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## Evolution Strategies with Cumulative Step Length Adaptation on the Noisy Parabolic Ridge (2006)

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### Citations

831 |
Evolutionsstrategie: optimierung technischer systeme nach prinzipien der biologischen evolution. Frommann-Hozboog,
- Rechenberg
- 1973
(Show Context)
Citation Context ...ork has gone into the analysis of the behaviour of evolution strategies on simple objective functions, such as the sphere model [1, 5, 11, 23], ellipsoidal fitness landscapes [12], the corridor model =-=[22]-=-, and the ridge function class [10, 19, 20, 21, 23]. While the sphere and the ellipsoids serve as models for fitness landscapes in the vicinity of local optima, both the corridor and the ridge strive ... |

672 |
Order Statistics
- David
- 1970
(Show Context)
Citation Context ... µ/µ,λ A = � 1 + (σ2 + σ2 ɛ )/ρ2σ2 = ρc µ/µ,λ � . (13) 1 + ϑ2 + ρ2 Finally, for the squared length of the combined central and lateral components of the progress vector, �z (avg) 2...N �2 N N→∞ = 1 µ =-=(14)-=- holds in analogy to the corresponding result in Eq. (8) for the sphere model. 3.2. Distance from the Ridge Axis and Progress Rate The expected values of the signed lengths of the axial and central co... |

549 | Completely derandomized selfadaptation in evolution strategies,”
- Hansen, Ostermeier
- 2001
(Show Context)
Citation Context ...s has been presented in [10]. And finally, as pointed out by Whitley, Lunacek, and Knight [26], the ridge is a prime example for the usefulness of nonisotropic mutations. Strategies such as the CMAES =-=[16]-=- are capable of learning the direction of the ridge axis. After adaptation of the covariance matrix is complete, the CMA-ES can track main.tex; 17/02/2006; 10:22; p.28sthe ridge by generating mutation... |

268 |
Evolution Strategies: A Comprehensive Introduction /
- Beyer, Schwefel
- 2002
(Show Context)
Citation Context ...ɛz ɛ )/ρσ, it follows from Lemma 2 that the expected value of the signed length of the central component of the progress vector is � E z (avg) � N→∞ c µ/µ,λ A = � 1 + (σ2 + σ2 ɛ )/ρ2σ2 = ρc µ/µ,λ � . =-=(13)-=- 1 + ϑ2 + ρ2 Finally, for the squared length of the combined central and lateral components of the progress vector, �z (avg) 2...N �2 N N→∞ = 1 µ (14) holds in analogy to the corresponding result in E... |

84 | Toward a theory of evolution strategies: On the benefit of sex — the (µ/µ, λ)-theory.
- Beyer
- 1995
(Show Context)
Citation Context ...etting Xi = z (i) A , Lemma 1 from Section 2.2 is thus immediately applicable and the expected signed length of the central component of the progress vector is c µ/µ,λ. Moreover, it has been shown in =-=[8]-=- that �z (avg) � 2 N N→∞ 9 1 = µ . (8) The reduction in the squared length by a factor of 1/µ compared to that of the mutation vectors being averaged results from the fact that the lateral components ... |

76 | Toward a theory of evolution strategies: Self-adaptation
- Beyer
- 1996
(Show Context)
Citation Context ...well as the treatment of fluctuations (i.e., of quantities that cannot simply be replaced by their average values). Second, different forms of step length adaptation, such as mutative self-adaptation =-=[9]-=- or hierarchically organised strategies [23] remain to be studied and compared with cumulative step length adaptation. For hierarchically organised strategies, Herdy [17] provides empirical evidence f... |

55 | Step-size adaptation based on non-local use of selection information,”
- Ostermeier, Gawelczyk, et al.
- 1994
(Show Context)
Citation Context ... to mathematical analysis. The following description of the algorithm is deliberately brief. See [13] for a more comprehensive discussion of evolution strategies and their naming conventions, and see =-=[18]-=- for a thorough motivation of cumulative step length adaptation. In every time step the (µ/µ, λ)-ES computes the centroid of the population of candidate solutions as a search point x ∈ IR N that mutat... |

50 |
The Theory of Evolution Strategies. Natural Computing Series
- Beyer
- 2001
(Show Context)
Citation Context ...r step length control. See [13] for a comprehensive introduction. Much work has gone into the analysis of the behaviour of evolution strategies on simple objective functions, such as the sphere model =-=[1, 5, 11, 23]-=-, ellipsoidal fitness landscapes [12], the corridor model [22], and the ridge function class [10, 19, 20, 21, 23]. While the sphere and the ellipsoids serve as models for fitness landscapes in the vic... |

