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Tables of Contents for The Theory of Evolution Strategies
Chapter/Section Title
Page #
Page Count
Introduction
1
25
A Short Characterization of the EA
1
3
The Evolution Strategy
4
10
The (μ/ρ+, λ)-ES Algorithm
4
3
The Genetic Operators of the ES
7
1
The Selection Operator
8
1
The Mutation Operator
9
2
The Reproduction Operator
11
1
The Recombination Operator
12
2
The Convergence of the Evolution Strategy
14
4
ES Convergence -Global Aspects
15
2
ES Convergence - Local Aspects
17
1
Basic Principles of Evolutionary Algorithms
18
4
Evolvability
18
2
EPP, GR, and MISR
20
1
EPP - the Evolutionary Progress Principle
20
1
GR - the Genetic Repair Hypothesis
21
1
The MISR Principle
21
1
The Analysis of the ES - an Overview
22
3
Concepts for the Analysis of the ES
25
26
Local Progress Measures
25
6
The Quality Gain Q
26
2
The Progress Rate ϕ
28
2
The Normal Progress ϕR
30
1
Models of Fitness Landscapes
31
6
The (Hyper-)Sphere Model
32
1
The Hyperplane
33
1
Corridor and Discus - Hyperplanes with Restrictions
34
1
Quadratic Functions and Landscapes of Higher-Order
35
1
The Bit Counting Function OneMax
36
1
Noisy Fitness Landscapes
36
1
The Differential-Geometrical Model for Non-Spherical Fitness Landscapes
37
10
Fundamentals of the Differential-Geometrical Model
38
1
The Local Hyperplane ∂Qy
38
1
The Metric Tensor gαβ
39
1
The Second Fundamental Form
40
1
The Mean Curvature ⟨x⟩
41
2
The Calculation of the Mean Radius R⟨x⟩
43
1
The Metric Tensor
43
1
The bαβ-Tensor
44
1
The Computation of the Mean Radius R⟨x⟩
45
2
The ES Dynamics
47
4
The R-Dynamics
48
1
The Special Case σ* (g) = Const
49
2
The Progress Rate of the (1 +, λ) -ES on the Sphere Model
51
62
The Exact (1+1)-ES Theory
51
10
The Progress Rate without Noise in Fitness Measurements
54
3
The Progress Rate at Disturbed Fitness Measurements
57
4
Asymptotic Formulae for the (1 +, λ)-ES
61
19
A Geometrical Analysis of the (1 + 1)-ES
61
1
The Asymptote of the Mutation Vector z
62
2
The Progress Rate of the (1+1)-ES on the Sphere Model
64
4
The Success Probability Ps1+1 and the EPP
68
1
The Asymptotic ϕ1+,λ Integral
69
2
The Analysis of the (1, λ)-ES
71
1
The Progress Rate ϕ1,λ
71
3
The Progress Coefficient c1,λ
74
3
The Analysis of the (1+λ)-ES
77
1
The Progress Rate
77
2
The Success Probability Ps1+λ
79
1
The Progress Function d1(1)+λ(x)
79
1
The Asymptotic Analysis of the (1 +, λ)-ES
80
24
The Theory of the (1 +, λ) Progress Rate
80
1
Progress Integrals and Acceptance Probabilities
80
3
The Asymptotic Fitness Model and the p(Q&varbar;x&varbar;x Density
83
2
The Calculation of P1(Q)
85
1
On the Analysis of the (1, λ)-ES
86
1
The Asymptotic ϕ1, λ Formula
86
3
The Dynamics of the (1, λ)-ES
89
4
On the Analysis of the (1 + λ)-ES
93
1
The Asymptotic ϕ*1, λ Integral and Ps1, λ
93
2
An (almost) Necessary Evolution Criterion for (1, λ)-ES
95
2
Some Aspects of the (1+1)-ES
97
5
Convergence Improvement by Inheriting Scaled Mutations
102
1
Theoretical Fundamentals
102
1
Discussion of the (1 + λ)-ES
103
1
The N-Dependent (1, λ) Progress Rate Formula
104
9
Motivation
104
1
The p(r) Density
105
2
The Derivation of the ϕ* Progress Rate Formula
107
2
Comparison with Experiments
109
2
A Normal Approximation for p(r)
111
2
The (1 +, λ) Quality Gain
113
30
The Theory of the (1 +, λ) Quality Gain
113
9
The Q1 +, λ Integral
113
2
On the Approximation of Pz
115
3
On Approximating the Quantile Function Pz(z)-1(f)
118
1
The Q1,λ Formula
119
1
The Q1,λ Formula
120
2
Fitness Models and Mutation Operators
122
9
The General Quadratic Model and Correlated Mutations
122
3
The Special Case of Isotropic Gaussian Mutations
125
1
Examples of Non-Quadratic Fitness Functions
126
1
Biquadratic Fitness with Isotropic Gaussian Mutations
126
1
The Counting Ones Function OneMax
127
4
Experiments and Interpretations of the Results
131
12
Normalization
131
1
Quadratic Fitness, Fitness, Isotropic Gaussian Mutations and the Differential-Geometric Model
131
3
The Normalization for the Biquadratic Case
134
1
Experiments and Approximation Quality
135
2
Quality Gain or Progress Rate?
