 1/n)t )n/2 1 − [1 − (1 − 1/n)t ]n/2 a given bit does not flip a given bit does not flip in t steps it flips at least once in t steps n/2 bits flip at least once in t steps at least one of the n/2 bits does not flip in t steps Set t = (n − 1) log n. Then: 1 − [1 − (1 − 1/n)t ]n/2 = 1 − [1 − (1 − 1/n)(n−1) log n ]n/2 ≥ ≥ 1 − [1 − (1/e)log n ]n/2 = 1 − [1 − 1/n]n/2 = = 1 − [1 − 1/n]n·1/2 ≥ 1 − (2e)−1/2 = c P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 41 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio AFL method for upper bounds Lower bound for OneMax (2) Theorem (Droste, Jansen, Wegener, 2002) The expected runtime of the (1+1)-EA for OneMax is Ω(n log n). Proof 1 At most n/2 one-bits are created during initialisation with probability at least 1/2 (By symmetry of the binomial distribution). 2 There is a constant probability that in cn log n steps one of the n/2 remaining zero-bits does not flip. P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 42 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio AFL method for upper bounds Lower bound for OneMax (2) Theorem (Droste, Jansen, Wegener, 2002) The expected runtime of the (1+1)-EA for OneMax is Ω(n log n). Proof 1 At most n/2 one-bits are created during initialisation with probability at least 1/2 (By symmetry of the binomial distribution). 2 There is a constant probability that in cn log n steps one of the n/2 remaining zero-bits does not flip. The Expected runtime is: E[T ] = ∞ X t · p(t) ≥ [(n − 1) log n] · p[t = (n − 1) log n] ≥ t=1 ≥ [(n − 1) log n] · [(1/2) · (1 − (2e)−1/2 ) = Ω(n log n) First inequality: law of total probability The upper bound given by artificial fitness levels is indeed tight! P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 42 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio AFL method for upper bounds Pn Qi Artificial Fitness Levels Exercises: LeadingOnes(x) = i=1 j=1 x[j] Theorem The expected runtime of RLS for LeadingOnes is O(n2 ). P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 43 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio AFL method for upper bounds Pn Qi Artificial Fitness Levels Exercises: LeadingOnes(x) = i=1 j=1 x[j] Theorem The expected runtime of RLS for LeadingOnes is O(n2 ). Proof Let partition Ai contain search points with exactly i leading ones To leave level Ai it suffices to flip the zero at position i + 1 and s−1 =n i P P −1 2 E(T ) ≤ n−1 s = n i=1 n = O(n ) i=1 i si = P. S. Oliveto & X. Yao 1 n (University of Birmingham) Runtime Analysis of EAs WCCI 2012 43 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio AFL method for upper bounds Pn Qi Artificial Fitness Levels Exercises: LeadingOnes(x) = i=1 j=1 x[j] Theorem The expected runtime of RLS for LeadingOnes is O(n2 ). Proof Let partition Ai contain search points with exactly i leading ones To leave level Ai it suffices to flip the zero at position i + 1 and s−1 =n i P P −1 2 E(T ) ≤ n−1 s = n i=1 n = O(n ) i=1 i si = 1 n Theorem The expected runtime of the (1+1)-EA for LeadingOnes is O(n2 ). P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 43 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio AFL method for upper bounds Pn Qi Artificial Fitness Levels Exercises: LeadingOnes(x) = i=1 j=1 x[j] Theorem The expected runtime of RLS for LeadingOnes is O(n2 ). Proof Let partition Ai contain search points with exactly i leading ones To leave level Ai it suffices to flip the zero at position i + 1 and s−1 =n i P P −1 2 E(T ) ≤ n−1 s = n i=1 n = O(n ) i=1 i si = 1 n Theorem The expected runtime of the (1+1)-EA for LeadingOnes is O(n2 ). Proof Left as Exercise. P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 43 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio AFL method for upper bounds Fitness Levels Advanced Exercises (Populations) Theorem The expected runtime of (1+λ)-EA for LeadingOnes is O(λn + n2 ) [Jansen et al., 2005]. P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 44 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio AFL method for upper bounds Fitness Levels Advanced Exercises (Populations) Theorem The expected runtime of (1+λ)-EA for LeadingOnes is O(λn + n2 ) [Jansen et al., 2005]. Proof Let partition Ai contain search points with exactly i leading ones To leave level Ai it suffices to flip the zero at position i + 1 „ «λ 1 si = 1 − 1 − en ≥ 1 − e−λ/(en) 1 2 si ≥ 1 − λ si ≥ 2en 1 e Case 1: λ ≥ en Case 2: λ < en „„ « „ Pn 1 Pn P −1 E(T ) ≤ λ · n−1 s ≤ λ + i=1 c i=1 i=1 i „ „ «« n2 O λ· n+ λ = O(λ · n + n2 ) P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs 2en λ «« = WCCI 2012 44 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio AFL method for upper bounds Fitness Levels Advanced Exercises (Populations) Theorem The expected runtime of the (µ+1)-EA for LeadingOnes is O(µ · n2 ). P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 45 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio AFL method for upper bounds Fitness Levels Advanced Exercises (Populations) Theorem The expected runtime of the (µ+1)-EA for LeadingOnes is O(µ · n2 ). Proof Left as Exercise. Theorem The expected runtime of the (µ+1)-EA for OneMax is O(µ · n log n). P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 45 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio AFL method for upper bounds Fitness Levels Advanced Exercises (Populations) Theorem The expected runtime of the (µ+1)-EA for LeadingOnes is O(µ · n2 ). Proof Left as Exercise. Theorem The expected runtime of the (µ+1)-EA for OneMax is O(µ · n log n). Proof Left as Exercise. P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 45 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio AFL method for parent populations Artificial Fitness Levels for Populations D. Sudholt, Tutorial 2011 Let: To be the expected time for a fraction χ(i) of the population to be in level Ai si be the probability to leave level Ai for Aj with j > i given χ(i) in level Ai Then: E(T ) ≤ m−1 X „ i=1 P. S. Oliveto & X. Yao (University of Birmingham) 1 + To si Runtime Analysis of EAs « WCCI 2012 46 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio AFL method for parent populations Applications to (µ+1)-EA Theorem The expected runtime of (µ+1)-EA for LeadingOnes is O(µn log n + n2 ) [Witt, 2006]. P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 47 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio AFL method for parent populations Applications to (µ+1)-EA Theorem The expected runtime of (µ+1)-EA for LeadingOnes is O(µn log n + n2 ) [Witt, 2006]. Proof Let partition Ai contain search points with exactly i leading ones P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 47 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio AFL method for parent populations Applications to (µ+1)-EA Theorem The expected runtime of (µ+1)-EA for LeadingOnes is O(µn log n + n2 ) [Witt, 2006]. Proof Let partition Ai contain search points with exactly i leading ones To leave level Ai it suffices to flip the zero at position i + 1 of the best individual P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 47 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio AFL method for parent populations Applications to (µ+1)-EA Theorem The expected runtime of (µ+1)-EA for LeadingOnes is O(µn log n + n2 ) [Witt, 2006]. Proof Let partition Ai contain search points with exactly i leading ones To leave level Ai it suffices to flip the zero at position i + 1 of the best individual We set χ(i) = n/ ln n P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 47 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio AFL method for parent populations Applications to (µ+1)-EA Theorem The expected runtime of (µ+1)-EA for LeadingOnes is O(µn log n + n2 ) [Witt, 2006]. Proof Let partition Ai contain