Natural Computing

By: Hans-Georg Beyer

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Hardcover | 27 March 2001

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406 Pages

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Evolutionary Algorithms, in particular Evolution Strategies, Genetic Algorithms, or Evolutionary Programming, have found wide acceptance as robust optimization algorithms in the last ten years. Compared with the broad propagation and the resulting practical prosperity in different scientific fields, the theory has not progressed as much. This monograph provides the framework and the first steps toward the theoretical analysis of Evolution Strategies (ES). The main emphasis is on understanding the functioning of these probabilistic optimization algorithms in real-valued search spaces by investigating the dynamical properties of some well-established ES algorithms. The book introduces the basic concepts of this analysis, such as progress rate, quality gain, and self-adaptation response, and describes how to calculate these quantities. Based on the analysis, functioning principles are derived, aiming at a qualitative understanding of why and how ES algorithms work.

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From the reviews:

"He gives an extensive mathematical treatment of idealised models of behaviour for several types of EA ... . The detail is extensive enough to guide and educate graduate students ... . The figures are clear and convincing. Part of the quality of the book is its aesthetically pleasing layout, for both figures and mathematics. ... The book is a desirable resource for all those, students and others, who need or wish to have a single portable source for the mathematically-based fundamentals of the subject." (John Campbell, Expert Update, Vol. 6 (1), 2003)

"Evolutionary algorithms (EA) have found a broad acceptance as robust optimization algorithms in the last ten years. ... The aim of this monograph is to provide a theoretical framework for the ES research field. ... The book contains references to open problems, to new problem formulations, and to future research directions at the relevant places." (Horst Hollatz, Zentralblatt MATH, Vol. 969, 2001)

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Non-Fiction Computing & I.T.Computer Science Artificial Intelligence Neural Networks & Fuzzy Systems Computer Programming & Software Development Algorithms & Data Structures Mathematics Optimisation Linear Programming Probability & Statistics Science

Physics Biology, Life Sciences Reference, Information & Interdisciplinary Subjects Research & Information Information theory Cybernetics & Systems Theory Science in General

