The Annals of Applied Probability

Multi-level stochastic approximation algorithms

Noufel Frikha

Full-text: Open access

Abstract

This paper studies multi-level stochastic approximation algorithms. Our aim is to extend the scope of the multi-level Monte Carlo method recently introduced by Giles [Oper. Res. 56 (2008) 607–617] to the framework of stochastic optimization by means of stochastic approximation algorithm. We first introduce and study a two-level method, also referred as statistical Romberg stochastic approximation algorithm. Then its extension to a multi-level method is proposed. We prove a central limit theorem for both methods and give optimal parameters. Numerical results confirm the theoretical analysis and show a significant reduction in the initial computational cost.

Article information

Source
Ann. Appl. Probab., Volume 26, Number 2 (2016), 933-985.

Dates
Received: October 2013
Revised: February 2015
First available in Project Euclid: 22 March 2016

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1458651825

Digital Object Identifier
doi:10.1214/15-AAP1109

Mathematical Reviews number (MathSciNet)
MR3476630

Zentralblatt MATH identifier
1344.93111

Subjects
Primary: 60F05: Central limit and other weak theorems 62K12 65C05: Monte Carlo methods 60H35: Computational methods for stochastic equations [See also 65C30]

Keywords
Multi-level Monte Carlo methods stochastic approximation Ruppert–Polyak averaging principle Euler scheme

Citation

Frikha, Noufel. Multi-level stochastic approximation algorithms. Ann. Appl. Probab. 26 (2016), no. 2, 933--985. doi:10.1214/15-AAP1109. https://projecteuclid.org/euclid.aoap/1458651825


Export citation

References

  • [1] Aldous, D. J. and Eagleson, G. K. (1978). On mixing and stability of limit theorems. Ann. Probab. 6 325–331.
  • [2] Bardou, O., Frikha, N. and Pagès, G. (2009). Computing VaR and CVaR using stochastic approximation and adaptive unconstrained importance sampling. Monte Carlo Methods Appl. 15 173–210.
  • [3] Bardou, O., Frikha, N. and Pagès, G. (2009). Recursive computation of value-at-risk and conditional value-at-risk using MC and QMC. In Monte Carlo and Quasi-Monte Carlo Methods 2008 193–208. Springer, Berlin.
  • [4] Bardou, O., Frikha, N. and Pagès, G. (2010). CVaR hedging using quantization based stochastic approximation algorithm. Math. Finance. To appear.
  • [5] Benveniste, A., Métivier, M. and Priouret, P. (1990). Adaptive Algorithms and Stochastic Approximations. Applications of Mathematics (New York) 22. Springer, Berlin.
  • [6] Ben Alaya, M. and Kebaier, A. (2015). Central limit theorem for the multilevel Monte Carlo Euler method. Ann. Appl. Probab. 25 211–234.
  • [7] Dereich, S. (2011). Multilevel Monte Carlo algorithms for Lévy-driven SDEs with Gaussian correction. Ann. Appl. Probab. 21 283–311.
  • [8] Duffie, D. and Glynn, P. (1995). Efficient Monte Carlo simulation of security prices. Ann. Appl. Probab. 5 897–905.
  • [9] Duflo, M. (1996). Algorithmes Stochastiques. Mathématiques & Applications (Berlin) [Mathematics & Applications] 23. Springer, Berlin.
  • [10] Fathi, M. and Frikha, N. (2013). Transport-entropy inequalities and deviation estimates for stochastic approximations schemes. Electron. J. Probab. 18 1–36.
  • [11] Frikha, N. (2014). Multilevel stochastic approximation. Preprint HAL (hal-00870585).
  • [12] Frikha, N. (2014). Shortfall risk minimization in discrete time financial market models. SIAM J. Financial Math. 5 384–414.
  • [13] Frikha, N. and Menozzi, S. (2012). Concentration bounds for stochastic approximations. Electron. Commun. Probab. 17 no. 47, 15.
  • [14] Giles, M. (2008). Improved multilevel Monte Carlo convergence using the Milstein scheme. In Monte Carlo and Quasi-Monte Carlo Methods 2006 343–358. Springer, Berlin.
  • [15] Giles, M. B. (2008). Multilevel Monte Carlo path simulation. Oper. Res. 56 607–617.
  • [16] Giles, M. B., Higham, D. J. and Mao, X. (2009). Analysing multi-level Monte Carlo for options with non-globally Lipschitz payoff. Finance Stoch. 13 403–413.
  • [17] Hall, P. and Heyde, C. C. (1980). Martingale Limit Theory and Its Application: Probability and Mathematical Statistics. Academic Press, New York.
  • [18] Heinrich, S. (2001). Multilevel Monte Carlo methods. In Large-Scale Scientific Computing. Springer, Berlin.
  • [19] Jacod, J. (1998). Rates of convergence to the local time of a diffusion. Ann. Inst. Henri Poincaré Probab. Stat. 34 505–544.
  • [20] Jacod, J. and Protter, P. (1998). Asymptotic error distributions for the Euler method for stochastic differential equations. Ann. Probab. 26 267–307.
  • [21] Kebaier, A. (2005). Statistical Romberg extrapolation: A new variance reduction method and applications to option pricing. Ann. Appl. Probab. 15 2681–2705.
  • [22] Kushner, H. J. and Yin, G. G. (2003). Stochastic Approximation and Recursive Algorithms and Applications, 2nd ed. Applications of Mathematics (New York) 35. Springer, New York.
  • [23] Pelletier, M. (1998). Weak convergence rates for stochastic approximation with application to multiple targets and simulated annealing. Ann. Appl. Probab. 8 10–44.
  • [24] Polyak, B. T. and Juditsky, A. B. (1992). Acceleration of stochastic approximation by averaging. SIAM J. Control Optim. 30 838–855.
  • [25] Rényi, A. (1963). On stable sequences of events. Sankhyā Ser. A 25 293–302.
  • [26] Robbins, H. and Monro, S. (1951). A stochastic approximation method. Ann. Math. Stat. 22 400–407.
  • [27] Ruppert, D. (1991). Stochastic approximation. In Handbook of Sequential Analysis. Statist. Textbooks Monogr. 118 503–529. Dekker, New York.