Evolution strategy

Evolution strategy (ES) from computer science is a subclass of evolutionary algorithms, which serves as an optimization technique. ^[1] It uses the major genetic operators mutation, recombination and selection of parents. ^[2]

History

The 'evolution strategy' optimization technique was created in the early 1960s and developed further in the 1970s and later by Ingo Rechenberg, Hans-Paul Schwefel and their co-workers. ^[1]

Timeline of ES - selected algorithms ^[1]

Year	Description	Reference
1973	ES introduced with mutation and selection	[3]
1994	Derandomized self-adaptation ES - Derandomized scheme of mutative step size control is used	[4]
1994	CSA-ES - usage information from the old generations	[5]
2001	CMA-ES	[6]
2006	Weighted multi-recombination ES - usage of weighted recombination	[7]
2007	Meta-ES - incremental aggregation of partial semantic structures	[8]
2008	Natural ES - usage of natural gradient	[9]
2010	Exponential natural ES - a simpler version of natural ES	[10]
2014	Limited memory CMA-ES - time–memory complexity reduction by covariance matrix decomposition	[11]
2016	Fitness inheritance CMA-ES - fitness evaluation computational cost reduction using fitness inheritance	[12]
2017	RS-CMSA ES - usage of subpopulations	[13]
2017	MA-ES - COV update and COV matrix square root are not used	[14]
2018	Weighted ES - weighted recombination of general convex quadratic functions	[15]

Methods

Evolution strategies use natural problem-dependent representations, so problem space and search space are identical. In common with evolutionary algorithms, the operators are applied in a loop. An iteration of the loop is called a generation. The sequence of generations is continued until a termination criterion is met.

The special feature of the ES is the self-adaptation of mutation step sizes and the coevolution associated with it. The ES is briefly presented using the standard form, ^{[16] [17] [18]} pointing out that there are many variants. ^{[19] [20] [21] [22]} The real-valued chromosome contains, in addition to the ${\displaystyle n}$ decision variables, ${\displaystyle n'}$ mutation step sizes ${\displaystyle {\sigma }_{j}}$, where: ${\displaystyle 1\leq j\leq n'\leq n}$. Often one mutation step size is used for all decision variables or each has its own step size. Mate selection to produce ${\displaystyle \lambda }$ offspring is random, i.e. independent of fitness. First, new mutation step sizes are generated per mating by intermediate recombination of the parental ${\displaystyle {\sigma }_{j}}$ with subsequent mutation as follows:

${\displaystyle {\sigma }'_{j}=\sigma _{j}\cdot e^{({\mathcal {N}}(0,1)-{\mathcal {N}}_{j}(0,1))}}$

where ${\displaystyle {\mathcal {N}}(0,1)}$ is a normally distributed random variable with mean ${\displaystyle 0}$ and standard deviation ${\displaystyle 1}$. ${\displaystyle {\mathcal {N}}(0,1)}$ applies to all ${\displaystyle {\sigma }'_{j}}$, while ${\displaystyle {\mathcal {N}}_{j}(0,1)}$ is newly determined for each ${\displaystyle {\sigma }'_{j}}$. Next, discrete recombination of the decision variables is followed by a mutation using the new mutation step sizes as standard deviations of the normal distribution. The new decision variables ${\displaystyle x_{j}'}$ are calculated as follows:

${\displaystyle x_{j}'=x_{j}+{\mathcal {N}}_{j}(0,{\sigma }_{j}')}$

This results in an evolutionary search on two levels: First, at the problem level itself and second, at the mutation step size level. In this way, it can be ensured that the ES searches for its target in ever finer steps. However, there is also the danger of being able to skip larger invalid areas in the search space only with difficulty.

Variants

The ES knows two variants of best selection for the generation of the next parent population (${\displaystyle \mu }$ - number of parents, ${\displaystyle \lambda }$ - number of offspring): ^[2]

• ${\displaystyle (\mu ,\lambda )}$: The ${\displaystyle \mu }$ best offspring are used for the next generation (usually ${\displaystyle \mu ={\frac {\lambda }{2}}}$).
• ${\displaystyle (\mu +\lambda )}$: The best are selected from a union of ${\displaystyle \mu }$ parents and ${\displaystyle \lambda }$ offspring.

