Abstract
Evolutionary algorithms make countless random decisions during selection, mutation and crossover operations. These random decisions require a steady stream of random numbers.
We analyze the expected number of random bits used throughout a run of an evolutionary algorithm and refer to this as the cost of randomness. We give general bounds on the cost of randomness for mutation-based evolutionary algorithms using 1-bit flips or standard mutations using either a naive or a common, more efficient implementation that uses \(\varTheta (\log n)\) random bits per mutation.
Uniform crossover is a potentially wasteful operator as the number of random bits used equals the Hamming distance of the two parents, which can be up to n. However, we show for a (2+1) Genetic Algorithm that is known to optimize the test function OneMax in roughly \((e/2)n \ln n\) expected evaluations, twice as fast as the fastest mutation-based evolutionary algorithms, that the total cost of randomness during all crossover operations on OneMax is only \(\varTheta (n)\). Consequently, the use of crossover can reduce the cost of randomness below that of the fastest evolutionary algorithms that only use standard mutations.
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Notes
- 1.
We use the term “genetic algorithm” (GA) for EAs that use crossover.
References
Böttcher, S., Doerr, B., Neumann, F.: Optimal fixed and adaptive mutation rates for the LeadingOnes problem. In: Schaefer, R., Cotta, C., Kołodziej, J., Rudolph, G. (eds.) PPSN 2010. LNCS, vol. 6238, pp. 1–10. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-15844-5_1
Bringmann, K., Friedrich, T.: Exact and efficient generation of geometric random variates and random graphs. In: Fomin, F.V., Freivalds, R., Kwiatkowska, M., Peleg, D. (eds.) ICALP 2013. LNCS, vol. 7965, pp. 267–278. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-39206-1_23
Corus, D., Lissovoi, A., Oliveto, P.S., Witt, C.: On steady-state evolutionary algorithms and selective pressure: why inverse rank-based allocation of reproductive trials is best. ACM Trans. Evol. Learn. Optim. 1(1), 2:1-2:38 (2021)
Corus, D., Oliveto, P.S.: Standard steady state genetic algorithms can hill climb faster than mutation-only evolutionary algorithms. IEEE Trans. Evol. Comput. 22(5), 720–732 (2018)
Corus, D., Oliveto, P.S.: On the benefits of populations for the exploitation speed of standard steady-state genetic algorithms. Algorithmica 82(12), 3676–3706 (2020)
Corus, D., Oliveto, P.S., Yazdani, D.: Fast immune system-inspired hypermutation operators for combinatorial optimization. IEEE Trans. Evol. Comput. 25(5), 956–970 (2021)
Dang, D.-C., et al.: Escaping local optima using crossover with emergent diversity. IEEE Trans. Evol. Comput. 22(3), 484–497 (2018)
Doerr, B.: Probabilistic tools for the analysis of randomized optimization heuristics. In: Doerr and Neumann [13], pp. 1–87
Doerr, B., Doerr, C., Ebel, F.: From black-box complexity to designing new genetic algorithms. Theoret. Comput. Sci. 567, 87–104 (2015)
Doerr, B., Fouz, M., Witt, C.: Sharp bounds by probability-generating functions and variable drift. In: Proceedings of the 13th Annual Genetic and Evolutionary Computation Conference (GECCO 2011), pp. 2083–2090. ACM Press (2011)
Doerr, B., Gießen, C., Witt, C., Yang, J.: The (1+\(\lambda \)) evolutionary algorithm with self-adjusting mutation rate. Algorithmica 81(2), 593–631 (2019)
Doerr, B., Phuoc Le, H., Makhmara, R., Nguyen, T.D.: Fast genetic algorithms. In: Proceedings of the Genetic and Evolutionary Computation Conference (GECCO 2017), pp. 777–784. ACM (2017)
Doerr, B., Neumann, F. (eds.): Theory of Evolutionary Computation - Recent Developments in Discrete Optimization. Natural Computing Series, Springer, Cham (2020). https://doi.org/10.1007/978-3-030-29414-4
