Some pathological traps for stochastic approximation

SIAM Journal on Control and Optimization

Volumn 36, Issue 4, 1998, Pages 1293-1314

  Brandière, Odile ^a,b  

^a UNIVERSITÉ GUSTAVE EIFFEL   (France)

^b UNIVERSITÉ PARIS SUD   (France)

Author keywords

Neural networks; Ordinary differential equations; Stochastic approximation; Traps

Indexed keywords

ALGORITHMS; DIFFERENTIAL EQUATIONS; EIGENVALUES AND EIGENFUNCTIONS; NEURAL NETWORKS; RANDOM PROCESSES;

PRINCIPAL COMPONENT ANALYSIS (PCA) ALGORITHMS;

APPROXIMATION THEORY;

EID: 0032122044     PISSN: 03630129     EISSN: None     Source Type: Journal    
DOI: 10.1137/S036301299630759X     Document Type: Article

References (28)

1. M. BENAÏM AND M. W. HIRSCH, Dynamics of Morse-Smale urn processes, Ergodic Theory Dynam. Systems, 15 (1995), pp. 1005-1030.
2. M. BENAÏM, A dynamical system approach to stochastic approximation, SIAM J. Control Optim., 34 (1996), pp. 437-472.
3. A. BENVENISTE, M. MÉTIVIER, AND P. PRIOURET, Algorithmes adaptatifs et approximations stochastiques, Masson, Paris, 1987.
4. M. BENZÉCRI, Approximation stochastique dans une algèbre normée non commutative. Bull. Soc. Math. France, 97 (1969), pp. 225-241.
5. O. BRANDIÈRE, Autour des pièges des algorithmes stochastiques, Thèse, Université de Marnela-Vallée, Noisy-le-Grand, France, 1996.
6. O. BRANDIÈRE AND M. DUFLO, Les algorithmes stochastiques contournent-ils les pièges?, Ann. Inst. H. Poincaré, 32 (1996), pp. 395-427.
7. B. DELYON, General convergence result on stochastic approximation, IEEE Trans. Automat. Control, 41 (1996), pp. 1245-1255.
8. M. DUFLO, Algorithmes Stochastiques, Math. Appl. 23, Springer-Verlag, Heidelberg, 1996.
9. P. HARTMAN, Ordinary Differential Equations, 2nd ed, Wiley, New York, 1982.
10. M. W. HIRSCH, Asymptotic phase, shadowing and reaction diffusion, in Control Theory, Dynamical Systems and Geometry of Dynamics, K. D. Elworthy and W. N. Everitts, eds., Marcel Dekker, New York, 1993, pp. 87-99.
11. K. HORNIK AND C. M. KUAN, Convergence analysis of local feature extraction algorithms, Neural Networks, 5 (1992), pp. 229-240.
12. J. KARHUNEN AND E. OJA, On stochastic approximation of the eigenvectors and eigenvalues of expectation of a random matrix, J. Math. Anal. Appl., 106 (1995), pp. 69-84.
13. G. D. KERSTING, Some results on the asymptotic behavior of the Robbins-Monro procedure, Bull. Int. Stat. Inst., 47 (1977), pp. 327-335.
14. H. J. KUSHNER AND D. S. CLARK, Stochastic Approximation for Constrained and Uncontrained Systems, Appl. Math. Sci. 26, Springer-Verlag, New York, 1978.
15. T. Z. LAI AND C. Z. WEI, A note on martingale difference sequences satisfying the local Marcinkiewicz-Zygmund condition, Bull. Inst. Math. Acad. Sinica, 11 (1983), pp. 1-13.
16. V. A. LAZAREV, Convergence, of stochastic approximation procedures in case of regression equation with several roots, Problemy Peredachi Informatsii, 28 (1992), pp. 75-88 (in Russian).
17. L. LEBART, Sur les calculs impliqués par la description de certains grands tableaux, Annales de l'INSEE 22-23, 1931.
18. P. LEVY, Sur les séries dont les termes sont des variables éventuellement indépendantes, Stud. Math., 3 (1931), pp. 119-155.
19. J. M. MONNEZ, Convergence d'un processus d'approximation stochastique en analyse factorielle, Internal Report, Institut de Statistique de l'Université de Paris, 38, 1994, pp. 37-56.
20. E. OJA, Simplified neuron model as a principal component analyser, J. Math. Biol., 15 (1982), pp. 267-273.
21. E. OJA, Subspace Methods of Pattern Recognition, Research Studies Press and Wiley, Letchworth, England, 1983.
22. E. OJA, C. Y. SUEN, AND L. XU, Modified Hebbian learning for curve and surface fitting, Neural Networks, 5 (1992), pp. 441-457.
23. E. OJA, Principal components, minor components, and linear neural networks, Neural Networks, 5 (1992), pp. 927-935.
24. E. OJA, H. OGAWA, AND J. WANGVIWATTANA, Principal component analysis by homogeneous neural networks, Parts I and II, IEICE Trans. Inf. Syst., E75-D, 3 (1992), pp. 316-382.
25. R. PEMANTLE, Nonconvergence to unstable points in urn models and stochastic approximations, Ann. Probab., 18 (1990), pp. 698-712.
26. T. D. SANGER, Optimal unsupervised learning in a single-layer linear feedforward network, Neural Networks, 2 (1989), pp. 459-473.
27. C. Z. WEI, Martingale transforms with non-atomic limits and stochastic approximation, Probab. Theory Related Fields, 95 (1993), pp. 103-114.
28. R. WILLIAMS, Feature Discovery through Error-Correcting Learning, Technical Report 8501, University of California, Institute of Cognitive Science, San Diego, CA, 1985.