메뉴 건너뛰기




Volumn 61, Issue 5, 2015, Pages 2788-2806

Optimal Rates for Zero-Order Convex Optimization: The Power of Two Function Evaluations

Author keywords

lower bounds; on line learning; Optimization; statistical learning; zero order optimization

Indexed keywords

ALGORITHMS; CONVEX OPTIMIZATION; FUNCTION EVALUATION; GRADIENT METHODS; INFORMATION THEORY; STOCHASTIC SYSTEMS;

EID: 84928632779     PISSN: 00189448     EISSN: None     Source Type: Journal    
DOI: 10.1109/TIT.2015.2409256     Document Type: Article
Times cited : (553)

References (39)
  • 5
    • 84860610528 scopus 로고    scopus 로고
    • Center Oper. Res. Econometrics, Univ. Catholique de Louvain, Louvain, Belgium, Tech. Rep.
    • Y. Nesterov, "Random gradient-free minimization of convex functions," Center Oper. Res. Econometrics, Univ. Catholique de Louvain, Louvain, Belgium, Tech. Rep. 2011001, 2011.
    • (2011) Random Gradient-Free Minimization of Convex Functions
    • Nesterov, Y.1
  • 6
    • 84892854517 scopus 로고    scopus 로고
    • Stochastic first- and zeroth-order methods for nonconvex stochastic programming
    • S. Ghadimi and G. Lan, "Stochastic first- and zeroth-order methods for nonconvex stochastic programming," SIAM J. Optim., vol. 23, no. 4, pp. 2341-2368, 2013.
    • (2013) SIAM J. Optim. , vol.23 , Issue.4 , pp. 2341-2368
    • Ghadimi, S.1    Lan, G.2
  • 7
    • 84877750537 scopus 로고    scopus 로고
    • Stochastic convex optimization with bandit feedback
    • A. Agarwal, D. P. Foster, D. Hsu, S. M. Kakade, and A. Rakhlin, "Stochastic convex optimization with bandit feedback," SIAM J. Optim., vol. 23, no. 1, pp. 213-240, 2013.
    • (2013) SIAM J. Optim. , vol.23 , Issue.1 , pp. 213-240
    • Agarwal, A.1    Foster, D.P.2    Hsu, D.3    Kakade, S.M.4    Rakhlin, A.5
  • 10
    • 65749118363 scopus 로고    scopus 로고
    • Graphical models, exponential families, and variational inference
    • Dec.
    • M. J. Wainwright and M. I. Jordan, "Graphical models, exponential families, and variational inference," Found. Trends Mach. Learn., vol. 1, nos. 1-2, pp. 1-305, Dec. 2008.
    • (2008) Found. Trends Mach. Learn. , vol.1 , Issue.1-2 , pp. 1-305
    • Wainwright, M.J.1    Jordan, M.I.2
  • 14
    • 84898041887 scopus 로고    scopus 로고
    • On the complexity of bandit and derivative-free stochastic convex optimization
    • O. Shamir, "On the complexity of bandit and derivative-free stochastic convex optimization," in Proc. 26th Annu. Conf. Comput. Learn. Theory, 2013, pp. 3-24.
    • Proc. 26th Annu. Conf. Comput. Learn. Theory, 2013 , pp. 3-24
    • Shamir, O.1
  • 15
    • 1942484421 scopus 로고    scopus 로고
    • Online convex programming and generalized infinitesimal gradient ascent
    • M. Zinkevich, "Online convex programming and generalized infinitesimal gradient ascent," in Proc. 20th Int. Conf. Mach. Learn., 2003, pp. 928-936.
    • Proc. 20th Int. Conf. Mach. Learn., 2003 , pp. 928-936
    • Zinkevich, M.1
  • 16
    • 0037403111 scopus 로고    scopus 로고
    • Mirror descent and nonlinear projected subgradient methods for convex optimization
    • May
    • A. Beck and M. Teboulle, "Mirror descent and nonlinear projected subgradient methods for convex optimization," Oper. Res. Lett., vol. 31, no. 3, pp. 167-175, May 2003.
    • (2003) Oper. Res. Lett. , vol.31 , Issue.3 , pp. 167-175
    • Beck, A.1    Teboulle, M.2
  • 17
    • 70450197241 scopus 로고    scopus 로고
    • Robust stochastic approximation approach to stochastic programming
    • A. Nemirovski, A. Juditsky, G. Lan, and A. Shapiro, "Robust stochastic approximation approach to stochastic programming," SIAM J. Optim., vol. 19, no. 4, pp. 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
  • 18
    • 78649396336 scopus 로고    scopus 로고
    • Dual averaging methods for regularized stochastic learning and online optimization
    • Oct.
    • L. Xiao, "Dual averaging methods for regularized stochastic learning and online optimization," J. Mach. Learn. Res., vol. 11, pp. 2543-2596, Oct. 2010.
    • (2010) J. Mach. Learn. Res. , vol.11 , pp. 2543-2596
    • Xiao, L.1
  • 20
    • 4544304381 scopus 로고    scopus 로고
    • On the generalization ability of on-line learning algorithms
    • Sep.
    • N. Cesa-Bianchi, A. Conconi, and C. Gentile, "On the generalization ability of on-line learning algorithms," IEEE Trans. Inf. Theory, vol. 50, no. 9, pp. 2050-2057, Sep. 2004.
    • (2004) IEEE Trans. Inf. Theory , vol.50 , Issue.9 , pp. 2050-2057
    • Cesa-Bianchi, N.1    Conconi, A.2    Gentile, C.3
  • 21
    • 84860244324 scopus 로고    scopus 로고