Doerr, B., Neumann, F.: A survey on recent progress in the theory of evolutionary algorithms for discrete optimization. ACM Trans. Evol. Learn. Optim. 1(4), 1–43 (2021)
Eiben, A.E., Smith, J.E.: Introduction to Evolutionary Computing, 1st edn. Springer, Cham (2015). https://doi.org/10.1007/978-3-662-44874-8
Feller, W.: An Introduction to Probability Theory and Its Applications, vol. 2. Wiley, New York (1971)
Friedrich, T., Kötzing, T., Krejca, M.S., Sutton, A.M.: The benefit of recombination in noisy evolutionary search. In: Elbassioni, K., Makino, K. (eds.) ISAAC 2015. LNCS, vol. 9472, pp. 140–150. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-48971-0_13
Gießen, C., Kötzing, T.: Robustness of populations in stochastic environments. Algorithmica 75(3), 462–489 (2016)
Grefenstette, J.: Efficient implementation of algorithms. In: Handbook of Evolutionary Computation, pp. E2.1:1–E2.1:6. IOP Publishing Ltd., 1st edn. (1997)
Thomas Jansen. Analyzing Evolutionary Algorithms - The Computer Science Perspective. Springer, 2013
Jansen, T., Zarges, C.: Analysis of evolutionary algorithms: from computational complexity analysis to algorithm engineering. In: Proceedings of the 11th Workshop on Foundations of Genetic Algorithms (FOGA 2011), pp. 1–14. ACM (2011)
Jansen, T., Zarges, C.: Analyzing different variants of immune inspired somatic contiguous hypermutations. Theoret. Comput. Sci. 412(6), 517–533 (2011)
Lässig, J., Sudholt, D.: General upper bounds on the running time of parallel evolutionary algorithms. Evol. Comput. 22(3), 405–437 (2014)
Mambrini, A., Sudholt, D.: Design and analysis of schemes for adapting migration intervals in parallel evolutionary algorithms. Evol. Comput. 23(4), 559–582 (2015)
Matsumoto, M., Nishimura, T.: Mersenne twister: a 623-dimensionally equidistributed uniform pseudo-random number generator. ACM Trans. Model. Comput. Simul. 8(1), 3–30 (1998)
Neumann, F., Witt, C.: Bioinspired Computation in Combinatorial Optimization - Algorithms and Their Computational Complexity. NCS, 1st edn. Springer, Cham (2010). https://doi.org/10.1007/978-3-642-16544-3
Nguyen, P.T.H., Sudholt, D.: Memetic algorithms outperform evolutionary algorithms in multimodal optimisation. Artif. Intell. 287, 103345 (2020)
Oliveto, P.S., Sudholt, D., Witt, C.: Tight bounds on the expected runtime of a standard steady state genetic algorithm. Algorithmica 84(6), 1603–1658 (2022)
Qian, C., Bian, C., Yang, Yu., Tang, K., Yao, X.: Analysis of noisy evolutionary optimization when sampling fails. Algorithmica 83(4), 940–975 (2021)
Route, M.: Radio-flaring ultracool dwarf population synthesis. Astrophys. J. 845(1), 66 (2017)
Rudolph, G., Ziegenhirt, J.: Computation time of evolutionary operators. In: Handbook of Evolutionary Computation, pp. E2.2:1–E2.2:4. IOP Publishing Ltd. 1st edn. (1997)
Sudholt, D.: A new method for lower bounds on the running time of evolutionary algorithms. IEEE Trans. Evol. Comput. 17(3), 418–435 (2013)
Sudholt, D.: How crossover speeds up building-block assembly in genetic algorithms. Evol. Comput. 25(2), 237–274 (2017)
Acknowledgment
The authors thank participants of Dagstuhl seminar 22081 “Theory of Randomized Optimization Heuristics” for fruitful discussions.
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Kneissl, C., Sudholt, D. (2023). The Cost of Randomness in Evolutionary Algorithms: Crossover can Save Random Bits. In: Pérez Cáceres, L., Stützle, T. (eds) Evolutionary Computation in Combinatorial Optimization. EvoCOP 2023. Lecture Notes in Computer Science, vol 13987. Springer, Cham. https://doi.org/10.1007/978-3-031-30035-6_12
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