We study the convergence rate of randomly truncated stochastic algorithms, which consist in the truncation of the standard Robbins-Monro procedure on an increasing sequence of compact sets. Such a truncation is often required in practice to ensure convergence when standard algorithms fail because the expected-value function grows too fast. In this work, we give a self contained proof of a central limit theorem for this algorithm under local assumptions on the expected-value function, which are fairly easy to check in practice.
Keywords: stochastic approximation, central limit theorem, randomly truncated stochastic algorithms, martingale arrays
@article{PS_2013__17__105_0,
author = {Lelong, J\'er\^ome},
title = {Asymptotic normality of randomly truncated stochastic algorithms},
journal = {ESAIM: Probability and Statistics},
pages = {105--119},
year = {2013},
publisher = {EDP Sciences},
volume = {17},
doi = {10.1051/ps/2011110},
mrnumber = {3021311},
language = {en},
url = {https://www.numdam.org/articles/10.1051/ps/2011110/}
}
TY - JOUR AU - Lelong, Jérôme TI - Asymptotic normality of randomly truncated stochastic algorithms JO - ESAIM: Probability and Statistics PY - 2013 SP - 105 EP - 119 VL - 17 PB - EDP Sciences UR - https://www.numdam.org/articles/10.1051/ps/2011110/ DO - 10.1051/ps/2011110 LA - en ID - PS_2013__17__105_0 ER -
Lelong, Jérôme. Asymptotic normality of randomly truncated stochastic algorithms. ESAIM: Probability and Statistics, Tome 17 (2013), pp. 105-119. doi: 10.1051/ps/2011110
[1] , Adaptative Monte Carlo method, a variance reduction technique. Monte Carlo Methods Appl. 10 (2004) 1-24. | Zbl | MR
[2] , and , Adaptive algorithms and stochastic approximations. Springer-Verlag, Berlin. Appl. Math. 22 (1990). Translated from the French by Stephen S. Wilson. | Zbl | MR
[3] , Approximation Gaussienne d'algorithmes stochastiques à dynamique Markovienne. Ph.D. Thesis, Université Pierre et Marie Curie, Paris 6 (1985). | Zbl
[4] and , Rate of convergence for constrained stochastic approximation algorithms. SIAM J. Control Optim. 40 (2001) 1011-1041 (electronic). | Zbl | MR
[5] , Stochastic approximation and its applications, Kluwer Academic Publishers, Dordrecht. Nonconvex Optim. Appl. 64 (2002). | Zbl | MR
[6] and , Stochastic Approximation Procedure with randomly varying truncations. Scientia Sinica Series (1986). | Zbl | MR
[7] , General results on the convergence of stochastic algorithms. IEEE Trans. Automat. Contr. 41 (1996) 1245-1255. | Zbl | MR
[8] , Algorithmes stochastiques (Mathématiques et Applications). Springer (1996). | Zbl | MR
[9] , Random Iterative Models. Springer-Verlag Berlin and New York (1997). | Zbl | MR
[10] and , Stochastic approximation and recursive algorithms and applications, Applications of Mathematics. Springer-Verlag, New York, 2nd edition 2003. Stoch. Model. Appl. Probab. 35 (2003). | Zbl | MR
[11] and , A framework for adaptive Monte-Carlo procedures. Monte Carlo Methods Appl. (2011). | Zbl
[12] , Almost sure convergence of randomly truncated stochastic agorithms under verifiable conditions. Stat. Probab. Lett. 78 (2009). | Zbl | MR
[13] and , Unconstrained Recursive Importance Sampling. Ann. Appl. Probab. 20 (2010) 1029-1067. | Zbl | MR
[14] , Weak convergence rates for stochastic approximation with application to multiple targets and simulated annealing. Ann. Appl. Probab. 8 (1998) 10-44. | Zbl | MR
[15] and , A stochastic approximation method. Ann. Math. Statistics 22 (1951) 400-407. | Zbl | MR
Cité par Sources :
