메뉴 건너뛰기




Volumn 25, Issue 4, 2015, Pages 781-795

Scalable estimation strategies based on stochastic approximations: classical results and new insights

Author keywords

Asymptotic analysis; Big data; Implicit stochastic gradient descent methods; Maximum likelihood; Optimal learning rate; Recursive estimation

Indexed keywords


EID: 84932192001     PISSN: 09603174     EISSN: 15731375     Source Type: Journal    
DOI: 10.1007/s11222-015-9560-y     Document Type: Article
Times cited : (42)

References (93)
  • 1
    • 0000396062 scopus 로고    scopus 로고
    • Natural gradient works efficiently in learning
    • Amari, S.-I.: Natural gradient works efficiently in learning. Neural Comput. 10(2), 251–276 (1998)
    • (1998) Neural Comput. , vol.10 , Issue.2 , pp. 251-276
    • Amari, S.-I.1
  • 2
    • 0034201611 scopus 로고    scopus 로고
    • Adaptive method of realizing natural gradient learning for multilayer perceptrons
    • Amari, S.-I., Park, H., Kenji, F.: Adaptive method of realizing natural gradient learning for multilayer perceptrons. Neural Comput. 12(6), 1399–1409 (2000)
    • (2000) Neural Comput. , vol.12 , Issue.6 , pp. 1399-1409
    • Amari, S.-I.1    Park, H.2    Kenji, F.3
  • 4
    • 0037403111 scopus 로고    scopus 로고
    • Mirror descent and nonlinear projected subgradient methods for convex optimization
    • Beck, A., Teboulle, M.: Mirror descent and nonlinear projected subgradient methods for convex optimization. Oper. Res. Lett. 31(3), 167–175 (2003)
    • (2003) Oper. Res. Lett. , vol.31 , Issue.3 , pp. 167-175
    • Beck, A.1    Teboulle, M.2
  • 5
    • 69349090197 scopus 로고    scopus 로고
    • Learning deep architectures for ai. Foundations and trends®
    • Bengio, Y.: Learning deep architectures for ai. Foundations and trends $$\textregistered $$®. Mach. Learn. 2, 1–127 (2009)
    • (2009) Mach. Learn. , vol.2
    • Bengio, Y.1
  • 6
    • 67651049775 scopus 로고    scopus 로고
    • Justifying and generalizing contrastive divergence
    • Bengio, Y., Delalleau, O.: Justifying and generalizing contrastive divergence. Neural Comput. 21(6), 1601–1621 (2009)
    • (2009) Neural Comput. , vol.21 , Issue.6 , pp. 1601-1621
    • Bengio, Y.1    Delalleau, O.2
  • 9
    • 68949096711 scopus 로고    scopus 로고
    • Sgd-qn: careful quasi-Newton stochastic gradient descent
    • Bordes, A., Bottou, L., Gallinari, P.: Sgd-qn: careful quasi-Newton stochastic gradient descent. J. Mach. Learn. Res. 10, 1737–1754 (2009)
    • (2009) J. Mach. Learn. Res. , vol.10 , pp. 1737-1754
    • Bordes, A.1    Bottou, L.2    Gallinari, P.3
  • 10
    • 84904136037 scopus 로고    scopus 로고
    • Large-scale machine learning with stochastic gradient descent
    • Springer, New York
    • Bottou, L.: Large-scale machine learning with stochastic gradient descent. In: Proceedings of COMPSTAT’2010, pp. 177–186. Springer, New York (2010)
    • (2010) Proceedings of COMPSTAT’2010 , pp. 177-186
    • Bottou, L.1
  • 11
    • 17444425307 scopus 로고    scopus 로고
    • On-line learning for very large data sets
    • Bottou, L., Le Cun, Y.: On-line learning for very large data sets. Appl. Stoch. Models Bus. Ind. 21(2), 137–151 (2005)