137
6
The Analysis of the (μ, λ)-ES
143
60
Fundamentals and Theoretical Framework
143
25
Preliminaries
143
1
The (μ,λ) Algorithm and the ϕμ,λ Definition
144
2
The ϕμ,λ Integral
146
1
Formal Approximation of the Offspring Distribution p(r)
147
3
Estimation of r, s, and γ and of r&varbar;Rm, M2&varbar;Rm, and M3&varbar;Rm
150
3
The Simplification of r&varbar;Rm, M2&varbar;Rm, and M3&varbar;Rm
153
3
The Statistical Approximation of r, s, and γ
156
3
The Integral Expression of &lrang;(ΔR)2&rrang;
159
3
The Intergral Expression of &lrang;(ΔR)3&rrang;
162
4
Approximation of the Stationary State-the Self-Consistent Method
166
2
On the Analysis in the Linear γ Approximation
168
17
The Linear γ Approximation
168
1
The Approximation of the ϕμ,λ Integral
169
3
The Approximation of s(g+1)
172
1
The IA Integral
173
2
The IB Integral
175
1
Composing the s(g+1) Formula
176
1
The Approximation of γ(g+1)
176
1
The IC Integral
177
1
The ID Integral
178
3
The IE Integral
181
1
Composing the γg+1) Formula
182
1
The Self-Consistent Method and the ϕμ,λ Formulae
183
2
The Discussion of the (ϕ,λ)-ES
185
18
The Comparison with Experiments
185
1
Data Extraction from ES Runs
185
1
The ES Simulations for σ const and σ* = const
186
2
Simplified ϕ*μ,λ Formulae and the Progress Coefficient cμ,λ
188
1
The Derivation of the ϕ*μ,λ Formulae
188
1
EPP, the Properties of cμ,λ and the Fitness Efficiency
189
3
The (μ,λ)-ES on the Hyperplane
192
2
The Exploration Behavior of the (μ,λ)-ES
194
1
Evolution in the (r,γi) Picture
194
1
The Random Walk in the Angular Space
195
4
Experimental Verification and Discussion
199
1
Final Remarks on the Search Behavior of the (μλ)-ES
199
4
The (μ/μ,λ) Strategies -or Why ``Sex'' May be Good
203
54
The Intermediate (μ/μ,I,λ)-ES
203
29
Foundations of the (μ/μ,I,λ) Theory
204
1
The (μ/μ,I,λ) Algorithm
204
1
The Definition of the Progress Rate ϕμ/μ,λ
205
1
The Statistical Approximation of ϕμ/μ,λ
206
4
The Calculation of ϕμ/μ,I,λ
210
1
The Derivation of the Expected Value &lrang;x&rrang;
210
3
The Derivation of the Expected Value &lrang;h2&rrang;
213
2
The (μ/μ,I,λ) Progress Rate
215
1
The Discussion of the (μ/μ,I,λ) Theory
216
1
The Comparison with Experiments and the Case N → ∞
216
2
On the Benefit of Recombination -- or Why ``Sex'' May be Good
218
4
System Conditions of Recombination
222
2
The (μ/ρ,I,λ)-ES and the Optimal μ Choice
224
5
The Exploration Behavior of the (μ/μ,I,λ)-ES
229
3
The Dominant (μ/μ,D,λ)-ES
232
14
A First Approach to the Analysis of the (μ/μ,D,λ)-ES
232
1
The (μ/μ,D,λ)-ES Algorithm and the Definition of ϕμ/μ,D,λ
232
1
A Simple Model for the Analysis of the (μ/μ,D,λ)-ES
233
2
The Isotropic Surrogate Mutations
235
2
The Progress Rateϕ*μ/μDλ
237
2
The Discussion of the (&mu/μD,λ)-ES
239
1
The Comparison with Experiments and the Case N → ∞
239
1
The MISR Principle, the Genetic Drift, and the GR Hypothesis
240
6
The Asymptotic Properties of the (&mu/μD,λ)-ES
246
11
The (c&mu/μ,λ) Coefficient
247
1
The Asymptotic Expansion of the c&mu/μ,λ Coefficient
247
3
The Asymptotic Order of the c&mu/μ,λ Coefficients
250
1
The Asymptotic Progress Law
250
2
Fitness Efficiency and the Optimal η&mu/μ,λ Choice
252
1
Asymptotic Fitness Efficiency (&mu/μ,λ) of the (c&mu/μ,λ)-ES
252
1
The Relation to the (1+1)-ES
252
3
The Dynamics and the Time Complexity of the (c&mu/μ,λ)-ES
255
2