Introduction	p. 1
A Short Characterization of the EA	p. 1
The Evolution Strategy	p. 4
The <$>(\mu/\rho {\mathop {,}^{+}} \lambda)<$>-ES Algorithm	p. 4
The Genetic Operators of the ES	p. 7
The Selection Operator	p. 8
The Mutation Operator	p. 9
The Reproduction Operator	p. 11
The Recombination Operator	p. 12
The Convergence of the Evolution Strategy	p. 14
ES Convergence Global Aspects	p. 15
ES Convergence - Local Aspects	p. 17
Basic Principles of Evolutionary Algorithms	p. 18
Evolvability	p. 18
EPP, GR, and MISR	p. 20
EPP - the Evolutionary Progress Principle	p. 20
GR - the Genetic Repair Hypothesis	p. 21
The MISR Principle	p. 21
The Analysis of the ES - an Overview	p. 22
Concepts for the Analysis of the ES	p. 25
Local Progress Measures	p. 25
The Quality Gain <$>\overline {Q}<$>	p. 26
The Progress Rate <$>\varphi<$>	p. 28
The Normal Progress <$>\varphi_R<$>	p. 30
Models of Fitness Landscapes	p. 31
The (Hyper-)Sphere Model	p. 32
The Hyperplane	p. 33
Corridor and Discus - Hyperplanes with Restrictions	p. 34
Quadratic Functions and Landscapes of Higher-Order	p. 35
The Bit Counting Function OneMax	p. 36
Noisy Fitness Landscapes	p. 36
The Differential-Geometrical Model for Non-Spherical Fitness Landscapes	p. 37
Fundamentals of the Differential-Geometrical Model	p. 38
The Local Hyperplane &partial;Q_y	p. 38
The Metric Tensor g_¿ß	p. 39
The Second Fundamental Form	p. 40
The Mean Curvature 〈ℵ〉	p. 41
The Calculation of the Mean Radius R_〈ℵ〉	p. 43
The Metric Tensor	p. 43
The b_¿ß-Tensor	p. 44
The Computation of the Mean Radius R_〈ℵ〉	p. 45
The ES Dynamics	p. 47
The R-Dynamics	p. 48
The Special Case ¿*(g) = const	p. 49
The Progress Rate of the <$>(1 {\mathop {,}^{+}} \lambda<$>)-ES on the Sphere Model	p. 51
The Exact (1 + 1)-ES Theory	p. 51
The Progress Rate without Noise in Fitness Measurements	p. 54
The Progress Rate at Disturbed Fitness Measurements	p. 57
Asymptotic Formulae for the <$>(1 {\mathop {,}^{+}} \lambda)<$>-ES	p. 61
A Geometrical Analysis of the (1 + 1)-ES	p. 61
The Asymptote of the Mutation Vector z	p. 62
The Progress Rate of the (1 + 1)-ES on the Sphere Model	p. 64
The Success Probability P_s1+1 and the EPP	p. 68
The Asymptotic <$>\varphi_{1 {\mathop {,}^{+}} \lambda}^{\ast}<$> Integral	p. 69
The Analysis of the (1, ¿)-ES	p. 71
The Progress Rate <$>\varphi_{1,\lambda}^{\ast}<$>	p. 71
The Progress Coefficient c_1,¿	p. 74
The Analysis of the (1 + ¿)-ES	p. 77
The Progress Rate	p. 77
The Success Probability P_s1+¿	p. 79
The Progress Function <$>d_{1+\lambda}^{(1)} (x)<$>	p. 79
The Asymptotic Analysis of the <$>(\tilde {1} {\mathop {,}^{+}} \tilde {\lambda})<$>-ES	p. 80
The Theory of the <$>(\tilde {1} {\mathop {,}^{+}} \tilde {\lambda})<$> Progress Rate	p. 80
Progress Integrals and Acceptance Probabilities	p. 80
The Asymptotic Fitness Model and the <$>p(\tilde {Q}_{ x} x)<$> Density	p. 83
The Calculation of <$>P_1(\tilde {Q})<$>	p. 85
On the Analysis of the <$>(\tilde {1}, \tilde {\lambda})<$>-ES	p. 86
The Asymptotic <$>\varphi_{\tilde {1},\tilde {\lambda}}^{\ast}<$> Formula	p. 86
The Dynamics of the <$>(\tilde {1}, \tilde {\lambda})<$>-ES	p. 89
On the Analysis of the <$>(\tilde {1} + \tilde {\lambda})<$>-ES	p. 93