    • (2005) Appl. Stoch. Models Bus. Ind. , vol.21 , Issue.2 , pp. 137-151
    • Bottou, L.1    Le Cun, Y.2
  • 13
    • 84968510937 scopus 로고
    • A class of methods for solving nonlinear simultaneous equations
    • Broyden, C.G.: A class of methods for solving nonlinear simultaneous equations. Math. Comput. 19, 577–593 (1965)
    • (1965) Math. Comput. , vol.19 , pp. 577-593
    • Broyden, C.G.1
  • 14
    • 80052231929 scopus 로고    scopus 로고
    • Online em algorithm for hidden Markov models
    • Cappé, O.: Online em algorithm for hidden Markov models. J. Comput. Graph. Stat. 20(3), 728–749 (2011)
    • (2011) J. Comput. Graph. Stat. , vol.20 , Issue.3 , pp. 728-749
    • Cappé, O.1
  • 15
    • 66849104300 scopus 로고    scopus 로고
    • On-line expectation-maximization algorithm for latent data models
    • Cappé, O., Moulines, M.: On-line expectation-maximization algorithm for latent data models. J. R. Stat. Soc. 71(3), 593–613 (2009)
    • (2009) J. R. Stat. Soc. , vol.71 , Issue.3 , pp. 593-613
    • Cappé, O.1    Moulines, M.2
  • 18
    • 0000792517 scopus 로고
    • On a stochastic approximation method
    • Chung, K.L.: On a stochastic approximation method. Ann. Math. Stat. 25, 463–483 (1954)
    • (1954) Ann. Math. Stat. , vol.25 , pp. 463-483
    • Chung, K.L.1
  • 19
    • 0002629270 scopus 로고
    • Maximum likelihood from incomplete data via the EM algorithm
    • Dempster, A., Laird, N., Rubin, D.: Maximum likelihood from incomplete data via the EM algorithm. J. R. Stat. Soc. Ser. B 39, 1–38 (1977)
    • (1977) J. R. Stat. Soc. Ser. B , vol.39
    • Dempster, A.1    Laird, N.2    Rubin, D.3
  • 20
    • 80052250414 scopus 로고    scopus 로고
    • Adaptive subgradient methods for online learning and stochastic optimization
    • Duchi, J., Hazan, E., Singer, Y.: Adaptive subgradient methods for online learning and stochastic optimization. J. Mach. Learn. Res. 999999, 2121–2159 (2011)
    • (2011) J. Mach. Learn. Res. , pp. 2121-2159
    • Duchi, J.1    Hazan, E.2    Singer, Y.3
  • 21
    • 0026205085 scopus 로고
    • On sampling controlled stochastic approximation
    • Dupuis, P., Simha, R.: On sampling controlled stochastic approximation. IEEE Trans. Autom. Control 36(8), 915–924 (1991)
    • (1991) IEEE Trans. Autom. Control , vol.36 , Issue.8 , pp. 915-924
    • Dupuis, P.1    Simha, R.2
  • 22
    • 62349116164 scopus 로고    scopus 로고
    • Spectrum estimation for large dimensional covariance matrices using random matrix theory
    • El Karoui, N.: Spectrum estimation for large dimensional covariance matrices using random matrix theory. Ann. Stat. 36, 2757–2790 (2008)
    • (2008) Ann. Stat. , vol.36 , pp. 2757-2790
    • El Karoui, N.1
  • 23
    • 0001214703 scopus 로고
    • On asymptotic normality in stochastic approximation
    • Fabian, V.: On asymptotic normality in stochastic approximation. Ann. Math. Stat. 39, 1327–1332 (1968)
    • (1968) Ann. Math. Stat. , vol.39 , pp. 1327-1332
    • Fabian, V.1
  • 24
    • 68949134067 scopus 로고
    • Asymptotically efficient stochastic approximation; the RM case
    • Fabian, V.: Asymptotically efficient stochastic approximation; the RM case. Ann. Stat. 1, 486–495 (1973)
    • (1973) Ann. Stat. , vol.1 , pp. 486-495
