Abstract
A simple success-based step-size adaptation rule for single-parent Evolution Strategies is formulated, and the setting of the corresponding parameters is considered. Theoretical convergence on the class of strictly unimodal functions of one variable that are symmetric around the optimum is investigated using a stochastic Lyapunov function method developed by Semenov and Terkel [5] in the context of martingale theory. General expressions for the conditional expectations of the next values of step size and distance to the optimum under \((1\mathop {,}\limits ^{+}\lambda )\)-selection are analytically derived, and an appropriate Lyapunov function is constructed. Convergence rate upper bounds, as well as adaptation parameter values, are obtained through numerical optimization for increasing values of \(\lambda \). By selecting the number of offspring that minimizes the bound on the convergence rate with respect to the number of function evaluations, all strategy parameter values result from the analysis.
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Notes
- 1.
Equivalently to \(V(x^*)=0\) and \(V(x)>0\) \(\forall \,x \ne x^*\), one may require that \(V(x) \rightarrow -\infty \) only when \(x \rightarrow x^*\).
- 2.
An event holds asymptotically almost surely if it holds with probability \(1- o(1)\), i.e. the probability of success goes to 1 in the limit as \(n \rightarrow \infty \) [14].
- 3.
A stochastic process \(V_t\) is said to be a supermartingale if \(E^{A_t}(V_{t+1}) \le V_t\).
References
Hansen, N., Ostermeier, A.: Completely derandomized self-adaptation in evolution strategies. Evol. Comput. 9(2), 159–195 (2001)
Doerr, B., Doerr, C.: Optimal parameter choices through self-adjustment: applying the \(1/5\)-th rule in discrete settings. In: Proceedings of the 2015 ACM-GECCO Genetic and Evolutionary Computation Conference, pp. 1335–1342. ACM (2015)
Doerr, B., Doerr, C.: A tight runtime analysis of the \((1+(\lambda ,\lambda ))\) genetic algorithm on onemax. In: Proceedings of the 2015 ACM-GECCO Genetic and Evolutionary Computation Conference, pp. 1423–1430. ACM (2015)
Wanner, E.F., Fonseca, C.M., Cardoso, R.T.N., Takahashi, R.H.C.: Lyapunov stability analysis and adaptation law synthesis of a derandomized self-adaptive \((1,2)\)-ES. Under review
Semenov, M.A., Terkel, D.A.: Analysis of convergence of an evolutionary algorithm with self-adaptation using a stochastic Lyapunov function. Evol. Comput. 11(4), 363–379 (2003)
Jägersküpper, J.: A blend of Markov-chain and drift analysis. In: Rudolph, G., Jansen, T., Lucas, S., Poloni, C., Beume, N. (eds.) PPSN 2008. LNCS, vol. 5199, pp. 41–51. Springer, Heidelberg (2008)
He, J., Yao, X.: Drift analysis and average time complexity of evolutionary algorithms. Artif. Intell. 127, 57–85 (2001)
He, J., Yao, X.: A study of drift analysis for estimating computational time of evolutionary algorithms. Natural Comput. 3, 21–35 (2004)
Doerr, B., Johannsen, D., Winzen, C.: Multiplicative drift analysis. Algorithmica 64(4), 673–697 (2011)
Witt, C.: Tight bounds on the optimization time of a randomized search heuristic on linear function. Comb. Probab. Comput. 22(02), 294–318 (2013)
Hart, W.E.: Rethinking the design of real-coded evolutionary algorithms: making discrete choices in continuous search domains. Soft Comput. J. 9, 225–235 (2002)
Lyapunov, A.M.: The general problem of stability of motion (reprint of the original paper of 1892). Int. J. Control 55(3), 531–773 (1992)
Hahn, W.: Stability of Motion. Springer, Heidelberg (1967)
Janson, S., Luczak, T., Rucinski, A.: Random Graphs. Wiley, Hoboken (2000)
Acknowledgment
This work was partially supported by national funds through the Portuguese Foundation for Science and Technology (FCT) and by the European Regional Development Fund (FEDER) through COMPETE 2020 – Operational Program for Competitiveness and Internationalization (POCI). The authors also would like to thank the Brazilian funding agencies, CAPES, CNPq and FAPEMIG.
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Correa, C.R., Wanner, E.F., Fonseca, C.M. (2016). Lyapunov Design of a Simple Step-Size Adaptation Strategy Based on Success. In: Handl, J., Hart, E., Lewis, P., López-Ibáñez, M., Ochoa, G., Paechter, B. (eds) Parallel Problem Solving from Nature – PPSN XIV. PPSN 2016. Lecture Notes in Computer Science(), vol 9921. Springer, Cham. https://doi.org/10.1007/978-3-319-45823-6_10
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