The Asymptotic <$>\varphi_{\tilde {1}+\tilde {\lambda}}^{\ast}<$> Integral and <$>P_{{\rm s}\tilde {1}+\tilde {\lambda}}<$>	p. 93
An (Almost) Necessary Evolution Criterion for <$>(\tilde {1} + \tilde {\lambda})<$>-ES	p. 95
Some Aspects of the <$>(\tilde {1} + \tilde {1})<$>-ES	p. 97
Convergence Improvement by Inheriting Scaled Mutations	p. 102
Theoretical Fundamentals	p. 102
Discussion of the <$>(\tilde {1},\tilde {\lambda})<$>-ES	p. 103
The N-Dependent (1,¿) Progress Rate Formula	p. 104
Motivation	p. 104
The p(r) Density	p. 105
The Derivation of the <$>\varphi^\ast<$> Progress Rate Formula	p. 107
Comparison with Experiments	p. 109
A Normal Approximation for p(r)	p. 111
The <$>(1 {\mathop {,}^{+}} \lambda)<$> Quality Gain	p. 113
The Theory of the <$>(1 {\mathop {,}^{+}} \lambda)<$> Quality Gain	p. 113
The <$>\overline {Q}_{1 {\mathop {,}^{+}} \lambda}<$> Integral	p. 113
On the Approximation of P_z(z)	p. 115
On Approximating the Quantile Function <$>P_z^{-1}(f)<$>	p. 118
The <$>\overline {Q}_{1, \lambda}<$> Formula	p. 119
The <$>\overline {Q}_{1+\lambda}<$> Formula	p. 120
Fitness Models and Mutation Operators	p. 122
The General Quadratic Model and Correlated Mutations	p. 122
The Special Case of Isotropic Gaussian Mutations	p. 125
Examples of Non-Quadratic Fitness Functions	p. 126
Biquadratic Fitness with Isotropic Gaussian Mutations	p. 126
The Counting Ones Function OneMax	p. 127
Experiments and Interpretations of the Results	p. 131
Normalization	p. 131
Quadratic Fitness, Isotropic Gaussian Mutations and the Differential-Geometric Model	p. 131
The Normalization for the Biquadratic Case	p. 134
Experiments and Approximation Quality	p. 135
Quality Gain or Progress Rate?	p. 137
The Analysis of the (¿, ¿)-ES	p. 143
Fundamentals and Theoretical Framework	p. 143
Preliminaries	p. 143
The (¿, ¿) Algorithm and the <$>\varphi_{\mu,\lambda}<$> Definition	p. 144
The <$>\varphi_{\mu,\lambda}<$> Integral	p. 146
Formal Approximation of the Offspring Distribution p(r)	p. 147
Estimation of <$>\overline {r}<$>, s, and ¿, and of <$>\overline {r}_{ R_m}<$>, <$>M_{2 R_m}<$>, and <$>M_{3 R_m}<$>	p. 150
The Simplification of <$>\overline {r}_{ R_m}<$>, <$>M_{2 R_m}<$>, and <$>M_{3 R_m}<$>	p. 153
The Statistical Approximation of <$>\overline {r}<$>, s, and ¿	p. 156
The Integral Expression of <$>\overline {\langle (\Delta R)^2 \rangle}<$>	p. 159
The Integral Expression of <$>\overline {\langle (\Delta R)^3\rangle}<$>	p. 162
Approximation of the Stationary State the Self-Consistent Method	p. 166
On the Analysis in the Linear ¿ Approximation	p. 168
The Linear ¿ Approximation	p. 168
The Approximation of the <$>\varphi_{\mu,\lambda}<$> Integral	p. 169
The Approximation of s^{(g + 1)}	p. 172
The I_A Integral	p. 173
The I_B Integral	p. 175
Composing the s^(g+1) Formula	p. 176
The Approximation of ¿^(g+1)	p. 176
The I_C Integral	p. 177
The I_D Integral	p. 178
The I_E Integral	p. 181
Composing the ¿^(g+1) Formula	p. 182
The Self-Consistent Method and the <$>\varphi_{\mu,\lambda}^{\ast}<$> Formulae	p. 183
The Discussion of the (¿, ¿)-ES	p. 185
The Comparison with Experiments	p. 185
Data Extraction from ES Runs	p. 185
The ES Simulations for ¿ = const and ¿* = const	p. 186