    • Fabian, V.1
  • 25
    • 0001735517 scopus 로고
    • On the mathematical foundations of theoretical statistics
    • Fisher, R.A.: On the mathematical foundations of theoretical statistics. Philos. Trans. R. Soc. Lond. Ser. A 222, 309–368 (1922)
    • (1922) Philos. Trans. R. Soc. Lond. Ser. A , vol.222 , pp. 309-368
    • Fisher, R.A.1
  • 28
    • 0021518209 scopus 로고
    • Stochastic relaxation, gibbs distributions, and the Bayesian restoration of images
    • Geman, S., Geman, D.: Stochastic relaxation, gibbs distributions, and the Bayesian restoration of images. IEEE Trans. Pattern Anal. Mach. Intell. 6, 721–741 (1984)
    • (1984) IEEE Trans. Pattern Anal. Mach. Intell. , vol.6 , pp. 721-741
    • Geman, S.1    Geman, D.2
  • 29
    • 33748998787 scopus 로고    scopus 로고
    • Adaptive stepsizes for recursive estimation with applications in approximate dynamic programming
    • George, A.P., Powell, W.B.: Adaptive stepsizes for recursive estimation with applications in approximate dynamic programming. Machine Learn. 65(1), 167–198 (2006)
    • (2006) Machine Learn. , vol.65 , Issue.1 , pp. 167-198
    • George, A.P.1    Powell, W.B.2
  • 30
    • 79952295497 scopus 로고    scopus 로고
    • Riemann manifold Langevin and Hamiltonian Monte Carlo methods
    • Girolami, M.: Riemann manifold Langevin and Hamiltonian Monte Carlo methods. J. R. Stat. Soc. Ser. B 73(2), 123–214 (2011)
    • (2011) J. R. Stat. Soc. Ser. B , vol.73 , Issue.2 , pp. 123-214
    • Girolami, M.1
  • 31
    • 67649964731 scopus 로고    scopus 로고
    • Reinforcement learning: a tutorial survey and recent advances
    • Gosavi, A.: Reinforcement learning: a tutorial survey and recent advances. INFORMS J. Comput. 21(2), 178–192 (2009)
    • (2009) INFORMS J. Comput. , vol.21 , Issue.2 , pp. 178-192
    • Gosavi, A.1
  • 32
    • 0001648516 scopus 로고
    • Iteratively reweighted least squares for maximum likelihood estimation, and some robust and resistant alternatives
    • Green, P.J.: Iteratively reweighted least squares for maximum likelihood estimation, and some robust and resistant alternatives. J. R. Stat. Soc. Ser. B 46, 149–192 (1984)
    • (1984) J. R. Stat. Soc. Ser. B , vol.46 , pp. 149-192
    • Green, P.J.1
  • 34
    • 84876217045 scopus 로고    scopus 로고
    • Quasi-Newton methods: a new direction
    • Hennig, P., Kiefel, M.: Quasi-Newton methods: a new direction. J. Mach. Learn. Res. 14(1), 843–865 (2013)
    • (2013) J. Mach. Learn. Res. , vol.14 , Issue.1 , pp. 843-865
    • Hennig, P.1    Kiefel, M.2
  • 35
    • 0013344078 scopus 로고    scopus 로고
    • Training products of experts by minimizing contrastive divergence
    • Hinton, G.E.: Training products of experts by minimizing contrastive divergence. Neural Comput. 14(8), 1771–1800 (2002)
    • (2002) Neural Comput. , vol.14 , Issue.8 , pp. 1771-1800
    • Hinton, G.E.1
  • 37
    • 0003157339 scopus 로고
    • Robust estimation of a location parameter
    • Huber, P.J., et al.: Robust estimation of a location parameter. Ann. Math. Stat. 35(1), 73–101 (1964)
    • (1964) Ann. Math. Stat. , vol.35 , Issue.1 , pp. 73-101
    • Huber, P.J.1
  • 39
    • 84898963415 scopus 로고    scopus 로고
