Log-linear Convergence and Optimal Bounds for the $(1+1)$-ES

Mohamed Jebalia 1 Anne Auger 1 Pierre Liardet 2
1 TANC - Algorithmic number theory for cryptology
LIX - Laboratoire d'informatique de l'École polytechnique [Palaiseau], Inria Saclay - Ile de France
Abstract : The $(1+1)$-ES is modeled by a general stochastic process whose asymptotic behavior is investigated. Under general assumptions, it is shown that the convergence of the related algorithm is sub-log-linear, bounded below by an explicit log-linear rate. For the specific case of spherical functions and scale-invariant algorithm, it is proved using the Law of Large Numbers for orthogonal variables, that the linear convergence holds almost surely and that the best convergence rate is reached. Experimental simulations illustrate the theoretical results.
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https://hal.inria.fr/inria-00173483
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Mohamed Jebalia, Anne Auger, Pierre Liardet. Log-linear Convergence and Optimal Bounds for the $(1+1)$-ES. Evolution Artificielle, Oct 2007, Tours, France. pp.207-218, ⟨10.1007/978-3-540-79305-2⟩. ⟨inria-00173483v4⟩

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