Simplified <$>\varphi_{\mu,\lambda}^{\ast}<$> Formulae and the Progress Coefficient c_¿,¿	p. 188
The Derivation of the <$>\varphi_{\mu,\lambda}^{\ast}<$> Formulae	p. 188
EPP, the Properties of c_¿,¿, and the Fitness Efficiency	p. 189
The (¿,¿)-ES on the Hyperplane	p. 192
The Exploration Behavior of the (¿,¿)-ES	p. 194
Evolution in the (r,¿_i) Picture	p. 194
The Random Walk in the Angular Space	p. 195
Experimental Verification and Discussion	p. 199
Final Remarks on the Search Behavior of the (¿,¿)-ES	p. 199
The (¿/¿,¿) Strategies - or Why "Sex" May be Good	p. 203
The Intermediate (¿/¿_I,¿)-ES	p. 203
Foundations of the (¿/¿,¿) Theory	p. 204
The (¿/¿_I,¿) Algorithm	p. 204
The Definition of the Progress Rate <$>\varphi_{\mu/\mu,\lambda}<$>	p. 205
The Statistical Approximation of <$>\varphi_{\mu/\mu,\lambda}<$>	p. 206
The Calculation of <$>\varphi_{\mu/\mu I,\lambda<$>	p. 210
The Derivation of the Expected Value <$>\overline {\langle x \rangle}<$>	p. 210
The Derivation of the Expected Value <$>\overline {\langle {\rm h}^2\rangle}<$>	p. 213
The (¿/¿_I,¿) Progress Rate	p. 215
The Discussion of the (¿/¿_I,¿) Theory	p. 216
The Comparison with Experiments and the Case N → ∞	p. 216
On the Benefit of Recombination - or Why "Sex" May be Good	p. 218
System Conditions of Recombination	p. 222
The (¿/¿_I, ¿)-ES and the Optimal ¿ Choice	p. 224
The Exploration Behavior of the (¿/¿_I, ¿)-ES	p. 229
The Dominant (¿/¿_D, ¿)-ES	p. 232
A First Approach to the Analysis of the (¿/¿_D, ¿)-ES	p. 232
The (¿/¿_D, ¿)-ES Algorithm and the Definition of <$>\varphi_{\mu/\mu_D, \lambda}<$>	p. 232
A Simple Model for the Analysis of the (¿/¿_D, ¿)-ES	p. 233
The Isotropic Surrogate Mutations	p. 235
The Progress Rate <$>\varphi_{\mu/\mu_D, \lambda}^{\ast}<$>	p. 237
The Discussion of the (¿/¿_D, ¿)-ES	p. 239
The Comparison with Experiments and the Case N → ∞	p. 239
The MISR Principle, the Genetic Drift, and the GR Hypothesis	p. 240
The Asymptotic Properties of the (¿/¿,¿)-ES	p. 246
The c_¿/¿,¿ Coefficient	p. 247
The Asymptotic Expansion of the c_¿/¿,¿ Coefficient	p. 247
The Asymptotic Order of the c_¿/¿,¿ Coefficients	p. 250
The Asymptotic Progress Law	p. 250
Fitness Efficiency and the Optimal <$>\vartheta<$> Choice	p. 252
Asymptotic Fitness Efficiency ¿_¿/¿,¿ of the (¿/¿, ¿)-ES	p. 252
The Relation to the (1 + 1)-ES	p. 252
The Dynamics and the Time Complexity of the (¿/¿,¿)-ES	p. 255
The (1, ¿)-¿-Self-Adaptation	p. 257
Introduction	p. 257
Concepts of ¿-Control	p. 257
The ¿-Self-Adaptation	p. 259
The (1, ¿)-¿SA Algorithm	p. 261
Operators for the Mutation of the Mutation Strength	p. 261
Theoretical Framework for the Analysis of the ¿SA	p. 263
The Evolutionary Dynamics of the (1, ¿)-¿SA-ES	p. 264
The r Evolution	p. 265
The <$>\varsigma<$> Evolution	p. 266
The Microscopic Aspects	p. 268
The Density p_1;1 (r) of a Single Descendant	p. 269
The Transition Density p_1;¿ (r)	p. 270
The Transition Density p_1;¿ <$>(\varsigma)<$>	p. 271
The <$>\varphi^{(k)}<$> and ¿^(k)-SAR Functions	p. 272
Determination of the Progress Rate and the SAR	p. 275
Progress Integrals <$>\varphi^{\ast(k)}<$>	p. 275
Numerical Examples for the Progress Rate <$>\varphi^{\ast}<$>	p. 275
An Analytic <$>\varphi^\ast<$> Formula for p_¿ = p_II, ¿ = 2	p. 276