    • Accelerating stochastic gradient descent using predictive variance reduction
    • Johnson, R., Zhang, T.: Accelerating stochastic gradient descent using predictive variance reduction. Adv. Neural Inf. Process. Syst. 26, 315–323 (2013)
    • (2013) Adv. Neural Inf. Process. Syst. , vol.26 , pp. 315-323
    • Johnson, R.1    Zhang, T.2
  • 40
    • 84932194480 scopus 로고    scopus 로고
    • Kivinen, J., Warmuth, M.K.: Additive versus exponentiated gradient updates for linear prediction. In: Proceedings of the Twenty-Seventh Annual ACM Symposium on Theory of Computing, pp. 209–218
    • Kivinen, J., Warmuth, M.K.: Additive versus exponentiated gradient updates for linear prediction. In: Proceedings of the Twenty-Seventh Annual ACM Symposium on Theory of Computing, pp. 209–218
  • 41
    • 33646032356 scopus 로고    scopus 로고
    • The p-norm generalization of the lms algorithm for adaptive filtering
    • Kivinen, J., Warmuth, M.K., Hassibi, B.: The p-norm generalization of the lms algorithm for adaptive filtering. IEEE Trans. Signal Process. 54(5), 1782–1793 (2006)
    • (2006) IEEE Trans. Signal Process. , vol.54 , Issue.5 , pp. 1782-1793
    • Kivinen, J.1    Warmuth, M.K.2    Hassibi, B.3
  • 44
    • 0000878355 scopus 로고
    • Adaptive design and stochastic approximation
    • Lai, T.L., Robbins, H.: Adaptive design and stochastic approximation. Ann. Stat. 7, 1196–1221 (1979)
    • (1979) Ann. Stat. , vol.7 , pp. 1196-1221
    • Lai, T.L.1    Robbins, H.2
  • 45
    • 0000808747 scopus 로고
    • A gradient algorithm locally equivalent to the EM algorithm
    • Lange, K.: A gradient algorithm locally equivalent to the EM algorithm. J. R. Stat. Soc. Ser. B 57, 425–437 (1995)
    • (1995) J. R. Stat. Soc. Ser. B , vol.57 , pp. 425-437
    • Lange, K.1
  • 49
    • 56449125197 scopus 로고    scopus 로고
    • Li, L.: A worst-case comparison between temporal difference and residual gradient with linear function approximation. In: Proceedings of the 25th International Conference on Machine Learning, ACM, pp. 560–567
    • Li, L.: A worst-case comparison between temporal difference and residual gradient with linear function approximation. In: Proceedings of the 25th International Conference on Machine Learning, ACM, pp. 560–567
  • 50
    • 26444444069 scopus 로고    scopus 로고
    • Online em algorithm for mixture with application to internet traffic modeling
    • Liu, Z., Almhana, J., Choulakian, V., McGorman, R.: Online em algorithm for mixture with application to internet traffic modeling. Comput. Stat. Data Anal. 50(4), 1052–1071 (2006)
    • (2006) Comput. Stat. Data Anal. , vol.50 , Issue.4 , pp. 1052-1071
    • Liu, Z.1    Almhana, J.2    Choulakian, V.3    McGorman, R.4
  • 52
    • 0016508280 scopus 로고
    • Robust estimation via stochastic approximation
    • Martin, R.D., Masreliez, C.: Robust estimation via stochastic approximation. IEEE Trans. Inf. Theory 21(3), 263–271 (1975)
    • (1975) IEEE Trans. Inf. Theory , vol.21 , Issue.3 , pp. 263-271
    • Martin, R.D.1    Masreliez, C.2
  • 53
    • 0001955526 scopus 로고    scopus 로고
    • Online Learning and Neural Networks, Cambridge University Press, Cambridge