39 |
Evolutionary Algorithms and Gradient Search: Similarities and Differences,
- Salomon
- 1998
(Show Context)
Citation Context ...anised strategies, Herdy [17] provides empirical evidence for their usefulness for step length adaptation on the ridge. Third, other strategy variants, such as evolutionary gradient search strategies =-=[24]-=- or the (λ)opt-ES studied in [2] on the sphere model remain to be considered. Of interest as well is the examination of ridge topologies other than the quadratic one. In the absence of noise and not c... |

38 |
A comparison of evolution strategies with other direct search methods in the presence of noise
- Arnold, Beyer
(Show Context)
Citation Context ...evolution strategies has extensively been studied on the quadratic sphere model with objective function N� f(x) = (xi − ˆxi) i=1 2 x ∈ IR N where ˆx is the optimiser and the task is minimisation. See =-=[4]-=- for a discussion of the usefulness of such considerations, and see [11] for main.tex; 17/02/2006; 10:22; p.6sPSfrag replacements x σzA σzB σz y R Figure 2. Decomposition of a vector z into central co... |

36 |
Verallgemeinerte individuelle Schrittweitenregelung in der Evolutionsstrategie. Eine Untersuchung zur entstochastisierten, koordinatensystemunabhängigen Adaptation der Mutationsverteilung. Mensch und Buch
- Hansen
- 1998
(Show Context)
Citation Context ...e superscripts indicate time. Multiplying by 4d 2 /N 2 in order to switch to standardised distances yields evolution equation � ρ (t+1)2 = ρ (t)2 − 4d N ρ (t) σz (avg) A − σ2d N �z(avg) 2...N �2 � 13 =-=(15)-=- for the dynamical system. Consider the case that ρ does not diverge. In that case, iterating Eq. (15), the squared standardised distance to the ridge axis tends towards and then fluctuates around a s... |

33 |
Reproductive Isolation as Strategy Parameter in Hierarchically Organized Evolution Strategies. In
- Herdy
- 1992
(Show Context)
Citation Context ...hat have been published so far are purely empirical and include the aforementioned paper by Salomon [25] as well as interesting results regarding hierarchically organised strategies provided by Herdy =-=[17]-=-. This paper studies the performance of evolution strategies with intermediate multirecombination and cumulative step length adaptation on the noisy parabolic ridge function class. It thus extends pre... |

20 |
Performance analysis of evolutionary optimization with cumulative step length adaptation
- Arnold, Beyer
(Show Context)
Citation Context ...r step length control. See [13] for a comprehensive introduction. Much work has gone into the analysis of the behaviour of evolution strategies on simple objective functions, such as the sphere model =-=[1, 5, 11, 23]-=-, ellipsoidal fitness landscapes [12], the corridor model [22], and the ridge function class [10, 19, 20, 21, 23]. While the sphere and the ellipsoids serve as models for fitness landscapes in the vic... |

18 | Where elitists start limping: Evolution strategies at ridge functions
- Oyman, Beyer, et al.
(Show Context)
Citation Context ... 0 where ζ ∗ = ζd/µc µ/µ,λ is the normalised noise level. Solving for the square of the standardised distance from the ridge axis yields � ρ 2 N→∞ = σ ∗2 2(1 − ζ ∗2 ) + σ ∗4 4(1 − ζ ∗2 ) 2 17 σ∗2 + . =-=(20)-=- 1 − ζ∗2 For ζ ∗ ≥ 1, there is no real-valued solution and the strategy fails to track the ridge for any nonzero value of the mutation strength. In that case, the distance to the ridge axis diverges t... |

16 |
Local Performance of the (μ/μI , λ)-ES in a Noisy Environment.
- Arnold, Beyer
- 2001
(Show Context)
Citation Context ...alues of Xi is � µ� � 1 E Y [λ+1−k:λ] µ k=1 = c µ/µ,λ √ 1 + ϑ2 where the progress coefficient c µ/µ,λ was defined above. The derivation of this result is straightforward using the approach pursued in =-=[3, 6]-=-. The quantity ϑ is referred to as the noise-to-signal ratio of the selection process. 2.3. The Quadratic Sphere Model Since the early work of Rechenberg [22], the performance of evolution strategies ... |

13 | Ruffled by ridges: How evolutionary algorithms can fail
- Whitley, Lunacek, et al.
- 2004
(Show Context)
Citation Context ...bute to the continued improvement of existing and the design of new strategy variants. Ridge functions are known to pose significant problems for optimisation strategies. Whitley, Lunacek, and Knight =-=[26]-=- point out that while the difficulties of optimising ridges “are relatively well documented in the mathematical literature on derivative free minimization algorithms [. . . ], there is little discussi... |

12 | On the performance of the (1,λ)-evolution strategies for the ridge function class
- Beyer
- 2001
(Show Context)
Citation Context ...he behaviour of evolution strategies on simple objective functions, such as the sphere model [1, 5, 11, 23], ellipsoidal fitness landscapes [12], the corridor model [22], and the ridge function class =-=[10, 19, 20, 21, 23]-=-. While the sphere and the ellipsoids serve as models for fitness landscapes in the vicinity of local optima, both the corridor and the ridge strive to model features of such landscapes in greater dis... |