search points with exactly i leading ones To leave level Ai it suffices to flip the zero at position i + 1 of the best individual We set χ(i) = n/ ln n Given „ j copies «n of the best individual another replica is created with probability j j 1 1 − ≥ 2eµ µ n P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 47 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio AFL method for parent populations Applications to (µ+1)-EA Theorem The expected runtime of (µ+1)-EA for LeadingOnes is O(µn log n + n2 ) [Witt, 2006]. Proof Let partition Ai contain search points with exactly i leading ones To leave level Ai it suffices to flip the zero at position i + 1 of the best individual We set χ(i) = n/ ln n Given „ j copies «n of the best individual another replica is created with probability j j 1 1 − ≥ 2eµ µ n To ≤ P. S. Oliveto & X. Yao Pn/ ln n j=1 2eµ j = 2eµ ln n (University of Birmingham) Runtime Analysis of EAs WCCI 2012 47 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio AFL method for parent populations Applications to (µ+1)-EA Theorem The expected runtime of (µ+1)-EA for LeadingOnes is O(µn log n + n2 ) [Witt, 2006]. Proof Let partition Ai contain search points with exactly i leading ones To leave level Ai it suffices to flip the zero at position i + 1 of the best individual We set χ(i) = n/ ln n Given „ j copies «n of the best individual another replica is created with probability j j 1 1 − ≥ 2eµ µ n To ≤ P. S. Oliveto & X. Yao Pn/ ln n j=1 1 si ≥ 2 si ≥ 2eµ j n/ ln n µ n/ ln n µ · · = 2eµ ln n 1 en 1 en (University of Birmingham) = ≥ 1 eµ ln n 1 en n ln n n ln n Case 1: µ > Case 2: µ ≤ Runtime Analysis of EAs WCCI 2012 47 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio AFL method for parent populations Applications to (µ+1)-EA Theorem The expected runtime of (µ+1)-EA for LeadingOnes is O(µn log n + n2 ) [Witt, 2006]. Proof Let partition Ai contain search points with exactly i leading ones To leave level Ai it suffices to flip the zero at position i + 1 of the best individual We set χ(i) = n/ ln n Given „ j copies «n of the best individual another replica is created with probability j j 1 1 − ≥ 2eµ µ n To ≤ Pn/ ln n j=1 1 si ≥ 2 si ≥ 2eµ j n/ ln n µ n/ ln n µ · · = 2eµ ln n 1 en 1 en = ≥ 1 eµ ln n 1 en n ln n n ln n Case 1: µ > Case 2: µ ≤ „ « ` ´ E(T ) ≤ i=1 (To + s−1 = i=1 2eµ ln n + en + eµ ln n i )≤ „ « ` ´ n · 2eµ ln n + en + eµ ln n = O(nµ ln n + n2 ) Pn−1 P. S. Oliveto & X. Yao (University of Birmingham) Pn Runtime Analysis of EAs WCCI 2012 47 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio AFL method for parent populations Populations Fitness Levels: Exercise Theorem The expected runtime of the (µ+1)-EA for OneMax is O(µn + n log n). P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 48 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio AFL method for parent populations Populations Fitness Levels: Exercise Theorem The expected runtime of the (µ+1)-EA for OneMax is O(µn + n log n). Proof Left as Exercise. P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 48 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio AFL for non-elitist EAs Advanced: Fitness Levels for non-Elitist Populations [Lehre, 2011] P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 49 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio AFL for lower bounds Advanced: Fitness Levels for Lower Bounds [Sudholt, 2010] P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 50 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio AFL for lower bounds Artificial Fitness Levels: Conclusions It’s a powerful general method to obtain (often) tight upper bounds on the runtime of simple EAs; For offspring populations tight bounds can often be achieved with the general method; For parent populations takeover times have to be introduced; Recent methods have been presented to deal with non-elitism and for lower bounds. P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 51 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Drift Analysis: Example 1 Friday night dinner at the restaurant. Peter walks back from the restaurant to the hotel. The restaurant is n meters away from the hotel; Peter moves towards the hotel of 1 meter in each step Question How many steps does Peter need to reach his hotel? P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 52 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Drift Analysis: Example 1 Friday night dinner at the restaurant. Peter walks back from the restaurant to the hotel. The restaurant is n meters away from the hotel; Peter moves towards the hotel of 1 meter in each step Question How many steps does Peter need to reach his hotel? n steps P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 52 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Drift Analysis: Formalisation Define a distance function d(x) to measure the distance from the hotel; d(x) = x, x ∈ {0, . . . , n} (In our case the distance is simply the number of metres from the hotel). Estimate the expected “speed” (drift), the expected decrease in distance from the goal; ( 0, if Xt = 0, d(Xt ) − d(Xt+1 ) = 1, if Xt ∈ {1, . . . , n} Time Then the expected time to reach the hotel (goal) is: E(T ) = P. S. Oliveto & X. Yao (University of Birmingham) n maximum distance = =n drif t 1 Runtime Analysis of EAs WCCI 2012 53 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Drift Analysis: Example 2 Friday night dinner at the restaurant. Peter walks back from the restaurant to the hotel but had a few drinks. The restaurant is n meters away from the hotel; Peter moves towards the hotel of 1 meter in each step with probability 0.6. Peter moves away from the hotel of 1 meter in each step with probability 0.4. Question How many steps does Peter need to reach his hotel? P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 54 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Drift Analysis: Example 2 Friday night dinner at the restaurant. Peter walks back from the restaurant to the hotel but had a few drinks. The restaurant is n meters away from the hotel; Peter moves towards the hotel of 1 meter in each step with probability 0.6. Peter moves away from the hotel of 1 meter in each step with probability 0.4. Question How many steps does Peter need to reach his hotel? 5n steps Let us calculate this through drift analysis. P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 54 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Drift Analysis (2): Formalisation Define the same distance function d(x) as before to measure the distance from the hotel; d(x) = x, x ∈ {0, . . . , n} (simply the number of metres from the hotel). Estimate the expected “speed” (drift), the expected decrease in distance from the goal; 8 > <0, if Xt = 0, d(Xt ) − d(Xt+1 ) = 1, if Xt ∈ {1, . . . , n}with probability 0.6 > : −1, if Xt ∈ {1, . . . , n}with probability 0.4 The expected dicrease in distance (drift) is: E[d(Xt ) − d(Xt+1 )] = 0.6 · 1 + 0.4 · (−1) = 0.6 − 0.4 = 0.2 Time Then the expected time to reach the hotel (goal) is: E(T ) = P. S. Oliveto & X. Yao (University of Birmingham) maximum distance n = = 5n drif t 0.2 Runtime Analysis of EAs WCCI 2012 55 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Additive Drift Theorem Additive Drift Theorem Theorem (Additive Drift Theorem for Upper Bounds [He and Yao, 2001]) Let {Xt }t≥0 be a Markov process over a set of states S, and d : S → R+ 0 a function that assigns a non-negative real number to every state. Let the time to reach the optimum be T := min{t ≥ 0 : d(Xt ) = 0}. If there exists δ > 0 such that at any time step t ≥ 0 and at any state Xt > 0 the following condition holds: E(∆(t)|d(Xt ) > 0) = E(d(Xt ) − d(Xt+1 ) | d(Xt ) > 0) ≥ δ then E(T | X0 > 0) ≤ and E(T ) ≤ P. S. Oliveto & X. Yao (University of Birmingham) d(X0 ) δ E(d(X0 )) . δ Runtime Analysis of EAs (1) (2) (3) WCCI 2012 56 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Additive Drift Theorem Drift Analysis for Leading Ones Theorem The expected time for the (1+1)-EA to optimise LeadingOnes is O(n2 ) Proof P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 57 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Additive Drift Theorem Drift Analysis for Leading Ones Theorem The expected time for the (1+1)-EA to optimise LeadingOnes is O(n2 ) Proof 1 Let g(Xt ) = i where i is the number of missing leading ones; P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 57 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Additive Drift Theorem Drift Analysis for Leading Ones Theorem The expected time for the (1+1)-EA to optimise LeadingOnes is O(n2 ) Proof 1 Let g(Xt ) = i where i is the number of missing leading ones; 2 The negative drift is 0 since if a leading one is removed from the current solution the new point will not be accepted; P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 57 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Additive Drift Theorem Drift Analysis for Leading Ones Theorem The expected time for the (1+1)-EA to optimise LeadingOnes is O(n2 ) Proof 1 Let g(Xt ) = i where i is the number of missing leading ones; 2 The negative drift is 0 since if a leading one is removed from the current solution the new point will not be accepted; 3 A positive drift (i.e. of length 1) is achieved as long as the first 0 is flipped and the leading ones are remained unchanged: E(∆+ (t)) = n−i X ` k · (p(∆+ (t)) = k) ≥ 1 · 1/n · 1 − 1/n)n−1 ≥ 1/(en) k=1 P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 57 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Additive Drift Theorem Drift Analysis for Leading Ones Theorem The expected time for the (1+1)-EA to optimise LeadingOnes is O(n2 ) Proof 1 Let g(Xt ) = i where i is the number of missing leading ones; 2 The negative drift is 0 since if a leading one is removed from the current solution the new point will not be accepted; 3 A positive drift (i.e. of length 1) is achieved as long as the first 0 is flipped and the leading ones are remained unchanged: E(∆+ (t)) = n−i X ` k · (p(∆+ (t)) = k) ≥ 1 · 1/n · 1 − 1/n)n−1 ≥ 1/(en) k=1 4 Hence, E[∆(t)] = E(∆+ (t)) − E(∆− (t)) ≥ 1/(en) = δ P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 57 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Additive Drift Theorem Drift Analysis for Leading Ones Theorem The expected time for the (1+1)-EA to optimise LeadingOnes is O(n2 ) Proof 1 Let g(Xt ) = i where i is the number of missing leading ones; 2 The negative drift is 0 since if a leading one is removed from the current solution the new point will not be accepted; 3 A positive drift (i.e. of length 1) is achieved as long as the first 0 is flipped and the leading ones are remained unchanged: E(∆+ (t)) = n−i X ` k · (p(∆+ (t)) = k) ≥ 1 · 1/n · 1 − 1/n)n−1 ≥ 1/(en) k=1 4 Hence, E[∆(t)] = E(∆+ (t)) − E(∆− (t)) ≥ 1/(en) = δ 5 The expected runtime is (i.e. Eq. (6)): E(T | d(X0 ) > 0) ≤ P. S. Oliveto & X. Yao (University of Birmingham) g(X0 ) n ≤ = e · n2 = O(n2 ) δ 1/(en) Runtime Analysis of EAs WCCI 2012 57 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Additive Drift Theorem Exercises Theorem The expected time for RLS to optimise LeadingOnes is O(n2 ) Proof P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 58 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Additive Drift Theorem Exercises Theorem The expected time for RLS to optimise LeadingOnes is O(n2 ) Proof Left as exercise. Theorem Let λ ≥ en. Then the expected time for the (1+λ)-EA to optimise LeadingOnes is O(λn) Proof P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 58 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Additive Drift Theorem Exercises Theorem The expected time for RLS to optimise LeadingOnes is O(n2 ) Proof Left as exercise. Theorem Let λ ≥ en. Then the expected time for the (1+λ)-EA to optimise LeadingOnes is O(λn) Proof Left as exercise. Theorem Let λ < en. Then the expected time for the (1+λ)-EA to optimise LeadingOnes is O(n2 ) Proof P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 58 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Additive Drift Theorem Exercises Theorem The expected time for RLS to optimise LeadingOnes is O(n2 ) Proof Left as exercise. Theorem Let λ ≥ en. Then the expected time for the (1+λ)-EA to optimise LeadingOnes is O(λn) Proof Left as exercise. Theorem Let λ < en. Then the expected time for the (1+λ)-EA to optimise LeadingOnes is O(n2 ) Proof Left as exercise. P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 58 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Additive Drift Theorem (1,λ)-EA Analysis for LeadingOnes Theorem Let λ = n. Then the expected time for the (1,λ)-EA to optimise LeadingOnes is O(n2 ) (Jansen and Neumann, Tutorial) Proof P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 59 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Additive Drift Theorem (1,λ)-EA Analysis for LeadingOnes Theorem Let λ = n. Then the expected time for the (1,λ)-EA to optimise LeadingOnes is O(n2 ) (Jansen and Neumann, Tutorial) Proof Distance: let d(x) = n − i where i is the number of leading ones; Drift: E[d(Xt ) − d(Xt+1 )] „ „ « « „ „ « « 1 n 1 n n ≥1· 1− 1− −n· 1− 1− en n = c1 − n · cn 2 = Ω(1) Hence, E(generations) ≤ max n distance = = O(n) drif t Ω(1) and, E(T ) ≤ n · E(generations) = O(n2 ) P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 59 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Additive Drift Theorem Additive Drift Theorem Theorem (Additive Drift Theorem for Lower Bounds [He and Yao, 2004]) Let {Xt }t≥0 be a Markov process over a set of states S, and d : S → R+ 0 a function that assigns a non-negative real number to every state. Let the time to reach the optimum be T := min{t ≥ 0 : d(Xt ) = 0}. If there exists δ > 0 such that at any time step t ≥ 0 and at any state Xt > 0 the following condition holds: E(∆(t)|d(Xt ) > 0) = E(d(Xt ) − d(Xt+1 ) | d(Xt ) > 0) ≤ δ then E(T | X0 > 0) ≥ and E(T ) ≥ P. S. Oliveto & X. Yao (University of Birmingham) d(X0 ) δ E(d(X0 )) . δ Runtime Analysis of EAs (4) (5) (6) WCCI 2012 60 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Additive Drift Theorem Theorem The expected time for the (1+1)-EA to optimise LeadingOnes is Ω(n2 ). P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 61 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Additive Drift Theorem Theorem The expected time for the (1+1)-EA to optimise LeadingOnes is Ω(n2 ). Sources of progress 1 Flipping the leftmost zero-bit; 2 Bits to right of the leftmost zero-bit that are one-bits (free riders). Proof P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 61 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Additive Drift Theorem Theorem The expected time for the (1+1)-EA to optimise LeadingOnes is Ω(n2 ). Sources of progress 1 Flipping the leftmost zero-bit; 2 Bits to right of the leftmost zero-bit that are one-bits (free riders). Proof 1 Let the current solution have n − i leading ones (i.e. 1n−i 0∗). P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 61 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Additive Drift Theorem Theorem The expected time for the (1+1)-EA to optimise LeadingOnes is Ω(n2 ). Sources of progress 1 Flipping the leftmost zero-bit; 2 Bits to right of the leftmost zero-bit that are one-bits (free riders). Proof 1 Let the current solution have n − i leading ones (i.e. 1n−i 0∗). 