    • Murata, N.: A Statistical Study of On-line Learning. Online Learning and Neural Networks. Cambridge University Press, Cambridge (1998)
    • (1998) A Statistical Study of On-line Learning
    • Murata, N.1
  • 54
    • 33846451627 scopus 로고
    • A learning method for system identification
    • Nagumo, J.-I., Noda, A.: A learning method for system identification. IEEE Trans. Autom. Control 12(3), 282–287 (1967)
    • (1967) IEEE Trans. Autom. Control , vol.12 , Issue.3 , pp. 282-287
    • Nagumo, J.-I.1    Noda, A.2
  • 55
    • 85052723106 scopus 로고    scopus 로고
    • National Research Council The National Academies Press, Washington, DC
    • National Research Council: Frontiers in Massive Data Analysis. The National Academies Press, Washington, DC (2013)
    • (2013) Frontiers in Massive Data Analysis
  • 56
    • 0002788893 scopus 로고    scopus 로고
    • A view of the em algorithm that justifies incremental, sparse, and other variants
    • Springer, New York
    • Neal, R.M., Hinton, G.E.: A view of the em algorithm that justifies incremental, sparse, and other variants. In: Learning in Graphical Models, pp. 355–368. Springer, New York (1998)
    • (1998) Learning in Graphical Models , pp. 355-368
    • Neal, R.M.1    Hinton, G.E.2
  • 59
    • 70450197241 scopus 로고    scopus 로고
    • Robust stochastic approximation approach to stochastic programming
    • Nemirovski, A., Juditsky, A., Lan, G., Shapiro, A.: Robust stochastic approximation approach to stochastic programming. SIAM J. Optim. 19(4), 1574–1609 (2009)
    • (2009) SIAM J. Optim. , vol.19 , Issue.4 , pp. 1574-1609
    • Nemirovski, A.1    Juditsky, A.2    Lan, G.3    Shapiro, A.4
  • 62
    • 84884129062 scopus 로고    scopus 로고
    • Proximal algorithms
    • Parikh, N., Boyd, S.: Proximal algorithms. Found. Trends Optim. 1(3), 123–231 (2013)
    • (2013) Found. Trends Optim. , vol.1 , Issue.3 , pp. 123-231
    • Parikh, N.1    Boyd, S.2
  • 63
    • 84932194804 scopus 로고    scopus 로고
    • Pillai, N.S., Smith, A.: Ergodicity of approximate mcmc chains with applications to large data sets. arXiv preprint (2014)
    • Pillai, N.S., Smith, A.: Ergodicity of approximate mcmc chains with applications to large data sets. arXiv preprint http://arxiv.org/abs/1405.0182 (2014)
  • 64
    • 0002410521 scopus 로고
    • Adaptive algorithms of estimation (convergence, optimality, stability)
    • Polyak, B.T., Tsypkin, Y.Z.: Adaptive algorithms of estimation (convergence, optimality, stability). Autom. Remote Control 3, 74–84 (1979)
    • (1979) Autom. Remote Control , vol.3 , pp. 74-84
    • Polyak, B.T.1    Tsypkin, Y.Z.2
  • 65
    • 0026899240 scopus 로고
    • Acceleration of stochastic approximation by averaging. SIAM
    • Polyak, B.T., Juditsky, A.B.: Acceleration of stochastic approximation by averaging. SIAM J. Control Optim. 30(4), 838–855 (1992)
    • (1992) J. Control Optim , vol.30 , Issue.4 , pp. 838-855
    • Polyak, B.T.1    Juditsky, A.B.2
  • 66
    • 0000016172 scopus 로고
    • A stochastic approximation method
    • Robbins, H., Monro, S.: A stochastic approximation method. Ann. Math. Stat. 22, 400–407 (1951)
    • (1951) Ann. Math. Stat. , vol.22 , pp. 400-407
    • Robbins, H.1    Monro, S.2
  • 67