12 |
Evolutionsstrategie ’94. Friedrich Frommann Holzboog
- Rechenberg
- 1994
(Show Context)
Citation Context ...r step length control. See [13] for a comprehensive introduction. Much work has gone into the analysis of the behaviour of evolution strategies on simple objective functions, such as the sphere model =-=[1, 5, 11, 23]-=-, ellipsoidal fitness landscapes [12], the corridor model [22], and the ridge function class [10, 19, 20, 21, 23]. While the sphere and the ellipsoids serve as models for fitness landscapes in the vic... |

9 | Optimal weighted recombination
- Arnold
- 2005
(Show Context)
Citation Context ...vides empirical evidence for their usefulness for step length adaptation on the ridge. Third, other strategy variants, such as evolutionary gradient search strategies [24] or the (λ)opt-ES studied in =-=[2]-=- on the sphere model remain to be considered. Of interest as well is the examination of ridge topologies other than the quadratic one. In the absence of noise and not considering step length adaptatio... |

8 |
Noisy Optimization with Evolution Strategies. Genetic Algorithms and Evolutionary Computation
- Arnold
- 2002
(Show Context)
Citation Context |

7 |
Analysis of the (µ/µ, λ)-ES on the parabolic ridge
- Oyman, Beyer
(Show Context)
Citation Context ...he behaviour of evolution strategies on simple objective functions, such as the sphere model [1, 5, 11, 23], ellipsoidal fitness landscapes [12], the corridor model [22], and the ridge function class =-=[10, 19, 20, 21, 23]-=-. While the sphere and the ellipsoids serve as models for fitness landscapes in the vicinity of local optima, both the corridor and the ridge strive to model features of such landscapes in greater dis... |

5 |
Order Statistics: An Introduction
- Balakrishnan, Rao
- 1998
(Show Context)
Citation Context ...ge the Xi in nondecreasing order such that X1:λ ≤ X2:λ ≤ · · · ≤ Xλ:λ. The kth smallest of the Xi is denoted by Xk:λ and referred to as the kth order statistic of the sample. See Balakrishnan and Rao =-=[7]-=- for an introduction to the area of order statistics. The following lemma gives an expression for the expected value of the mean of the µ largest of the Xi for the case that the sample members are ind... |

4 |
Analysis of the (1,λ)-ES on the parabolic ridge
- Oyman, Beyer, et al.
(Show Context)
Citation Context ...he behaviour of evolution strategies on simple objective functions, such as the sphere model [1, 5, 11, 23], ellipsoidal fitness landscapes [12], the corridor model [22], and the ridge function class =-=[10, 19, 20, 21, 23]-=-. While the sphere and the ellipsoids serve as models for fitness landscapes in the vicinity of local optima, both the corridor and the ridge strive to model features of such landscapes in greater dis... |

3 |
A new approach for predicting the final outcome of evolution strategy optimization under noise. Genetic Programming and Evolvable Machines
- Beyer, Arnold, et al.
- 2005
(Show Context)
Citation Context ...ticular, letting Yi = z (i) 1 and Zi = (ρσz (i) A it follows from Lemma 2 that the expected value of the signed length of the axial component of the progress vector is � E z (avg) � N→∞ c µ/µ,λ 1 = � =-=(12)-=- 1 + ϑ2 + ρ2 +σɛz (i) ɛ )/σ, where ϑ = σɛ/σ denotes the noise-to-signal ratio that the strategy operates under. Similarly, letting Yi = z (i) A and Zi = (σz (i) (i) 1 +σɛz ɛ )/ρσ, it follows from Lemm... |

2 | Expected sample moments of concomitants of selected order statistics
- Arnold, Beyer
- 2002
(Show Context)
Citation Context ...alues of Xi is � µ� � 1 E Y [λ+1−k:λ] µ k=1 = c µ/µ,λ √ 1 + ϑ2 where the progress coefficient c µ/µ,λ was defined above. The derivation of this result is straightforward using the approach pursued in =-=[3, 6]-=-. The quantity ϑ is referred to as the noise-to-signal ratio of the selection process. 2.3. The Quadratic Sphere Model Since the early work of Rechenberg [22], the performance of evolution strategies ... |

2 | The curse of high-dimensional search spaces: Observing premature convergence in unimodal functions
- Salomon
- 2004
(Show Context)
Citation Context ...e often shortsighted and generate step lengths much shorter than optimal. This deficiency on ridges is the cause of the premature convergence of evolution strategies that has been observed by Salomon =-=[25]-=- on a unimodal objective function. Several steps have been made toward a quantitative understanding of the behaviour of evolution strategies on ridge functions. Rechenberg [23] provides a formula for ... |