## Completely Derandomized Self-Adaptation in Evolution Strategies (2001)

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Venue: | Evolutionary Computation |

Citations: | 330 - 36 self |

### BibTeX

@ARTICLE{Hansen01completelyderandomized,

author = {Nikolaus Hansen and Andreas Ostermeier},

title = {Completely Derandomized Self-Adaptation in Evolution Strategies},

journal = {Evolutionary Computation},

year = {2001},

volume = {9},

pages = {159--195}

}

### Years of Citing Articles

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### Abstract

This paper puts forward two useful methods for self-adaptation of the mutation distribution -- the concepts of derandomization and cumulation. Principle shortcomings of the concept of mutative strategy parameter control and two levels of derandomization are reviewed. Basic demands on the self-adaptation of arbitrary (normal) mutation distributions are developed. Applying arbitrary, normal mutation distributions is equivalent to applying a general, linear problem encoding.

### Citations

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Citation Context ...of strategy parameters is available. With respect to the mutation distribution, there is typically only a small width of strategy parameter settings where substantial search progress can be observed (=-=Rechenberg, 1973-=-). Good parameter settings differ remarkably from problem to problem. Even worse, they usually change during the search process (possibly by several orders of magnitude). For this reason, self-adaptat... |

510 | Numerical Optimization of Computer Models - Schwefel - 1981 |

209 | Evolutionary computation: Comments on the history and current states
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Citation Context ...erate any n-dimensional normal distribution with zero mean (Rudolph, 1992). It is generally recognized that the typical values for = 15 and = 100 are not sufficient for this adaptation mechanism (Ba=-=ck et al., 19-=-97). Due to the mutative approach, the parent number has presumably to scale with n 2 . Choosing n 2 =2, which is roughly the number of free strategy parameters, performance on the sphere problem de... |

155 | Adapting arbitrary normal mutation distributions in evolutionstrategies: the covariance matrix adaptation
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Citation Context ...ansen, 1999). To satisfy the performance demand the concept of MSC has to be modified. 3.2 A Completely Derandomized Approach: The Covariance Matrix Adaptation (CMA) The covariance matrix adaptation (=-=Hansen and Ostermeier, 1996-=-) is a second-level (i.e., completely) derandomized self-adaptation scheme. First, it directly implements the 10 Evolutionary Computation Volume 9, Number 2 Derandomized Self-Adaptation aim of MSC to ... |

108 |
Evolution and Optimum Seeking. Sixth-Generation Computer Technology
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Citation Context ...y) on the mutation strength on the strategy parameter level. Based on this observation, considerable theoretical efforts were made to calculate the optimal mutation strength for the global step size (=-=Schwefel, 1995-=-; Beyer, 1996b). But, in general, the conflict between an optimal change rate versus a significant selection difference (see above) cannot be resolved by choosing an ambiguous compromise for the mutat... |

104 | Numerical Recipes in C: The Art of Scientific Computing (Second edition
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Citation Context ...he principal axes of the mutation distribution ellipsoid (compare Figure 3). Storage requirements are O(n 2 ). We note, that "generally, for any moderate n, this is an entirely trivial disadvanta=-=ge" (Press et al., 1992-=-). For computational and numerical requirements, refer to Sections 5.2 and 7.1. 4 Utilizing the Evolution Path: Cumulation The concept of MSC utilizes selection information of a single generation step... |

77 | Toward a Theory of Evolution Strategies: On the Benefits of Sex- the (μ/μ,λ
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Citation Context ...use it yields comparable values independent of H, b, c, n, and k. For H = I and large n, this measure yields values close to the common normalized progress measure ' ? := n r (g) (r (g) E[r (g+1) ]) (=-=Beyer, 1995-=-), where r is the distance to the optimum. 10 Not taking into account weighted recombination, the theoretically optimal (1 + 1)-ES yields approximatelys0:20. 8 Evolutionary Computation Volume 9, Numbe... |

66 | Toward a theory of evolution strategies: Self–adaptation
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(Show Context)
Citation Context ...on strength on the strategy parameter level. Based on this observation, considerable theoretical efforts were made to calculate the optimal mutation strength for the global step size (Schwefel, 1995; =-=Beyer, 1996-=-b). But, in general, the conflict between an optimal change rate versus a significant selection difference (see above) cannot be resolved by choosing an ambiguous compromise for the mutation strength ... |