    • 0016985417 scopus 로고
    • Monotone operators and the proximal point algorithm
    • Rockafellar, R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14(5), 877–898 (1976)
    • (1976) SIAM J. Control Optim. , vol.14 , Issue.5 , pp. 877-898
    • Rockafellar, R.T.1
  • 68
    • 84932194717 scopus 로고    scopus 로고
    • Rosasco, L., Villa, S., Công Vũ, B.: Convergence of stochastic proximal gradient algorithm. arXiv preprint , 2014
    • Rosasco, L., Villa, S., Công Vũ, B.: Convergence of stochastic proximal gradient algorithm. arXiv preprint http://arxiv.org/abs/1403.5074, 2014
  • 70
    • 84932197319 scopus 로고    scopus 로고
    • Ryu, E.K., Boyd, S.: Stochastic proximal iteration: a non-asymptotic improvement upon stochastic gradient descent. Working paper. (2014)
    • Ryu, E.K., Boyd, S.: Stochastic proximal iteration: a non-asymptotic improvement upon stochastic gradient descent. Working paper. http://web.stanford.edu/~eryu/papers/spi.pdf (2014)
  • 71
    • 0000431134 scopus 로고
    • Asymptotic distribution of stochastic approximation procedures
    • Sacks, J.: Asymptotic distribution of stochastic approximation procedures. Ann. Math. Stat. 29(2), 373–405 (1958)
    • (1958) Ann. Math. Stat. , vol.29 , Issue.2 , pp. 373-405
    • Sacks, J.1
  • 72
    • 0007229977 scopus 로고
    • Efficient recursive estimation; application to estimating the parameters of a covariance function
    • Sakrison, D.J.: Efficient recursive estimation; application to estimating the parameters of a covariance function. Int. J. Eng. Sci. 3(4), 461–483 (1965)
    • (1965) Int. J. Eng. Sci. , vol.3 , Issue.4 , pp. 461-483
    • Sakrison, D.J.1
  • 74
    • 0034131785 scopus 로고    scopus 로고
    • On-line em algorithm for the normalized Gaussian network
    • Sato, M.-A., Ishii, S.: On-line em algorithm for the normalized Gaussian network. Neural Comput. 12(2), 407–432 (2000)
    • (2000) Neural Comput. , vol.12 , Issue.2 , pp. 407-432
    • Sato, M.-A.1    Ishii, S.2
  • 75
    • 84932199211 scopus 로고    scopus 로고
    • Approximation analysis of stochastic gradient langevin dynamics by using Fokker-Planck equation and ito process
    • Sato, I., Nakagawa, H.: Approximation analysis of stochastic gradient langevin dynamics by using Fokker-Planck equation and ito process. JMLR W&CP 32(1), 982–990 (2014)
    • (2014) JMLR W&CP , vol.32 , Issue.1 , pp. 982-990
    • Sato, I.1    Nakagawa, H.2
  • 76
    • 0013419177 scopus 로고    scopus 로고
    • On the worst-case analysis of temporal-difference learning algorithms
    • Schapire, R.E., Warmuth, M.K.: On the worst-case analysis of temporal-difference learning algorithms. Mach. Learn. 22(1–3), 95–121 (1996)
    • (1996) Mach. Learn. , vol.22 , Issue.1-3 , pp. 95-121
    • Schapire, R.E.1    Warmuth, M.K.2
  • 77
    • 84932193261 scopus 로고    scopus 로고
    • Schaul, T., Zhang, S., LeCun, Y.: No more pesky learning rates. arXiv preprint. , 2012
    • Schaul, T., Zhang, S., LeCun, Y.: No more pesky learning rates. arXiv preprint. http://arxiv.org/abs/1206.1106, 2012
  • 79
    • 0027667902 scopus 로고
    • On the convergence behavior of the LMS and the normalized LMS algorithms