Bäck and Schwefel recommend that the value of ${\displaystyle \lambda }$ should be seven times the population size ${\displaystyle \mu }$, ^[17] whereby ${\displaystyle \mu }$ must not be chosen too small because of the strong selection pressure. Suitable values for ${\displaystyle \mu }$ are application-dependent and must be determined experimentally.

Individual step sizes for each coordinate, or correlations between coordinates, which are essentially defined by an underlying covariance matrix, are controlled in practice either by self-adaptation or by covariance matrix adaptation (CMA-ES). ^[21] When the mutation step is drawn from a multivariate normal distribution using an evolving covariance matrix, it has been hypothesized that this adapted matrix approximates the inverse Hessian of the search landscape. This hypothesis has been proven for a static model relying on a quadratic approximation. ^[23]

The selection of the next generation in evolution strategies is deterministic and only based on the fitness rankings, not on the actual fitness values. The resulting algorithm is therefore invariant with respect to monotonic transformations of the objective function. The simplest and oldest ^[1] evolution strategy operates on a population of size two: the current point (parent) and the result of its mutation. Only if the mutant's fitness is at least as good as the parent one, it becomes the parent of the next generation. Otherwise the mutant is disregarded. This is a ${\displaystyle {\mathit {(1+1)}}}$. More generally, ${\displaystyle \lambda }$ mutants can be generated and compete with the parent, called ${\displaystyle (1+\lambda )}$. In ${\displaystyle (1,\lambda )}$ the best mutant becomes the parent of the next generation while the current parent is always disregarded. For some of these variants, proofs of linear convergence (in a stochastic sense) have been derived on unimodal objective functions. ^{[24] [25]}

See also

• Covariance matrix adaptation evolution strategy (CMA-ES)
• Derivative-free optimization
• Evolutionary computation
• Genetic algorithm
• Natural evolution strategy
• Evolutionary game theory

