Evolutionary Computation Theory. Jun He School of Computer Science University of Birmingham Web: jxh

Transcription

1 Evolutionary Computation Theory Jun He School of Computer Science University of Birmingham Web: jxh

2 Outline Motivation History Schema Theorem Convergence and Convergence Rate Computational Complexity No Free Lunch Theorem Fitness Landscape 04/12/2006 p.1

3 Motivation Question: why we need some theory of evolutionary computation? Many reasons: To understand evolutionary algorithms. To guide the application of EAs. Experiment is not enough sometimes. To need it to write a PhD thesis However, theory itself might be hard to be understood due to too many mathematics inside. 04/12/2006 p.2

4 Experiment vs Theory Experimental study: run an algorithm to evaluate the performance of EAs. Easy to implement, an intuitive guidance in practice. Only cover limited instances. Theory: make a mathematical analysis to evaluate the performance of EAs. Cover all instances. Analysis is difficult. 04/12/2006 p.3

5 Main Questions Question 1: whether can an EA find the optimal solution eventually? Question 2: if it can, then how many generations should the EA take? Question 3: if it cannot or needs a long time, then can an EA find an good approximation solution very quickly? Question 4: what kind of problems is hard to EAs? and what is not? or what kind of EAs are efficient? 04/12/2006 p.4

6 History s: schema theorem (Q.?) 1980s-2000s: fitness landscapes (Q.4) 1990s: convergence and Convergence rate (Q.1) 1990s: no free lunch theorem (Q. 4) 2000s: computational complexity (Q.2 and Q.3) 04/12/2006 p.5

7 Theoretical Tools Question: how to analyze evolutionary computation from a theoretical viewpoint? 1. Describe EAs in a mathematical way. 2. Describe question in a mathematical way. 3. Prove answers in a mathematical way. Tools: Probability theory and stochastic processes, e.g. Markov chains. Orthogonal function analysis, e.g. Fourier, Walsh transformations. Statistical physics.. 04/12/2006 p.6

8 Schema Theorem Questions: Given f(101 ) > f(100 ), which schema will be selected more often? how about schemas 101 and ? Answer: Schema Theorem: first proposed by John Holland (1975) to explain how schema propagates from generation to generation. Original form: one-point crossover, bit mutation, roulette wheel selection, binary strings. Development: other forms of EAs, e.g. genetic programming. 04/12/2006 p.7

9 Notations Schema H: string consists of 1, 0,. Schema order o(h): the number of fixed (0/1) positions in H. Schema length δ(h): the distance between the first and last fixed position. m(h,t) number of of individuals in the population at time t which contains schema H. f(h): average fitness of individuals representation H at time t; f: average fitness of population at time t; l: length of an individual string. p c,p m : crossover rate and mutation rate. 04/12/2006 p.8

10 Schema Theorem Formula: m(h,t + 1) m(h,t) f(h) f [ ] δ(h) 1 p c l 1 o(h)p m Short, low-order and above average schema increases exponentially quickly Given f(101 ) > f(100 ), which schema will be selected more often? how about schemas 101 and ? Application: to describe how pattern increases, but provide no answer to Question /12/2006 p.9

11 Convergence Question: can an EA find the optimal solution eventually? How to answer this question? First model the population sequence X (1),X (2), by a Markov chain. A population is regarded as a state x. The transition from current population X (t) = x to the next population X (t+1) = y is described by a probability transition: p(x,y; t) = P(X (t+1) = y X (t) = x). 04/12/2006 p.10

12 Convergence Condition Convergence: starting from any initial population X(0) = x, the population X(t) will be in the set of optimal solution S opt (as t ) with probability 1. Conditions: The best individual in a population should be kept (elitist section). And Through mutation and crossover, the population sequence can visit any other population (global search) or there is a path from any initial population x to the optimal set S opt. 04/12/2006 p.11

13 Convergence Rate Question: how quickly the EA can converge to the optima? Too difficult to answer. Why? 04/12/2006 p.12

14 Computational Complexity Question: how many generations are needed for an EA to find the optimal solution? Depend on the problem and EAs, and initial population. Two different measures: average-case and worst-case. worst-case: the worst number of generations (you choose the worst initial population). average-case: the average number of generations (you choose the initial population at random). 04/12/2006 p.13

15 Drift Analysis Define a distance d(x) for each population: to measure how far it is away from the optimal solution. Calculate the drift (moving speed) towards the optimal solution. drift = y (d(x) d(y))p(x, y) Then obtain the first hitting time to the optimal solution: time = distance speed. 04/12/2006 p.14

16 Case Study How many generations is needed for a (1+1) EA (one-bit mutation and elitism selection) to solve the OneMax problem? max n i=1 s i. where the solution is (1 1). Distance: For an individual, define its distance to be the Hamming distance H(x, 1) = n i=1 s i 1 n. Drift: at each generation, speed is at least 1/n. Time: T n 2. 04/12/2006 p.15

17 No Free Lunch Theorem For any algorithm, its performance gain over a class of problem of problems is offset by its performance loss over a different class problem (No algorithm is best for every problem) What are implications? You cannot design a general-purpose and powerful evolutionary algorithms. You should design a problem-specific algorithm. Question: how to prove no free lunch theorem? 04/12/2006 p.16

18 Fitness Landscape Question: given an EA, what kind of problems is hard for it? Given a distance, under this distance, some populations are far (exponential in the input size) away from the optimal solution Drift (speed) towards the optimal solution is no more than a positive constant. A wide-gap far-distance problem. A narrow-gap far-distance problem (long-path problem). 04/12/2006 p.17

19 Exercises 1. Prove the EA, using elitism selection, bitwise mutation, and one-point crossover, is convergent when solving any optimization problem. (ref.[5]) 2. Prove the EA, using roulette wheel selection, bitwise mutation, is not convergent when solving any optimization problem. (ref.[5]) 3. Prove the (1+1) EA, using elitist selection, bitwise mutation, can solve the OneMax problem in O(n log n) generations. (ref.[6]) 04/12/2006 p.18

20 Further Readings [1] T. Bäck, D. B. Fogel, and Z. Michalewicz, editors. Handbook of Evolutionary Computation. Oxford University Press, Oxford, [2] A. E. Eiben and G. Rudolph. Theory of evolutionary algorithms: A bird eye view. Theoretical Computer Science, 229(1):3 9, [3] J. He and X. Yao. Drift analysis and average time complexity of evolutionary algorithms. Artificial Intelligence, 127(1):57 85, [4] D. H. Wolpert and W. G. Macready. No free lunch theorem for optimization. IEEE Trans. on Evolutionary Computation, 1(1):67 82, [5] G. Rudolph. Convergence analysis of canonical genetic algorithms. IEEE Trans. on Neural Networks, 5(1):96 101, [6] J. He and X. Yao. A study of drift analysis for estimating computation time of evolutionary algorithms. Natural Computing, 3(1):21 35, /12/2006 p.19