    • Slock, D.T.M.: On the convergence behavior of the LMS and the normalized LMS algorithms. IEEE Trans. Signal Process. 41(9), 2811–2825 (1993)
    • (1993) IEEE Trans. Signal Process , vol.41 , Issue.9 , pp. 2811-2825
    • Slock, D.T.M.1
  • 80
    • 33847202724 scopus 로고
    • Learning to predict by the methods of temporal differences
    • Sutton, R.S.: Learning to predict by the methods of temporal differences. Mach. Learn. 3(1), 9–44 (1988)
    • (1988) Mach. Learn. , vol.3 , Issue.1 , pp. 9-44
    • Sutton, R.S.1
  • 83
    • 0001593436 scopus 로고
    • Recursive parameter estimation using incomplete data
    • Titterington, M.D.: Recursive parameter estimation using incomplete data. J. R. Stat. Soc. Ser. B 46, 257–267 (1984)
    • (1984) J. R. Stat. Soc. Ser. B , vol.46 , pp. 257-267
    • Titterington, M.D.1
  • 84
    • 84932198982 scopus 로고    scopus 로고
    • Toulis, P., Airoldi, E.M.: Implicit stochastic gradient descent for principled estimation with large datasets. arXiv preprint , 2014
    • Toulis, P., Airoldi, E.M.: Implicit stochastic gradient descent for principled estimation with large datasets. arXiv preprint http://arxiv.org/abs/1408.2923, 2014
  • 85
    • 85028606750 scopus 로고    scopus 로고
    • Statistical analysis of stochastic gradient methods for generalized linear models
    • Toulis, P., Airoldi, E., Rennie, J.: Statistical analysis of stochastic gradient methods for generalized linear models. JMLR W&CP 32(1), 667–675 (2014)
    • (2014) JMLR W&CP , vol.32 , Issue.1 , pp. 667-675
    • Toulis, P.1    Airoldi, E.2    Rennie, J.3
  • 86
    • 0002010858 scopus 로고
    • An extension of the robbins-monro procedur
    • Venter, J.H.: An extension of the robbins-monro procedur. Ann. Math. Stat. 38, 181–190 (1967)
    • (1967) Ann. Math. Stat. , vol.38 , pp. 181-190
    • Venter, J.H.1
  • 88
    • 84899690779 scopus 로고    scopus 로고
    • Stabilization of stochastic iterative methods for singular and nearly singular linear systems
    • Wang, M., Bertsekas, D.P.: Stabilization of stochastic iterative methods for singular and nearly singular linear systems. Math. Oper. Res. 39(1), 1–30 (2013)
    • (2013) Math. Oper. Res. , vol.39 , Issue.1
    • Wang, M.1    Bertsekas, D.P.2
  • 89
    • 0000221062 scopus 로고
    • Multivariate adaptive stochastic approximation
    • Wei, C.Z.: Multivariate adaptive stochastic approximation. Ann. Stat. 3, 1115–1130 (1987)
    • (1987) Ann. Stat. , vol.3 , pp. 1115-1130
    • Wei, C.Z.1
  • 91
    • 84932196044 scopus 로고    scopus 로고
    • Xu, W.: Towards optimal one pass large scale learning with averaged stochastic gradient descent. arXiv preprint , 2011
    • Xu, W.: Towards optimal one pass large scale learning with averaged stochastic gradient descent. arXiv preprint http://arxiv.org/abs/1107.2490, 2011
  • 92
    • 33644756784 scopus 로고    scopus 로고
    • On the convergence of markovian stochastic algorithms with rapidly decreasing ergodicity rates
    • Younes, L.: On the convergence of markovian stochastic algorithms with rapidly decreasing ergodicity rates. Stochastics 65(3–4), 177–228 (1999)
    • (1999) Stochastics , vol.65 , Issue.3-4 , pp. 177-228
    • Younes, L.1


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.