50 | Convergence Properties of Evolution Strategies with the Derandomized Covariance Matrix Adaptation: The (µ/µI,λ)-CMA-ES
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Citation Context ... n. In the ( I ; )-CMA-ES, in most cases, 0:27 will suffice (Beyer, 1996a; Herdy, 1993). To provide a robust strategy, large and if need be a larger ratio of = up to 0:5 are preferable (Hansen and=-= Ostermeier, 1-=-997). In particular for n 5, even = 1 can occasionally be the best choice. 17 The algorithm is independent from multiplication of w = (w 1 ; : : : ; w ) with a real number greater than zero. Evoluti... |

47 | N.: Step-size adaptation based on nonlocal use of selection information
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Citation Context ... between an optimal change rate versus a significant selection difference (see above) cannot be resolved by choosing an ambiguous compromise for the mutation strength on the strategy parameter level (=-=Ostermeier et al., 1994-=-a). The mutation strength that achieves an optimal change rate is usually smaller than the mutation strength that achieves a suitable selection difference. The discrepancy increases with increasing pr... |

42 | A Derandomized Approach to Self-Adaptation of Evolution Strategies
- Ostermeier, Gawelczyk, et al.
(Show Context)
Citation Context ... between an optimal change rate versus a significant selection difference (see above) cannot be resolved by choosing an ambiguous compromise for the mutation strength on the strategy parameter level (=-=Ostermeier et al., 1994-=-a). The mutation strength that achieves an optimal change rate is usually smaller than the mutation strength that achieves a suitable selection difference. The discrepancy increases with increasing pr... |

32 |
Verallgemeinerte individuelle Schrittweitenregelung in der Evolutionsstrategie
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(Show Context)
Citation Context ...n distribution with the distribution of the selected mutation steps. If both distributions become alike, as under random selection, in expectation, no further change of the distributions takes place (=-=Hansen, 199-=-8). This illustrative but also formally precise description of the CMA differs in three points from the CMA-ES formalized in Section 5. These extensions are as follows: Apart from the adaptation of t... |

25 | On correlated mutations in evolution strategies
- Rudolph
(Show Context)
Citation Context ...imensional subspaces spanned by canonical unit vectors. This complete rotation matrix is denoted with R(:). This algorithm allows one to generate any n-dimensional normal distribution with zero mean (=-=Rudolph, 19-=-92). It is generally recognized that the typical values for = 15 and = 100 are not sufficient for this adaptation mechanism (Back et al., 1997). Due to the mutative approach, the parent number has p... |

11 |
Optimisation of a stator blade used in a transonic compressor cascade with evolution strategies
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(Show Context)
Citation Context ...unctions. In addition, there are reported successful applications of the CMA-ES to real-world search problems (Alvers, 1998; Holste, 1998; Meyer, 1998; Lutz and Wagner, 1998a; Lutz and Wagner, 1998b; =-=Olhofer et al., 2000-=-; Bergener et al., 2001; Cerveri et al., 2001; Igel and von Seelen, 2001; Igel et al., 2001). Consequently, comparable to quasi-Newton methods, we expect this algorithm, or at least some quite similar... |

10 |
An evolution strategy with momentum adaptation of the random number distribution,” in Parallel Problem Solving from Nature 2
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(Show Context)
Citation Context ... compared to an ES with derandomized strategy parameter control of the complete covariance matrix, the advantage from extrapolation seems small. Correspondingly, we find algorithms with nonzero mean (=-=Ostermeier, 1992-=-; Ghozeil and Fogel, 1996; Hildebrand et al., 1999) unpromising. In our opinion, any adaptation mechanism that adapts a general linear encoding /decoding has to meet the following fundamental demands:... |

9 | On the asymptotic behavior of multirecombinant evolution strategies
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(Show Context)
Citation Context ...on strength on the strategy parameter level. Based on this observation, considerable theoretical efforts were made to calculate the optimal mutation strength for the global step size (Schwefel, 1995; =-=Beyer, 1996-=-b). But, in general, the conflict between an optimal change rate versus a significant selection difference (see above) cannot be resolved by choosing an ambiguous compromise for the mutation strength ... |

8 | Self-Adaptation and Correlated Mutations in Evolution Strategies
- Hansen, “Invariance
- 2000
(Show Context)
Citation Context ...ecause rotation planes are chosen with respect to the given coordinate system. In simulations, we found the algorithm to be dependent even on permutations of the coordinate axes (Hansen et al., 1995; =-=Hansen, 2000-=-)! Furthermore, its performance depends highly on the orientation of the given coordinate system (compare Section 7.2). The invariance demand is not met. Taking into account these difficulties, it is ... |

6 | On the desired behaviors of self-adaptive evolutionary algorithms - Beyer, Deb - 2000 |