2 We define the distance function as the number of missing leading ones, i.e. g(X) = i. P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 61 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Additive Drift Theorem Theorem The expected time for the (1+1)-EA to optimise LeadingOnes is Ω(n2 ). Sources of progress 1 Flipping the leftmost zero-bit; 2 Bits to right of the leftmost zero-bit that are one-bits (free riders). Proof 1 Let the current solution have n − i leading ones (i.e. 1n−i 0∗). 2 We define the distance function as the number of missing leading ones, i.e. g(X) = i. 3 The n − i + 1 bit is a zero; P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 61 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Additive Drift Theorem Theorem The expected time for the (1+1)-EA to optimise LeadingOnes is Ω(n2 ). Sources of progress 1 Flipping the leftmost zero-bit; 2 Bits to right of the leftmost zero-bit that are one-bits (free riders). Proof 1 Let the current solution have n − i leading ones (i.e. 1n−i 0∗). 2 We define the distance function as the number of missing leading ones, i.e. g(X) = i. 3 The n − i + 1 bit is a zero; 4 let E[Y ] be the expected number of one-bits after the first zero (i.e. the free riders). P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 61 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Additive Drift Theorem Theorem The expected time for the (1+1)-EA to optimise LeadingOnes is Ω(n2 ). Sources of progress 1 Flipping the leftmost zero-bit; 2 Bits to right of the leftmost zero-bit that are one-bits (free riders). Proof 1 Let the current solution have n − i leading ones (i.e. 1n−i 0∗). 2 We define the distance function as the number of missing leading ones, i.e. g(X) = i. 3 The n − i + 1 bit is a zero; 4 let E[Y ] be the expected number of one-bits after the first zero (i.e. the free riders). 5 Such i − 1 bits are uniformely distributed at initialisation and still are! P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 61 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Additive Drift Theorem Theorem The expected time for the (1+1)-EA to optimise LeadingOnes is Ω(n2 ). Sources of progress 1 Flipping the leftmost zero-bit; 2 Bits to right of the leftmost zero-bit that are one-bits (free riders). Proof 1 Let the current solution have n − i leading ones (i.e. 1n−i 0∗). 2 We define the distance function as the number of missing leading ones, i.e. g(X) = i. 3 The n − i + 1 bit is a zero; 4 let E[Y ] be the expected number of one-bits after the first zero (i.e. the free riders). 5 Such i − 1 bits are uniformely distributed at initialisation and still are! P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 61 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Additive Drift Theorem Drift Theorem for LeadingOnes (lower bound) Theorem The expected time for the (1+1)-EA to optimise LeadingOnes is Ω(n2 ). The expected number of free riders is: E[Y ] = i−1 X k · P r(Y = k) = k=1 P. S. Oliveto & X. Yao (University of Birmingham) i−1 X P r(Y ≥ k) = k=1 Runtime Analysis of EAs i−1 X (1/2)k ≤ 1 k=1 WCCI 2012 62 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Additive Drift Theorem Drift Theorem for LeadingOnes (lower bound) Theorem The expected time for the (1+1)-EA to optimise LeadingOnes is Ω(n2 ). The expected number of free riders is: E[Y ] = i−1 X k · P r(Y = k) = k=1 i−1 X P r(Y ≥ k) = k=1 i−1 X (1/2)k ≤ 1 k=1 The negative drift is 0; P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 62 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Additive Drift Theorem Drift Theorem for LeadingOnes (lower bound) Theorem The expected time for the (1+1)-EA to optimise LeadingOnes is Ω(n2 ). The expected number of free riders is: E[Y ] = i−1 X k · P r(Y = k) = k=1 i−1 X P r(Y ≥ k) = k=1 i−1 X (1/2)k ≤ 1 k=1 The negative drift is 0; Let p(A) be the probability that the first zero-bit flips into a one-bit. P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 62 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Additive Drift Theorem Drift Theorem for LeadingOnes (lower bound) Theorem The expected time for the (1+1)-EA to optimise LeadingOnes is Ω(n2 ). The expected number of free riders is: E[Y ] = i−1 X k · P r(Y = k) = k=1 i−1 X P r(Y ≥ k) = k=1 i−1 X (1/2)k ≤ 1 k=1 The negative drift is 0; Let p(A) be the probability that the first zero-bit flips into a one-bit. The positive drift (i.e. the decrease in distance) is bounded as follows: E(∆+ (t)) ≤ p(A) · E[∆+ (t)|A] = 1/n · (1 + E[Y ]) ≤ 2/n = δ P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 62 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Additive Drift Theorem Drift Theorem for LeadingOnes (lower bound) Theorem The expected time for the (1+1)-EA to optimise LeadingOnes is Ω(n2 ). The expected number of free riders is: E[Y ] = i−1 X k · P r(Y = k) = k=1 i−1 X P r(Y ≥ k) = k=1 i−1 X (1/2)k ≤ 1 k=1 The negative drift is 0; Let p(A) be the probability that the first zero-bit flips into a one-bit. The positive drift (i.e. the decrease in distance) is bounded as follows: E(∆+ (t)) ≤ p(A) · E[∆+ (t)|A] = 1/n · (1 + E[Y ]) ≤ 2/n = δ Since, also at initialisation the expected number of free riders is less than 2, it follows that E[d(X0 )] ≥ n − 2, By applying the Drift Theorem we get E(T ) ≥ P. S. Oliveto & X. Yao (University of Birmingham) E(d(X0 ) n−2 = = Ω(n2 ) δ 2/n Runtime Analysis of EAs WCCI 2012 62 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Multiplicative Drift Theorem Drift Analysis for OneMax Lets calculate the runtime of the (1+1)-EA using the additive Drift Theorem. 1 Let g(Xt ) = i where i is the number of zeroes in the bitstring; P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 63 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Multiplicative Drift Theorem Drift Analysis for OneMax Lets calculate the runtime of the (1+1)-EA using the additive Drift Theorem. 1 Let g(Xt ) = i where i is the number of zeroes in the bitstring; 2 The negative drift is 0 since solution with less one-bits will not be accepted; P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 63 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Multiplicative Drift Theorem Drift Analysis for OneMax Lets calculate the runtime of the (1+1)-EA using the additive Drift Theorem. 1 Let g(Xt ) = i where i is the number of zeroes in the bitstring; 2 The negative drift is 0 since solution with less one-bits will not be accepted; 3 A positive drift is achieved as long as a 0 is flipped and the ones remain unchanged: E(∆+ (t)) = E[d(Xt ) − d(Xt+1 )] ≥ 1 · P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs „ « i i 1 n−1 1 1− ≥ ≥ n n en en WCCI 2012 63 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Multiplicative Drift Theorem Drift Analysis for OneMax Lets calculate the runtime of the (1+1)-EA using the additive Drift Theorem. 1 Let g(Xt ) = i where i is the number of zeroes in the bitstring; 2 The negative drift is 0 since solution with less one-bits will not be accepted; 3 A positive drift is achieved as long as a 0 is flipped and the ones remain unchanged: E(∆+ (t)) = E[d(Xt ) − d(Xt+1 )] ≥ 1 · 4 „ « i i 1 n−1 1 1− ≥ ≥ n n en en Hence, E[∆(t)] = E(∆+ (t)) − E(∆− (t)) ≥ 1/(en) = δ P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 63 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Multiplicative Drift Theorem Drift Analysis for OneMax Lets calculate the runtime of the (1+1)-EA using the additive Drift Theorem. 