On the adaptation of noise level for stochastic optimization

This paper deals with the optimization of noisy fitness functions, where the noise level can be reduced by increasing the computational effort. We theoretically investigate the question of the control of the noise level. We analyse two different schemes for an adaptive control and prove sufficient conditions ensuring the existence of an homogeneous Markov chain, which is the first step to prove linear convergence when dealing with non-noisy fitness functions. We experimentally validate the relevance of the homogeneity criterion. Large-scale experiments conclude to the efficiency in a difficult framework.

[1]  Petros Koumoutsakos,et al.  Learning probability distributions in continuous evolutionary algorithms – a comparative review , 2004, Natural Computing.

[2]  Anne Auger,et al.  Reconsidering the progress rate theory for evolution strategies in finite dimensions , 2006, GECCO '06.

[3]  K. Marti.  Stochastic Optimization Methods , 2005 .

[4]  E. Braaten,et al.  An Improved Low-Discrepancy Sequence for Multidimensional Quasi-Monte Carlo Integration , 1979 .

[5]  J. Dennis,et al.  MANAGING APPROXIMATION MODELS IN OPTIMIZATION , 2007 .

[6]  B. Tuffin.  A new permutation choice in Halton sequences , 1998 .

[7]  Jati Kumar Sengupta,et al.  Stochastic programming: Methods and applications , 1972 .

[8]  Hongmei Chi,et al.  On the Scrambled Halton Sequence , 2004, Monte Carlo Methods Appl..

[9]  Russel E. Caflisch,et al.  Quasi-Random Sequences and Their Discrepancies , 1994, SIAM J. Sci. Comput..

[10]  Fred J. Hickernell,et al.  Randomized Halton sequences , 2000 .

[11]  H. Faure.  Good permutations for extreme discrepancy , 1992 .

[12]  Olivier François,et al.  Global convergence for evolution strategies in spherical problems: some simple proofs and difficulties , 2003, Theor. Comput. Sci..

[13]  Richard L. Tweedie,et al.  Markov Chains and Stochastic Stability , 1993, Communications and Control Engineering Series.

[14]  Brian T. Denton,et al.  Review of Stochastic optimization: Algorithms and applications by Stanislav Uryasev and Panos M. Pardalos, Kluwer Academic Publishers 2001 , 2003 .

[15]  S. Uryasev,et al.  Stochastic optimization : Algorithms and Applications , 2001 .

[16]  Ashok Srinivasan,et al.  Parallel Quasi-Monte Carlo Methods on a Heterogeneous Cluster , 2002 .

[17]  Olivier Teytaud,et al.  General Lower Bounds for Evolutionary Algorithms , 2006, PPSN.

[18]  Tony Warnock,et al.  Computational investigations of low-discrepancy point-sets. , 1972 .

[19]  S. Volkwein,et al.  Reduced order output feedback control design for PDE systems using proper orthogonal decomposition and nonlinear semidefinite programming , 2006 .

[20]  T. Warnock.  Computational Investigations of Low-Discrepancy Point Sets II , 1995 .

[21]  Hans-Georg Beyer,et al.  Local performance of the (1 + 1)-ES in a noisy environment , 2002, IEEE Trans. Evol. Comput..

[22]  Anne Auger,et al.  Convergence results for the (1, lambda)-SA-ES using the theory of phi-irreducible Markov chains , 2005, Theor. Comput. Sci..

[23]  Nikolaus Hansen,et al.  Completely Derandomized Self-Adaptation in Evolution Strategies , 2001, Evolutionary Computation.

[24]  R. Cranley,et al.  Randomization of Number Theoretic Methods for Multiple Integration , 1976 .

[25]  A. Auger.  Convergence results for the ( 1 , )-SA-ES using the theory of-irreducible Markov chains , 2005 .

[26]  Ashok Srinivasan.  Parallel and distributed computing issues in pricing financial derivatives through quasi Monte Carlo , 2002, Proceedings 16th International Parallel and Distributed Processing Symposium.

[27]  R. Cools,et al.  Good permutations for deterministic scrambled Halton sequences in terms of L2-discrepancy , 2006 .

[28]  R. Wets,et al.  Stochastic programming , 1989 .

[29]  E. Gobet,et al.  Stochastic Linear Programming , 2022 .