References

1. Slowik, Adam; Kwasnicka, Halina (1 August 2020). "Evolutionary algorithms and their applications to engineering problems". Neural Computing and Applications. 32 (16): 12363–12379. doi:10.1007/s00521-020-04832-8. ISSN   1433-3058.
2. Alrashdi, Zaid; Sayyafzadeh, Mohammad (1 June 2019). "(μ+λ) Evolution strategy algorithm in well placement, trajectory, control and joint optimisation". Journal of Petroleum Science and Engineering. 177: 1042–1058. doi:10.1016/j.petrol.2019.02.047. ISSN   0920-4105.
3. Vent, W. (January 1975). "Rechenberg, Ingo, Evolutionsstrategie — Optimierung technischer Systeme nach Prinzipien der biologischen Evolution. 170 S. mit 36 Abb. Frommann-Holzboog-Verlag. Stuttgart 1973. Broschiert". Feddes Repertorium. 86 (5): 337. doi:10.1002/fedr.19750860506.
4. Ostermeier, Andreas; Gawelczyk, Andreas; Hansen, Nikolaus (December 1994). "A Derandomized Approach to Self-Adaptation of Evolution Strategies". Evolutionary Computation. 2 (4): 369–380. doi:10.1162/evco.1994.2.4.369.
5. Ostermeier, Andreas; Gawelczyk, Andreas; Hansen, Nikolaus (1994). "Step-size adaptation based on non-local use of selection information". Parallel Problem Solving from Nature — PPSN III. Lecture Notes in Computer Science. Vol. 866. Springer. pp. 189–198. doi:10.1007/3-540-58484-6_263. ISBN   978-3-540-58484-1.
6. Hansen, Nikolaus; Ostermeier, Andreas (June 2001). "Completely Derandomized Self-Adaptation in Evolution Strategies". Evolutionary Computation. 9 (2): 159–195. doi:10.1162/106365601750190398. PMID   11382355.
7. Arnold, Dirk V. (28 August 2006). "Weighted multirecombination evolution strategies". Theoretical Computer Science. 361 (1): 18–37. doi:10.1016/j.tcs.2006.04.003. ISSN   0304-3975.
8. Jung, Jason J.; Jo, Geun-Sik; Yeo, Seong-Won (2007). "Meta-evolution Strategy to Focused Crawling on Semantic Web". Artificial Neural Networks – ICANN 2007. Lecture Notes in Computer Science. Vol. 4669. Springer. pp. 399–407. doi:10.1007/978-3-540-74695-9_41. ISBN   978-3-540-74693-5.
9. Wierstra, Daan; Schaul, Tom; Glasmachers, Tobias; Sun, Yi; Peters, Jan; Schmidhuber, Jürgen (1 January 2014). "Natural evolution strategies". J. Mach. Learn. Res. 15 (1): 949–980. ISSN   1532-4435.
10. Glasmachers, Tobias; Schaul, Tom; Yi, Sun; Wierstra, Daan; Schmidhuber, Jürgen (7 July 2010). "Exponential natural evolution strategies". Proceedings of the 12th annual conference on Genetic and evolutionary computation (PDF). Association for Computing Machinery. pp. 393–400. doi:10.1145/1830483.1830557. ISBN   978-1-4503-0072-8.
11. Loshchilov, Ilya (12 July 2014). "A computationally efficient limited memory CMA-ES for large scale optimization". Proceedings of the 2014 Annual Conference on Genetic and Evolutionary Computation. Association for Computing Machinery. pp. 397–404. arXiv: 1404.5520 . doi:10.1145/2576768.2598294. ISBN   978-1-4503-2662-9.
12. Liaw, Rung-Tzuo; Ting, Chuan-Kang (July 2016). "Enhancing covariance matrix adaptation evolution strategy through fitness inheritance". 2016 IEEE Congress on Evolutionary Computation (CEC): 1956–1963. doi:10.1109/CEC.2016.7744027. ISBN   978-1-5090-0623-6.
13. Ahrari, Ali; Deb, Kalyanmoy; Preuss, Mike (September 2017). "Multimodal Optimization by Covariance Matrix Self-Adaptation Evolution Strategy with Repelling Subpopulations". Evolutionary Computation. 25 (3): 439–471. doi:10.1162/evco_a_00182. hdl:1887/76707. PMID   27070282.
14. Beyer, Hans-Georg; Sendhoff, Bernhard (October 2017). "Simplify Your Covariance Matrix Adaptation Evolution Strategy". IEEE Transactions on Evolutionary Computation. 21 (5): 746–759. doi:10.1109/TEVC.2017.2680320.
15. Akimoto, Youhei; Auger, Anne; Hansen, Nikolaus (6 September 2020). "Quality gain analysis of the weighted recombination evolution strategy on general convex quadratic functions". Theoretical Computer Science. 832: 42–67. doi:10.1016/j.tcs.2018.05.015. ISSN   0304-3975.
16. Schwefel, Hans-Paul (1995). Evolution and Optimum Seeking. Sixth-generation computer technology series. New York: Wiley. ISBN   978-0-471-57148-3.
17. Bäck, Thomas; Schwefel, Hans-Paul (1993). "An Overview of Evolutionary Algorithms for Parameter Optimization". Evolutionary Computation. 1 (1): 1–23. doi:10.1162/evco.1993.1.1.1. ISSN   1063-6560.
18. Schwefel, Hans-Paul; Rudolph, Günter; Bäck, Thomas (1995), Morán, Frederico; Moreno, Alvaro; Merelo, J.J.; Chacón, Pablo (eds.), "Contemporary Evolution Strategies", Proceedings of the Third European Conference on Advances in Artificial Life, Berlin, Heidelberg: Springer, pp. 893–907, doi:10.1007/3-540-59496-5_351, ISBN   978-3-540-59496-3
19. Bäck, Thomas; Hoffmeister, Frank; Schwefel, Hans-Paul (1991), Belew, Richard K.; Booker, Lashon B. (eds.), "A Survey of Evolution Strategies", Proceedings of the Fourth International Conference on Genetic Algorithms (ICGA), San Mateo, Calif: Morgan Kaufmann, ISBN   978-1-55860-208-3
20. Coelho, V. N.; Coelho, I. M.; Souza, M. J. F.; Oliveira, T. A.; Cota, L. P.; Haddad, M. N.; Mladenovic, N.; Silva, R. C. P.; Guimarães, F. G. (2016). "Hybrid Self-Adaptive Evolution Strategies Guided by Neighborhood Structures for Combinatorial Optimization Problems". Evolutionary Computation. 24 (4): 637–666. doi:10.1162/EVCO_a_00187. ISSN   1063-6560.
21. Hansen, Nikolaus; Ostermeier, Andreas (2001). "Completely Derandomized Self-Adaptation in Evolution Strategies". Evolutionary Computation. 9 (2): 159–195. doi:10.1162/106365601750190398. ISSN   1063-6560.
22. Hansen, Nikolaus; Kern, Stefan (2004), Yao, Xin; Burke, Edmund K.; Lozano, José A.; Smith, Jim (eds.), "Evaluating the CMA Evolution Strategy on Multimodal Test Functions", Parallel Problem Solving from Nature - PPSN VIII, vol. 3242, Berlin, Heidelberg: Springer, pp. 282–291, doi:10.1007/978-3-540-30217-9_29, ISBN   978-3-540-23092-2
23. Shir, Ofer M.; Yehudayoff, Amir (January 2020). "On the covariance-Hessian relation in evolution strategies". Theoretical Computer Science. 801: 157–174. doi:10.1016/j.tcs.2019.09.002.
24. Auger, Anne (April 2005). "Convergence results for the ( 1 , λ ) -SA-ES using the theory of ϕ -irreducible Markov chains". Theoretical Computer Science. 334 (1–3): 35–69. doi:10.1016/j.tcs.2004.11.017.
25. Jägersküpper, Jens (August 2006). "How the (1+1) ES using isotropic mutations minimizes positive definite quadratic forms". Theoretical Computer Science. 361 (1): 38–56. doi:10.1016/j.tcs.2006.04.004.