1 Let g(Xt ) = i where i is the number of zeroes in the bitstring; 2 The negative drift is 0 since solution with less one-bits will not be accepted; 3 A positive drift is achieved as long as a 0 is flipped and the ones remain unchanged: E(∆+ (t)) = E[d(Xt ) − d(Xt+1 )] ≥ 1 · „ « i i 1 n−1 1 1− ≥ ≥ n n en en 4 Hence, E[∆(t)] = E(∆+ (t)) − E(∆− (t)) ≥ 1/(en) = δ 5 The expected initial distance is E(d(X0 )) = n/2 The expected runtime is (i.e. Eq. (6)): E(T | d(X0 ) > 0) ≤ E[(d(X0 )] n/2 ≤ = e/2 · n2 = O(n2 ) δ 1/(en) We need a different distance function! P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 63 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Multiplicative Drift Theorem Drift Analysis for OneMax 1 Let g(Xt ) = ln(i + 1) where i is the number of zeroes in the bitstring; P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 64 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Multiplicative Drift Theorem Drift Analysis for OneMax 1 Let g(Xt ) = ln(i + 1) where i is the number of zeroes in the bitstring; 2 The negative drift is 0 since solution with less one-bits will not be accepted; P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 64 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Multiplicative Drift Theorem Drift Analysis for OneMax 1 Let g(Xt ) = ln(i + 1) where i is the number of zeroes in the bitstring; 2 The negative drift is 0 since solution with less one-bits will not be accepted; 3 A positive drift is achieved as long as a 0 is flipped and the ones remain unchanged: E(∆+ (t)) = E[d(Xt )−d(Xt+1 )] ≥ ln(i+1)· P. S. Oliveto & X. Yao (University of Birmingham) „ « i 1 n−1 ln(i + 1) ln(2) 1− ≥ ≥ n n en en Runtime Analysis of EAs WCCI 2012 64 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Multiplicative Drift Theorem Drift Analysis for OneMax 1 Let g(Xt ) = ln(i + 1) where i is the number of zeroes in the bitstring; 2 The negative drift is 0 since solution with less one-bits will not be accepted; 3 A positive drift is achieved as long as a 0 is flipped and the ones remain unchanged: E(∆+ (t)) = E[d(Xt )−d(Xt+1 )] ≥ ln(i+1)· 4 „ « i 1 n−1 ln(i + 1) ln(2) 1− ≥ ≥ n n en en Hence, ∆(t) = E(∆+ (t)) − E(∆− (t)) ≥ ln(2)/(en) = δ P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 64 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Multiplicative Drift Theorem Drift Analysis for OneMax 1 Let g(Xt ) = ln(i + 1) where i is the number of zeroes in the bitstring; 2 The negative drift is 0 since solution with less one-bits will not be accepted; 3 A positive drift is achieved as long as a 0 is flipped and the ones remain unchanged: E(∆+ (t)) = E[d(Xt )−d(Xt+1 )] ≥ ln(i+1)· „ « i 1 n−1 ln(i + 1) ln(2) 1− ≥ ≥ n n en en 4 Hence, ∆(t) = E(∆+ (t)) − E(∆− (t)) ≥ ln(2)/(en) = δ 5 The distance is d(X0 ) ≤ ln(n + 1) The expected runtime is (i.e. Eq. (6)): E(T | d(X0 ) > 0) ≤ ln(n + 1) d(X0 ) ≤ = O(n ln n) δ ln(2)/(en) If the amount of progress depends on the distance from the optimum we need to use a logarithmic distance! P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 64 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Multiplicative Drift Theorem Multiplicative Drift Theorem Theorem (Multiplicative Drift, [Doerr et al., 2010]) Let {Xt }t∈N0 be random variables describing a Markov process over a finite state space S ⊆ R. Let T be the random variable that denotes the earliest point in time t ∈ N0 such that Xt = 0. If there exist δ, cmin , cmax > 0 such that 1 E[Xt − Xt+1 | Xt ] ≥ δXt and 2 cmin ≤ Xt ≤ cmax , for all t < T , then E[T ] ≤ P. S. Oliveto & X. Yao (University of Birmingham) „ « 2 cmax · ln 1 + δ cmin Runtime Analysis of EAs WCCI 2012 65 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Multiplicative Drift Theorem (1+1)-EA Analysis for OneMax Theorem The expected time for the (1+1)-EA to optimise OneMax is O(n ln n) Proof P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 66 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Multiplicative Drift Theorem (1+1)-EA Analysis for OneMax Theorem The expected time for the (1+1)-EA to optimise OneMax is O(n ln n) Proof Distance: let i be the number of zeroes; „ « „ « 1 1 i = i · 1 − en = Xt · 1 − en E[Xt+1 |Xt = i] ≥ i − 1 · en „ « 1 1 1 E[Xt − Xt+1 |Xt = i] ≤ Xt − Xt · 1 − en = en Xt (δ = en ) 1 = cmin ≤ Xt ≤ cmax = n Hence, E[T ] ≤ P. S. Oliveto & X. Yao „ « cmax 2 · ln 1 + = 2en ln(1 + n) = O(n ln n) δ cmin (University of Birmingham) Runtime Analysis of EAs WCCI 2012 66 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Multiplicative Drift Theorem Exercises Theorem The expected time for RLS to optimise OneMax is O(n log n) Proof P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 67 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Multiplicative Drift Theorem Exercises Theorem The expected time for RLS to optimise OneMax is O(n log n) Proof Left as exercise. Theorem Let λ ≥ en. Then the expected time for the (1+λ)-EA to optimise OneMax is O(λn) Proof P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 67 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Multiplicative Drift Theorem Exercises Theorem The expected time for RLS to optimise OneMax is O(n log n) Proof Left as exercise. Theorem Let λ ≥ en. Then the expected time for the (1+λ)-EA to optimise OneMax is O(λn) Proof Left as exercise. Theorem Let λ < en. Then the expected time for the (1+λ)-EA to optimise OneMax is O(n log n) Proof P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 67 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Multiplicative Drift Theorem Exercises Theorem The expected time for RLS to optimise OneMax is O(n log n) Proof Left as exercise. Theorem Let λ ≥ en. Then the expected time for the (1+λ)-EA to optimise OneMax is O(λn) Proof Left as exercise. Theorem Let λ < en. Then the expected time for the (1+λ)-EA to optimise OneMax is O(n log n) Proof Left as exercise. P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 67 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Simplified Negative Drift Theorem Drift Analysis: Example 3 Friday night dinner at the restaurant. Peter walks back from the restaurant to the hotel but had too many drinks. The restaurant is n meters away from the hotel; Peter moves towards the hotel of 1 meter in each step with probability 0.4. Peter moves away from the hotel of 1 meter in each step with probability 0.6. Question How many steps does Peter need to reach his hotel? P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 68 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Simplified Negative Drift Theorem Drift Analysis: Example 3 Friday night dinner at the restaurant. Peter walks back from the restaurant to the hotel but had too many drinks. The restaurant is n meters away from the hotel; Peter moves towards the hotel of 1 meter in each step with probability 0.4. Peter moves away from the hotel of 1 meter in each step with probability 0.6. Question How many steps does Peter need to reach his hotel? at least 2cn steps with overwhelming probability (exponential time) We need Negative-Drift Analysis. P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 68 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Simplified Negative Drift Theorem Simplified Drift Theorem Xt , t ≥ 0, (Markov process over S ⊆ [0, N ]) ∆t (i) := (Xt+1 − Xt | Xt = i) for i ∈ S and t ≥ 0 [a, b] T∗ (Interval in State space S) := min{t ≥ 0 : Xt ≤ a | X0 ≥ b} P. S. Oliveto & X. Yao (Drift) (University of Birmingham) (Time to reach a) Runtime Analysis of EAs WCCI 2012 69 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Simplified Negative Drift Theorem Simplified Drift Theorem Xt , t ≥ 0, (Markov process over S ⊆ [0, N ]) ∆t (i) := (Xt+1 − Xt | Xt = i) for i ∈ S and t ≥ 0 [a, b] T∗ (Drift) (Interval in State space S) := min{t ≥ 0 : Xt ≤ a | X0 ≥ b} (Time to reach a) Theorem (Simplified Negative-Drift Theorem, [Oliveto and Witt, 2011]) Suppose there exist three constants δ,,r such that for all t ≥ 0: 1 E(∆t (i)) ≥ for a < i < b, 2 Prob(∆t (i) = −j) ≤ Then 1 (1+δ)j−r for i > a and j ≥ 1. ∗ Prob(T ∗ ≤ 2c (b−a) ) = 2−Ω(b−a) . P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 69 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Simplified Negative Drift Theorem Simplified Drift Theorem 1 E(∆t (i)) ≥ for a < i < b, 2 