6 |
A preliminary investigation into directed mutations in evolutionary algorithms
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Citation Context ... with derandomized strategy parameter control of the complete covariance matrix, the advantage from extrapolation seems small. Correspondingly, we find algorithms with nonzero mean (Ostermeier, 1992; =-=Ghozeil and Fogel, 1996-=-; Hildebrand et al., 1999) unpromising. In our opinion, any adaptation mechanism that adapts a general linear encoding /decoding has to meet the following fundamental demands: Adaptation: The adaptati... |

6 |
Schrittweitenadaptation in der Evolutionsstrategie mit einem entstochastisierten Ansatz
- Ostermeier
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(Show Context)
Citation Context ...Equations (3) and (5)). b n = E[kN(0; I)k] = p 2 n+1 2 = n 2 , expectation of the length of a (0; I)- normally distributed random vector. A good approximation is b n p n 1 1 4n + 1 21n 2 (Osterme=-=ier, 1997-=-). Apart from omitting the transformation with D (g) , Equations (16) and (14) are identical: In (16) we use B (g) hzi (g+1) w for the cumulation, instead of B (g) D (g) hzi (g+1) w . Under random sel... |

5 | Optimization of neural field models
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Citation Context ...search problems (Alvers, 1998; Holste, 1998; Meyer, 1998; Lutz and Wagner, 1998a; Lutz and Wagner, 1998b; Olhofer et al., 2000; Bergener et al., 2001; Cerveri et al., 2001; Igel and von Seelen, 2001; =-=Igel et al., 2001-=-). Consequently, comparable to quasi-Newton methods, we expect this algorithm, or at least some quite similar method, based on its superior performance to become state-of-the-art for the application o... |

5 |
Drag Reduction and Shape Optimization of Airship Bodies
- Lutz, Wagner
(Show Context)
Citation Context ... inverse Hessian matrix of different objective functions. In addition, there are reported successful applications of the CMA-ES to real-world search problems (Alvers, 1998; Holste, 1998; Meyer, 1998; =-=Lutz and Wagner, 1998-=-a; Lutz and Wagner, 1998b; Olhofer et al., 2000; Bergener et al., 2001; Cerveri et al., 2001; Igel and von Seelen, 2001; Igel et al., 2001). Consequently, comparable to quasi-Newton methods, we expect... |

3 |
Directed mutation—a new self adaptation for evolution strategies
- Hildebrand, Reusch, et al.
- 1999
(Show Context)
Citation Context ...gy parameter control of the complete covariance matrix, the advantage from extrapolation seems small. Correspondingly, we find algorithms with nonzero mean (Ostermeier, 1992; Ghozeil and Fogel, 1996; =-=Hildebrand et al., 1999-=-) unpromising. In our opinion, any adaptation mechanism that adapts a general linear encoding /decoding has to meet the following fundamental demands: Adaptation: The adaptation must be successful in ... |

3 |
Analyse der Adaptation von Verteilungsparametern in der Evolutionsstrategie. Diploma thesis, Fachgebiet Bionik und Evolutionstechnik der Technischen Universität
- Holzheuer
- 1996
(Show Context)
Citation Context ... that the adaptation demand is not met. Progress rates on convex-quadratic functions with high axis ratios can be several orders of magnitude lower than progress rates achieved on the sphere problem (=-=Holzheuer, 1996-=-; Hansen, 2000, and Section 7.2). When the typical intermediate recombination is applied to the step sizes, they increase unbounded under random selection. The systematic drift is slow, usually causes... |

3 | An evolution strategy with coordinate system invariant adaptation of arbitrary normal mutation distributions within the concept of mutative strategy parameter control
- Ostermeier, Hansen
- 1999
(Show Context)
Citation Context ... the stationarity demand to be met. Using another parameterization together with a suitable mutation operator can solve the demands for adaptation and invariance without giving up the concept of MSC (=-=Ostermeier and Hansen, 1999-=-). To satisfy the performance demand the concept of MSC has to be modified. 3.2 A Completely Derandomized Approach: The Covariance Matrix Adaptation (CMA) The covariance matrix adaptation (Hansen and ... |

2 | Parameter optimization for visual obstacle detection using a derandomized evolution strategy
- Bergener, Bruckhoff, et al.
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(Show Context)
Citation Context ... there are reported successful applications of the CMA-ES to real-world search problems (Alvers, 1998; Holste, 1998; Meyer, 1998; Lutz and Wagner, 1998a; Lutz and Wagner, 1998b; Olhofer et al., 2000; =-=Bergener et al., 2001-=-; Cerveri et al., 2001; Igel and von Seelen, 2001; Igel et al., 2001). Consequently, comparable to quasi-Newton methods, we expect this algorithm, or at least some quite similar method, based on its s... |