Bibliography

• Ingo Rechenberg (1971): Evolutionsstrategie – Optimierung technischer Systeme nach Prinzipien der biologischen Evolution (PhD thesis). Reprinted by Frommann-Holzboog (1973). ISBN   3-7728-1642-8
• Hans-Paul Schwefel (1974): Numerische Optimierung von Computer-Modellen (PhD thesis). Reprinted by Birkhäuser (1977).
• Hans-Paul Schwefel: Evolution and Optimum Seeking . New York: Wiley & Sons 1995. ISBN   0-471-57148-2
• H.-G. Beyer and H.-P. Schwefel. Evolution Strategies: A Comprehensive Introduction . Journal Natural Computing, 1(1):3–52, 2002.
• Hans-Georg Beyer: The Theory of Evolution Strategies. Springer, April 27, 2001. ISBN   3-540-67297-4
• Ingo Rechenberg: Evolutionsstrategie '94. Stuttgart: Frommann-Holzboog 1994. ISBN   3-7728-1642-8
• J. Klockgether and H. P. Schwefel (1970). Two-Phase Nozzle And Hollow Core Jet Experiments. AEG-Forschungsinstitut. MDH Staustrahlrohr Project Group. Berlin, Federal Republic of Germany. Proceedings of the 11th Symposium on Engineering Aspects of Magneto-Hydrodynamics, Caltech, Pasadena, Cal., 24.–26.3. 1970.
• M. Emmerich, O.M. Shir, and H. Wang: Evolution Strategies. In: Handbook of Heuristics, 1-31. Springer International Publishing (2018).

Research centers

• Bionics & Evolutiontechnique at Technische Universität Berlin
• Chair of Algorithm Engineering (Ls11) – TU Dortmund University
• Collaborative Research Center 531 – TU Dortmund University