Prob(∆t (i) = −j) ≤ P. S. Oliveto & X. Yao (University of Birmingham) 1 (1+δ)j−r for i > a and j ≥ 1. Runtime Analysis of EAs WCCI 2012 70 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Simplified Negative Drift Theorem Negative-Drift Analysis: Example (3) Define the same distance function d(x) = x, x ∈ {0, . . . , n} (metres from the hotel) (b=n-1, a=1). P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 71 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Simplified Negative Drift Theorem Negative-Drift Analysis: Example (3) Define the same distance function d(x) = x, x ∈ {0, . . . , n} (metres from the hotel) (b=n-1, a=1). Estimate the increase in distance from the goal (negative drift); 8 > <0, if Xt = 0, d(Xt ) − d(Xt+1 ) = 1, if Xt ∈ {1, . . . , n}with probability 0.6 > : −1, if Xt ∈ {1, . . . , n}with probability 0.4 P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 71 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Simplified Negative Drift Theorem Negative-Drift Analysis: Example (3) Define the same distance function d(x) = x, x ∈ {0, . . . , n} (metres from the hotel) (b=n-1, a=1). Estimate the increase in distance from the goal (negative drift); 8 > <0, if Xt = 0, d(Xt ) − d(Xt+1 ) = 1, if Xt ∈ {1, . . . , n}with probability 0.6 > : −1, if Xt ∈ {1, . . . , n}with probability 0.4 The expected increase in distance (negative drift) is: (Condition 1) E[d(Xt ) − d(Xt+1 )] = 0.6 · 1 + 0.4 · (−1) = 0.6 − 0.4 = 0.2 P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 71 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Simplified Negative Drift Theorem Negative-Drift Analysis: Example (3) Define the same distance function d(x) = x, x ∈ {0, . . . , n} (metres from the hotel) (b=n-1, a=1). Estimate the increase in distance from the goal (negative drift); 8 > <0, if Xt = 0, d(Xt ) − d(Xt+1 ) = 1, if Xt ∈ {1, . . . , n}with probability 0.6 > : −1, if Xt ∈ {1, . . . , n}with probability 0.4 The expected increase in distance (negative drift) is: (Condition 1) E[d(Xt ) − d(Xt+1 )] = 0.6 · 1 + 0.4 · (−1) = 0.6 − 0.4 = 0.2 Probability of jumps (i.e. Prob(∆t (i) = −j) ≤ 1 (1+δ)j−r ) (set δ = r = 1) (Condition 2): ( P r(∆t (i) = −j) = P. S. Oliveto & X. Yao (University of Birmingham) 0 < (1/2)j−1 , if j > 1, 0.6 < (1/2)0 = 1, if j = 1 Runtime Analysis of EAs WCCI 2012 71 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Simplified Negative Drift Theorem Negative-Drift Analysis: Example (3) Define the same distance function d(x) = x, x ∈ {0, . . . , n} (metres from the hotel) (b=n-1, a=1). Estimate the increase in distance from the goal (negative drift); 8 > <0, if Xt = 0, d(Xt ) − d(Xt+1 ) = 1, if Xt ∈ {1, . . . , n}with probability 0.6 > : −1, if Xt ∈ {1, . . . , n}with probability 0.4 The expected increase in distance (negative drift) is: (Condition 1) E[d(Xt ) − d(Xt+1 )] = 0.6 · 1 + 0.4 · (−1) = 0.6 − 0.4 = 0.2 Probability of jumps (i.e. Prob(∆t (i) = −j) ≤ 1 (1+δ)j−r ) (set δ = r = 1) (Condition 2): ( P r(∆t (i) = −j) = 0 < (1/2)j−1 , if j > 1, 0.6 < (1/2)0 = 1, if j = 1 Then the expected time to reach the hotel (goal) is: P r(T ≤ 2c(b−a) ) = P r(T ≤ 2c(n−2) ) = 2−Ω(n) P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 71 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Simplified Negative Drift Theorem Needle in a Haystack Theorem Let η > 0 be constant. Then there is a constant c > 0 such that with probability 1 − 2−Ω(n) the (1+1)-EA on Needle creates only search points with at most n/2 + ηn ones in 2cn steps. P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 72 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Simplified Negative Drift Theorem Needle in a Haystack Theorem Let η > 0 be constant. Then there is a constant c > 0 such that with probability 1 − 2−Ω(n) the (1+1)-EA on Needle creates only search points with at most n/2 + ηn ones in 2cn steps. Proof Idea 1 By Chernoff bounds the probability that the initial bit string has less than n/2 − γn zeroes is e−Ω(n) . 2 we set b := n/2 − γn and a := n/2 − 2γn where γ := η/2; 3 Let Xt denote the number of zeroes in the bit string at time step t.; 4 Let ∆(i) denote the random increase of the number of zeroes (i.e. the drift); 5 Now we have to check that the two conditions of the Simplified drift theorem hold. P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 72 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Simplified Negative Drift Theorem Needle in a Haystack (2) Proof of Condition 1 Condition 1 holds if E(∆(i)) ≥ for some constant > 0. 1 The (1+1)-EA flips 0-bits and 1-bits independently with probability 1/n. 2 Since there are i zero-bits and n − i one-bits in the current string, the expected number of zero-bits flipped into one-bits is i/n and the viceversa is (n − i)/n. 3 Hence, E(∆(i)) = n−i i n − 2i − = ≥ 2γ = n n n and Condition 1 holds. P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 73 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Simplified Negative Drift Theorem Needle in a Haystack (3) Proof of Condition 2 Condition 2 is: P rob(∆(i) = −j) ≤ 1/(1 + δ)j−r for j ∈ N0 . 1 Since to reach state i − j or less from state i, at least j bits have to flip, the probability of a drift of length j can be bounded as follows: P rob(∆(i) ≤ −j) ≤ “n” „ 1 «j j n ≤ 1 ≤ j! „ «j−1 1 2 This proves Condition 2 by setting δ = r = 1. Applying the Simplified Drift Theorem ∗ For a constant c∗ > 0 the global optimum is found in 2c (b−a) = 2cn steps with −Ω(b−a) −Ω(n) probability at most 2 =2 . P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 74 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Simplified Negative Drift Theorem Exercise: Trap Functions Trap(x) = n+1 OneMax(x) if x = 0n otherwise. f(x) n+1 n 3 2 1 0123 P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs n ones(x) WCCI 2012 75 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Simplified Negative Drift Theorem Exercise: Trap Functions Trap(x) = n+1 OneMax(x) if x = 0n otherwise. f(x) n+1 n 3 2 1 0123 n ones(x) Theorem With overwhelming probability at least 1 − 2−Ω(n) the (1+1)-EA requires 2Ω(n) steps to optimise Trap . Proof Left as exercise. P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 75 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Simplified Negative Drift Theorem Drift Analysis Conclusion Overview Additive Drift Analysis (upper and lower bounds); Multiplicative Drift Analysis; Simplified Negative-Drift Theorem; Advanced Lower bound Drift Techniques Drift Analysis for Stochastic Populations (mutation) [Lehre, 2010]; Simplified Drift Theorem combined with bandwidth analysis (mutation + crossover stochastic populations = GAs) [Oliveto and Witt, 2012]; P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 76 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Typical run investigations Typical Runs have been introduced by considering that the “global behaviour of a process” is predictable with high probability; the “local behaviour” is quite unpredictable; P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 77 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Typical run investigations Typical Runs have been introduced by considering that the “global behaviour of a process” is predictable with high probability; the “local behaviour” is quite unpredictable; Method 1 Divide the process into k phases which are long enough to assure that some even happens with probability pk ; 2 Hence, does not happen with failure probability 1 − pk ; 3 The last phase should lead the EA to the global optimum; P The failure probability is 1 − ki=1 pi ; 4 5 The runtime is lower than the sum of each phase time. Generally Chernoff bounds are used to obtain the failure probabilities! P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 77 