The (1, λ)-σ-Self-Adaptation
257
70
Introduction
257
6
Concepts of σ-Control
257
2
The σ-Self Adaptation
259
2
The (1,λ) σSA Algorithm
261
1
Operators for the Mutation of the Mutation Strength
261
2
Theoretical Framework for the Analysis of the (1,λ)σSA-ESA
263
12
The Evolutionary Dynamics of the (1,λ)-σSA-ES
264
1
The r Evolution
265
1
The &etaxs; Evolution
266
2
The Microscopic Aspects
268
1
The Density p1;1(r) of a Single Descendant
269
1
The Transition Density p1;λ(r)
270
1
The Transition Density p1;λ(ς)
271
1
The ϕ(k) and ψ(k)-SAR Functions
272
3
Determination of the Progress Rate and the SAR
275
24
Progress Integrals ϕ*(k)
275
1
Numerical Examples for the Progress Rate ϕ*
275
1
An Analytic ϕ* Formula for pσ = pii, λ λ = 2
276
3
The τ → 0 and β → 0 Approximation for ϕ*
279
1
The SAR Functions ψ(k)
280
1
Numerical Examples for ψ(1) Discussion, and Comparison with Experiments
280
3
An Analytic ψ Formula for pσ = pii, λ = 2
283
3
Bounds for ψ
286
3
Analytic Approximation of ψ(k) for small ς*and τ - General Aspects
289
5
Approximations for ψ, ψ(2) and Dψ
294
5
The (1,λ)-σSA Evolution (I) -- Dynamics in the Deterministic Approximation
299
10
The Evolution Equations of the (1,λ)-σ-SA-ES
299
1
The ES in the Stationary State
300
1
Determining the Stationary State
300
2
Optimal ES Performance and the 1/√N Rule
302
1
The Differential Equation of the σ Evolution, the Transient Behavior for Small ς*(0) > ςss and the Stationary r Dynamics
303
5
Approaching the Steady-State from ς*» ς*ss
308
1
The (1,λ)-σ-SA Evolution (II) - Dynamics with Fluctuations
309
Motivation
309
1
Chapman-Kolmogorov Equation and Transition Densities
309
4
Mean Value Dynamics of the r Evolution
313
2
Mean Value Dynamics of the ς* Evolution
315
2
Approximate Equations for the Stationary σ*∞ State
317
1
The Integral Equation of the Stationary p(ς*) Density
317
2
An Approach for Solving the Momentum Equations
319
1
Discussion of the Stationary State and the 1/√N Rule
320
1
Comparison with ES Experiments
320
1
Special Analytical Cases
321
2
The τ-Scaling Rule
323
1
Final Remarks on the σ Self-Adaptation
324
3
Appendies
327
38
A. Integrals
329
8
A.1 Definite Integrals of the Normal Distribution
329
2
A.2 Indefinite Integrals of the Normal Distribution
331
1
A.2.1 Integrals of the Form
331
1
A.2.2 Integrals of the Form
332
2
A.3 Some Integral Identities
334
3
B. Approximations
337
14
B.1 Frequently Used Taylor Expansions
337
2
B.2 The Hermite Polynomials Heκ(x)
339
2
B.3 Cumulants, Moments, and Approximations
341
1
B.3.1 Fundamental Relations
341
3
B.3.2 The Weight Coefficients for the Density Approximation of a Standardized Random Variable
344
5
B.3.4 Approximation of the Quantile Function
349
2
C. The Normal Distribution
351
8
C.1 Distribution Function, Gaussian Integral, and Error Function
351
3
C.2 Asymptotic Order of the Moments of x/R
354
1
C.3 Product Moments of Correlated Gaussian Mutations
355
1
C.3.1 Fundamental Relations
355
1
C.3.2 Derivation of the Product Moments
356
3
D. (1, λ) Progress Coefficients
359
6
D.1 Asymptotics of the Progress Coefficients d(k)1,λ
359
1
D.1.1 An Asymptotic Expansion for the d(k)1,λ Coefficients
359
1
D.1.2 The Asymptotic c1,λ and d(k)1,λ Formulae
360
1
D.1.3 An Alternative Derivation for c1,λ
361
2
D.2 Table of Progress Coefficients of the (1,λ)-ES
363
2
References
365
8
Index
373