The ¿ → 0 and ß → 0 Approximation for <$>D^{\ast}_{\varphi}<$>	p. 279
The SAR Functions ¿^(k)	p. 280
Numerical Examples for ¿⁽¹⁾, Discussion, and Comparison with Experiments	p. 280
An Analytic ¿ Formula for p_¿ = p_II, ¿ = 2	p. 283
Bounds for ¿	p. 286
Analytic Approximation of ¿^(k) for Small <$>\varsigma^\ast<$> and ¿ - General Aspects	p. 289
Approximations for ¿, ¿⁽²⁾, and D_¿	p. 294
The (1, ¿)-¿SA Evolution (I) - Dynamics in the Deterministic Approximation	p. 299
The Evolution Equations of the (1, ¿)-¿SA-ES	p. 299
The ES in the Stationary State	p. 300
Determining the Stationary State	p. 300
Optimal ES Performance and the <$>1/\sqrt {N}<$> Rule	p. 302
The Differential Equation of the ¿ Evolution, the Transient Behavior for Small <$>\varsigma^{\ast(0)} < \varsigma_{ss}^{\ast}<$>, and the Stationary r Dynamics	p. 303
Approaching the Steady-State from <$>\varsigma^{\ast} >> \varsigma_{ss}^{\ast}<$>	p. 308
The (1, ¿)-¿SA Evolution (II) - Dynamics with Fluctuations	p. 309
Motivation	p. 309
Chapman-Kolmogorov Equation and Transition Densities	p. 309
Mean Value Dynamics of the r Evolution	p. 313
Mean Value Dynamics of the <$>\varsigma^{\ast}<$> Evolution	p. 315
Approximate Equations for the Stationary <$>\sigma_{\infty}^{\ast}<$> State	p. 317
The Integral Equation of the Stationary <$>p(\varsigma^{\ast})<$> Density	p. 317
An Approach for Solving the Momentum Equations	p. 319
Discussion of the Stationary State and the <$>1/\sqrt{N}<$> Rule	p. 320
Comparison with ES Experiments	p. 320
Special Analytical Cases	p. 321
The ¿-Scaling Rule	p. 323
Final Remarks on the ¿ Self-Adaptation	p. 324
Appendices	p. 327
Integrals	p. 329
Definite Integrals of the Normal Distribution	p. 329
Indefinite Integrals of the Normal Distribution	p. 331
Integrals of the Form <$>I^{\alpha,\beta}(x) = \int\nolimits_{-\infty}^{t=x} t^{\beta} \ {\rm e}^{-{{1+\alpha} \over 2}t^2} \ {\rm d}t<$>	p. 331
Integrals of the Form <$>I_{\phi}^{\beta}(x) = \int\nolimits_{-\infty}^{t=x} t^{\beta} \ {\rm e}^{-\textstyle{1}{2} t^2 \Phi(t) \ {\rm d}t<$>	p. 332
Some Integral Identities	p. 334
Approximations	p. 337
Frequently Used Taylor Expansions	p. 337
The Hermite Polynomials Hek(x)	p. 339
Cumulants, Moments, and Approximations	p. 341
Fundamental Relations	p. 341
The Weight Coefficients for the Density Approximation of a Standardized Random Variable	p. 344
Approximation of the Quantile Function	p. 349
The Normal Distribution	p. 351
<$>{\cal N}<$>(0,1) Distribution Function, Gaussian Integral, and Error Function	p. 351
Asymptotic Order of the Moments of <$>{x \over R}<$>	p. 354
Product Moments of Correlated Gaussian Mutations	p. 355
Fundamental Relations	p. 355
Derivation of the Product Moments	p. 356
(1, ¿)-Progress Coefficients	p. 359
Asymptotics of the Progress Coefficients <$>d_{1,\lambda}^{(k)}<$>	p. 359
An Asymptotic Expansion for the <$>d_{1,\lambda}^{(k)}<$> Coefficients	p. 359
The Asymptotic c_1,¿ and <$>d_{1,\lambda}^{(2)}<$> Formulae	p. 360
An Alternative Derivation for c_1,¿	p. 361
Table of Progress Coefficients of the (1, ¿)-ES	p. 363
References	p. 365
Index	p. 373
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Natural Computing

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