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio (1+1)-EA for Trap functions Theorem With overwhelming probability at least 1 − e−Ω(n) the (1+1)-EA requires nΩ(n) steps to optimise Trap . The deceptive trap function is the following: n+1 Trap(x) = OneMax(x) P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs if x = 0n otherwise. WCCI 2012 78 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio (1+1)-EA for Trap functions Theorem With overwhelming probability at least 1 − e−Ω(n) the (1+1)-EA requires nΩ(n) steps to optimise Trap . The deceptive trap function is the following: n+1 Trap(x) = OneMax(x) if x = 0n otherwise. Proof Idea We use the Typical run investigations method and split the analysis into the following three phases: P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 78 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio (1+1)-EA for Trap functions Theorem With overwhelming probability at least 1 − e−Ω(n) the (1+1)-EA requires nΩ(n) steps to optimise Trap . The deceptive trap function is the following: n+1 Trap(x) = OneMax(x) if x = 0n otherwise. Proof Idea We use the Typical run investigations method and split the analysis into the following three phases: 1 This phase ends when at least n/3 one-bits are in the current solution. P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 78 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio (1+1)-EA for Trap functions Theorem With overwhelming probability at least 1 − e−Ω(n) the (1+1)-EA requires nΩ(n) steps to optimise Trap . The deceptive trap function is the following: n+1 Trap(x) = OneMax(x) if x = 0n otherwise. Proof Idea We use the Typical run investigations method and split the analysis into the following three phases: 1 This phase ends when at least n/3 one-bits are in the current solution. 2 This phase ends when the 1n bitstring is the current solution. P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 78 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio (1+1)-EA for Trap functions Theorem With overwhelming probability at least 1 − e−Ω(n) the (1+1)-EA requires nΩ(n) steps to optimise Trap . The deceptive trap function is the following: n+1 Trap(x) = OneMax(x) if x = 0n otherwise. Proof Idea We use the Typical run investigations method and split the analysis into the following three phases: 1 This phase ends when at least n/3 one-bits are in the current solution. 2 This phase ends when the 1n bitstring is the current solution. 3 This phase ends when the global optimum is found. P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 78 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio (1+1)-EA for Trap functions (2) We need to show that the runtime required to complete the three phases is nΩ(n) and that the failure probability of each phase is at most e−Ω(n) . Phase 1 This phase ends when at least n/3 one-bits are in the current solution. Proof 1 By using Chernoff bounds we calculate the probability that the algorithm is initialised with less than n/3 one-bits. P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 79 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio (1+1)-EA for Trap functions (2) We need to show that the runtime required to complete the three phases is nΩ(n) and that the failure probability of each phase is at most e−Ω(n) . Phase 1 This phase ends when at least n/3 one-bits are in the current solution. Proof 1 By using Chernoff bounds we calculate the probability that the algorithm is initialised with less than n/3 one-bits. 2 n/2 expected one-bits after initialisation (i.e. E(X) = n · p = n/2). P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 79 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio (1+1)-EA for Trap functions (2) We need to show that the runtime required to complete the three phases is nΩ(n) and that the failure probability of each phase is at most e−Ω(n) . Phase 1 This phase ends when at least n/3 one-bits are in the current solution. Proof 1 By using Chernoff bounds we calculate the probability that the algorithm is initialised with less than n/3 one-bits. 2 n/2 expected one-bits after initialisation (i.e. E(X) = n · p = n/2). 3 Setting δ = 1/3 we get P (X ≤ n/3) ≤ e−E(X)δ P. S. Oliveto & X. Yao (University of Birmingham) 2 /2 = e−(n/2)·(1/9)·(1/2) = e−n/36 = e−Ω(n) Runtime Analysis of EAs WCCI 2012 79 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio (1+1)-EA for Trap functions (3) Phase 2 This phase ends when the 1n bitstring is the current solution. Proof P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 80 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio (1+1)-EA for Trap functions (3) Phase 2 This phase ends when the 1n bitstring is the current solution. Proof 1 The probability of reaching the 0n bitstring in one step is less than n−n/3 ; P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 80 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio (1+1)-EA for Trap functions (3) Phase 2 This phase ends when the 1n bitstring is the current solution. Proof 1 The probability of reaching the 0n bitstring in one step is less than n−n/3 ; 2 in expected time c · n log n the algorithm climbs up the OneMax . P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 80 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio (1+1)-EA for Trap functions (3) Phase 2 This phase ends when the 1n bitstring is the current solution. Proof 1 The probability of reaching the 0n bitstring in one step is less than n−n/3 ; 2 in expected time c · n log n the algorithm climbs up the OneMax . 3 By Markov’s inequality, ` ´ P X ≥ e · c · n log n ≤ P. S. Oliveto & X. Yao (University of Birmingham) c · n log n = 1/e e · c · n log n Runtime Analysis of EAs WCCI 2012 80 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio (1+1)-EA for Trap functions (3) Phase 2 This phase ends when the 1n bitstring is the current solution. Proof 1 The probability of reaching the 0n bitstring in one step is less than n−n/3 ; 2 in expected time c · n log n the algorithm climbs up the OneMax . 3 By Markov’s inequality, ` ´ P X ≥ e · c · n log n ≤ 4 c · n log n = 1/e e · c · n log n ` ´ By applying it iteratively: P X ≥ e · c · n2 log n ≤ e−n P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 80 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio (1+1)-EA for Trap functions (3) Phase 2 This phase ends when the 1n bitstring is the current solution. Proof 1 The probability of reaching the 0n bitstring in one step is less than n−n/3 ; 2 in expected time c · n log n the algorithm climbs up the OneMax . 3 By Markov’s inequality, ` ´ P X ≥ e · c · n log n ≤ 4 5 c · n log n = 1/e e · c · n log n ` ´ By applying it iteratively: P X ≥ e · c · n2 log n ≤ e−n The probability that in c · n2 log n steps n/3 precise bits flip is c · n2 log n · n−n/3 = n−Ω(n) . (union bound) P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 80 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio (1+1)-EA for Trap functions (4) Phase 3 This phase ends when the global optimum is found. Proof P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 81 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio (1+1)-EA for Trap functions (4) Phase 3 This phase ends when the global optimum is found. Proof 1 The expected time to reach the global optimum is nn because all the bits have to be flipped at the same time; P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 81 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio (1+1)-EA for Trap functions (4) Phase 3 This phase ends when the global optimum is found. Proof 1 The expected time to reach the global optimum is nn because all the bits have to be flipped at the same time; 2 By Chernoff bounds the probability that the time is less than (1/2)nn is (i.e. δ = 1/2): P (T ≤ (1/2)nn ) ≤ e−n P. S. Oliveto & X. Yao (University of Birmingham) n ·1/4·1/2 Runtime Analysis of EAs = e−n n ·1/8 = e−Ω(n) WCCI 2012 81 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio (1+1)-EA for Trap functions (4) Phase 3 This phase ends when the global optimum is found. Proof 1 The expected time to reach the global optimum is nn because all the bits have to be flipped at the same time; 2 By Chernoff bounds the probability that the time is less than (1/2)nn is (i.e. δ = 1/2): P (T ≤ (1/2)nn ) ≤ e−n n ·1/4·1/2 = e−n n ·1/8 = e−Ω(n) Summing up 1 the algorithm requires exponential runtime 1 + c · n2 log n + (1/2)nn = nΩ(n) 2 With probability at least 1 − e−Ω(n) . P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 81 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio ExpectationTrap ExpectationTrap(x) = n − 1/2 OneMax(x) if x = 0n otherwise. Theorem ` ´ The expected time for the (1+1)-EA to optimise ExpectationTrap is Ω (n/2)n . [Jansen and Neumann, Tutorial] P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 82 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio ExpectationTrap ExpectationTrap(x) = n − 1/2 OneMax(x) if x = 0n otherwise. Theorem ` ´ The expected time for the (1+1)-EA to optimise ExpectationTrap is Ω (n/2)n . [Jansen and Neumann, Tutorial] Observation Prob(Local) = 2−n (probability the algorithm is initialised with the 0n bitstring) E(T |X0 = 0n ) = nn (all bits need to be flipped) Proof P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 82 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio ExpectationTrap ExpectationTrap(x) = n − 1/2 OneMax(x) if x = 0n otherwise. Theorem ` ´ The expected time for the (1+1)-EA to optimise ExpectationTrap is Ω (n/2)n . [Jansen and Neumann, Tutorial] Observation Prob(Local) = 2−n (probability the algorithm is initialised with the 0n bitstring) E(T |X0 = 0n ) = nn (all bits need to be flipped) Proof E(T ) ≥ E(T |X0 = 0n ) = nn · 2−n = (n/2)n P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs (law of total probability) WCCI 2012 82 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Overview Final Overview Overview Basic Probability Theory Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Runs Other Techniques (Not covered) Family Trees [Witt, 2006] Gambler’s Ruin & Martingales [Jansen and Wegener, 2001] P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 83 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio State-of-the-art State of the Art in Computational Complexity of RSHs OneMax Linear Functions Max. Matching Sorting SS Shortest Path (1+1) EA (1+λ) EA (µ+1) EA 1-ANT (µ+1) IA (1+1) EA cGA Max. Clique (1+1) EA (1+1) EA (1+1) EA MO (1+1) EA (1+1) EA (1+λ) EA 1-ANT (1+1) EA (rand. planar) Eulerian Cycle Partition (16n+1) RLS (1+1) EA (1+1) EA Vertex Cover (1+1) EA Set Cover (1+1) EA SEMO (1+1) EA MST Intersection of p ≥ 3 matroids UIO/FSM conf. (1+1) EA O(n log n) O(λn + n log n) O(µn + n log n) O(n2 ) w.h.p. O(µn + n log n) Θ(n log n) Θ(n2+ε ), ε > 0 const. eΩ(n) , PRAS Θ(n2 log n) O(n3 log(nwmax )) O(n3 ) Θ(m2 log(nwmax )) O(n log(nwmax )) O(mn log(nwmax )) Θ(n5 ) Θ(n5/3 ) Θ(m2 log m) 4/3 approx., competitive avg. eΩ(n) , arb. bad approx. eΩ(n) , arb. bad approx. Pol. O(log n)-approx. 1/p-approximation in O(|E|p+2 log(|E|wmax )) eΩ(n) See [Oliveto et al., 2007] for an overview. P. K. Lehre, 2008 P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 84 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Further reading Further Reading [Auger and Doerr, 2011, Neumann and Witt, 2010] P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 85 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Further reading Coming up Conference P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 86 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Further reading References I Auger, A. and Doerr, B. (2011). Theory of Randomized Search Heuristics: Foundations and Recent Developments. World Scientific Publishing Co., Inc., River Edge, NJ, USA. Bäck, T. (1993). Optimal mutation rates in genetic search. In In Proceedings of the Fifth International Conference on Genetic Algorithms (ICGA), pages 2–8. Doerr, B., Johannsen, D., and Winzen, C. (2010). Multiplicative drift analysis. In Proceedings of the 12th annual conference on Genetic and evolutionary computation, GECCO ’10, pages 1449–1456. ACM. Droste, S., Jansen, T., and Wegener, I. (1998). A rigorous complexity analysis of the (1 + 1) evolutionary algorithm for separable functions with boolean inputs. Evolutionary Computation, 6(2):185–196. Droste, S., Jansen, T., and Wegener, I. (2002). On the analysis of the (1+1) evolutionary algorithm. Theoretical Computer Science, 276(1-2):51–81. Goldberg, D. E. (1989). Genetic Algorithms for Search, Optimization, and Machine Learning. Addison-Wesley. He, J. and Yao, X. (2001). Drift analysis and average time complexity of evolutionary algorithms. Artificial Intelligence, 127(1):57–85. P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 87 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Further reading References II He, J. and Yao, X. (2004). A study of drift analysis for estimating computation time of evolutionary algorithms. Natural Computing: an international journal, 3(1):21–35. Holland, J. H. (1992). Adaptation in Natural and Artificial Systems: An Introductory Analysis with Applications to Biology, Control, and Artificial Intelligence. The MIT Press. Jansen, T., Jong, K. A. D., and Wegener, I. A. (2005). On the choice of the offspring population size in evolutionary algorithms. Evolutionary Computation, 13(4):413–440. Jansen, T. and Wegener, I. (2001). Evolutionary algorithms - how to cope with plateaus of constant fitness and when to reject strings of the same fitness. IEEE Trans. Evolutionary Computation, 5(6):589–599. Lehre, P. K. (2010). Negative drift in populations. In PPSN (1), pages 244–253. Lehre, P. K. (2011). Fitness-levels for non-elitist populations. In Proceedings of the 13th annual conference on Genetic and evolutionary computation, GECCO ’11, pages 2075–2082. ACM. Neumann, F. and Witt, C. (2010). Bioinspired Computation in Combinatorial Optimization: Algorithms and Their Computational Complexity. Springer-Verlag New York, Inc., New York, NY, USA, 1st edition. P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 88 / 92 Motivation Basic Probability Theory Evolutionary Algorithms Tail Inequalities Artificial Fitness Levels Drift Analysis Typical Run Investigations Conclusio Further reading References III Oliveto, P. and Witt, C. (2012). On the analysis of the simple genetic algorithm (to appear). In Proceedings of the 12th annual conference on Genetic and evolutionary computation, GECCO ’12, pages –. ACM. Oliveto, P. S., He, J., and Yao, X. (2007). Time complexity of evolutionary algorithms for combinatorial optimization: A decade of results. International Journal of Automation and Computing, 4(3):281–293. Oliveto, P. S. and Witt, C. (2011). Simplified drift analysis for proving lower bounds inevolutionary computation. Algorithmica, 59(3):369–386. Reeves, C. R. and Rowe, J. E. (2002). Genetic Algorithms: Principles and Perspectives: A Guide to GA Theory. Kluwer Academic Publishers, Norwell, MA, USA. Rudolph, G. (1998). Finite Markov chain results in evolutionary computation: A tour d’horizon. Fundamenta Informaticae, 35(1–4):67–89. Sudholt, D. (2010). General lower bounds for the running time of evolutionary algorithms. In PPSN (1), pages 124–133. Witt, C. (2006). Runtime analysis of the (µ+1) ea on simple pseudo-boolean functions evolutionary computation. In GECCO ’06: Proceedings of the 8th annual conference on Genetic and evolutionary computation, pages 651–658, New York, NY, USA. ACM Press. P. S. Oliveto & X. Yao (University of Birmingham) Runtime Analysis of EAs WCCI 2012 89 / 92