2 | Design of a field model for early vision: A case study of evolutionary algorithms in neuroscience - Igel, Seelen - 2001 |

2 |
Parameteroptimierung dynamischer Systeme mit der Evolutionsstrategie. Diploma thesis, Fachgebiet Bionik und Evolutionstechnik der Technischen Universität
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(Show Context)
Citation Context ...roximates the inverse Hessian matrix of different objective functions. In addition, there are reported successful applications of the CMA-ES to real-world search problems (Alvers, 1998; Holste, 1998; =-=Meyer, 1998-=-; Lutz and Wagner, 1998a; Lutz and Wagner, 1998b; Olhofer et al., 2000; Bergener et al., 2001; Cerveri et al., 2001; Igel and von Seelen, 2001; Igel et al., 2001). Consequently, comparable to quasi-Ne... |

1 |
Zur Anwendung von Optimierungsstrategien auf Potentialfeldmodelle. Berliner geowissenschaftliche Abhandlungen, Reihe B: Geophysik. Selbstverlag Fachbereich Geowissenschaften, Freie Universität
- Alvers
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(Show Context)
Citation Context ...ons, the CMA-ES reliably approximates the inverse Hessian matrix of different objective functions. In addition, there are reported successful applications of the CMA-ES to real-world search problems (=-=Alvers, 1998-=-; Holste, 1998; Meyer, 1998; Lutz and Wagner, 1998a; Lutz and Wagner, 1998b; Olhofer et al., 2000; Bergener et al., 2001; Cerveri et al., 2001; Igel and von Seelen, 2001; Igel et al., 2001). Consequen... |

1 |
34 Evolutionary Computation Volume 9, Number 2 Self-Adaptation Bäck
- T, Schwefel
- 1993
(Show Context)
Citation Context ... = x (g) + (g+1) k z k ; (2) where: k 2 R, for k = 1; : : : ; independent realizations of a random number with zero mean. Typically, k is normally distributed with standard deviation 1= p 2n (Back=-= and Schwefel, -=-1993). We usually prefer to choose P( k = 0:3) = P( k = 0:3) = 1=2 (Rechenberg, 1994). z k N(0; I) 2 R n , for k = 1; : : : ; independent realizations of a (0; I)-normally distributed random vector,... |

1 | Mutate large, but inherit small - Beyer - 1998 |

1 |
Enhanced evolution strategies: A novel approach to stereo-camera calibration
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(Show Context)
Citation Context ...cessful applications of the CMA-ES to real-world search problems (Alvers, 1998; Holste, 1998; Meyer, 1998; Lutz and Wagner, 1998a; Lutz and Wagner, 1998b; Olhofer et al., 2000; Bergener et al., 2001; =-=Cerveri et al., 2001-=-; Igel and von Seelen, 2001; Igel et al., 2001). Consequently, comparable to quasi-Newton methods, we expect this algorithm, or at least some quite similar method, based on its superior performance to... |

1 | Evolutionary Computation Volume 9, Number 2 35 Hansen and - Herdy, M - 1993 |

1 |
Modellkalibrierung am Beispiel von Kläranlagenmodellen
- Holste
- 1998
(Show Context)
Citation Context ...S reliably approximates the inverse Hessian matrix of different objective functions. In addition, there are reported successful applications of the CMA-ES to real-world search problems (Alvers, 1998; =-=Holste, 1998-=-; Meyer, 1998; Lutz and Wagner, 1998a; Lutz and Wagner, 1998b; Olhofer et al., 2000; Bergener et al., 2001; Cerveri et al., 2001; Igel and von Seelen, 2001; Igel et al., 2001). Consequently, comparabl... |

1 |
Numerical shape optimization of natural laminar flow bodies
- Lutz, Wagner
- 1998
(Show Context)
Citation Context ... inverse Hessian matrix of different objective functions. In addition, there are reported successful applications of the CMA-ES to real-world search problems (Alvers, 1998; Holste, 1998; Meyer, 1998; =-=Lutz and Wagner, 1998-=-a; Lutz and Wagner, 1998b; Olhofer et al., 2000; Bergener et al., 2001; Cerveri et al., 2001; Igel and von Seelen, 2001; Igel et al., 2001). Consequently, comparable to quasi-Newton methods, we expect... |