    • Information-theoretic lower bounds on the oracle complexity of stochastic convex optimization
    • May
    • A. Agarwal, P. L. Bartlett, P. Ravikumar, and M. J. Wainwright, "Information-theoretic lower bounds on the oracle complexity of stochastic convex optimization," IEEE Trans. Inf. Theory, vol. 58, no. 5, pp. 3235-3249, May 2012.
    • (2012) IEEE Trans. Inf. Theory , vol.58 , Issue.5 , pp. 3235-3249
    • Agarwal, A.1    Bartlett, P.L.2    Ravikumar, P.3    Wainwright, M.J.4
  • 22
    • 0344875562 scopus 로고    scopus 로고
    • The robustness of the p-norm algorithms
    • Dec.
    • C. Gentile, "The robustness of the p-norm algorithms," Mach. Learn., vol. 53, no. 3, pp. 265-299, Dec. 2003.
    • (2003) Mach. Learn. , vol.53 , Issue.3 , pp. 265-299
    • Gentile, C.1
  • 23
    • 0008815681 scopus 로고    scopus 로고
    • Exponentiated gradient versus gradient descent for linear predictors
    • Jan.
    • J. Kivinen and M. K. Warmuth, "Exponentiated gradient versus gradient descent for linear predictors," Inf. Comput., vol. 132, no. 1, pp. 1-63, Jan. 1997.
    • (1997) Inf. Comput. , vol.132 , Issue.1 , pp. 1-63
    • Kivinen, J.1    Warmuth, M.K.2
  • 24
    • 0036342276 scopus 로고    scopus 로고
    • The ordered subsets mirror descent optimization method with applications to tomography
    • A. Ben-Tal, T. Margalit, and A. Nemirovski, "The ordered subsets mirror descent optimization method with applications to tomography," SIAM J. Optim., vol. 12, no. 1, pp. 79-108, 2001.
    • (2001) SIAM J. Optim. , vol.12 , Issue.1 , pp. 79-108
    • Ben-Tal, A.1    Margalit, T.2    Nemirovski, A.3
  • 25
    • 17444406259 scopus 로고    scopus 로고
    • Smooth minimization of non-smooth functions
    • May
    • Y. Nesterov, "Smooth minimization of non-smooth functions," Math. Program., vol. 103, no. 1, pp. 127-152, May 2005.
    • (2005) Math. Program. , vol.103 , Issue.1 , pp. 127-152
    • Nesterov, Y.1
  • 26
    • 84865694209 scopus 로고    scopus 로고
    • Randomized smoothing for stochastic optimization
    • J. C. Duchi, P. L. Bartlett, and M. J. Wainwright, "Randomized smoothing for stochastic optimization," SIAM J. Optim., vol. 22, no. 2, pp. 674-701, 2012.
    • (2012) SIAM J. Optim. , vol.22 , Issue.2 , pp. 674-701
    • Duchi, J.C.1    Bartlett, P.L.2    Wainwright, M.J.3
  • 28
  • 29
    • 84874045238 scopus 로고    scopus 로고
    • Regret analysis of stochastic and nonstochastic multi-armed bandit problems
    • S. Bubeck and N. Cesa-Bianchi, "Regret analysis of stochastic and nonstochastic multi-armed bandit problems," Found. Trends Mach. Learn., vol. 5, no. 1, pp. 1-122, 2012.
    • (2012) Found. Trends Mach. Learn. , vol.5 , Issue.1 , pp. 1-122
    • Bubeck, S.1    Cesa-Bianchi, N.2
  • 30
    • 84966203785 scopus 로고
    • Some aspects of the sequential design of experiments
    • H. Robbins, "Some aspects of the sequential design of experiments," Bull. Amer. Math. Soc., vol. 58, no. 5, pp. 527-535, 1952.
    • (1952) Bull. Amer. Math. Soc. , vol.58 , Issue.5 , pp. 527-535
    • Robbins, H.1
  • 31
    • 0002899547 scopus 로고
    • Asymptotically efficient adaptive allocation rules
    • Mar.
    • T. L. Lai and H. Robbins, "Asymptotically efficient adaptive allocation rules," Adv. Appl. Math., vol. 6, no. 1, pp. 4-22, Mar. 1985.
    • (1985) Adv. Appl. Math. , vol.6 , Issue.1 , pp. 4-22
    • Lai, T.L.1    Robbins, H.2
  • 33
    • 0033233737 scopus 로고    scopus 로고
    • Information-theoretic determination of minimax rates of convergence
    • Y. Yang and A. Barron, "Information-theoretic determination of minimax rates of convergence," Ann. Statist., vol. 27, no. 5, pp. 1564-1599, 1999.
    • (1999) Ann. Statist. , vol.27 , Issue.5 , pp. 1564-1599
    • Yang, Y.1    Barron, A.2
  • 34
    • 17644388356 scopus 로고    scopus 로고
    • Assouad, Fano, and Le Cam
    • New York, NY, USA: Springer-Verlag
    • B. Yu, "Assouad, Fano, and Le Cam," in Festschrift for Lucien Le Cam. New York, NY, USA: Springer-Verlag, 1997, pp. 423-435.
    • (1997) Festschrift for Lucien Le Cam , pp. 423-435
    • Yu, B.1
  • 35
    • 84871766619 scopus 로고    scopus 로고
    • On the fundamental limits of adaptive sensing
    • Jan.
    • E. Arias-Castro, E. J. Candés, and M. A. Davenport, "On the fundamental limits of adaptive sensing," IEEE Trans. Inf. Theory, vol. 59, no. 1, pp. 472-481, Jan. 2013.
    • (2013) IEEE Trans. Inf. Theory , vol.59 , Issue.1 , pp. 472-481
    • Arias-Castro, E.1    Candés, E.J.2    Davenport, M.A.3
  • 36
    • 33748311057 scopus 로고    scopus 로고
    • Metric Characterization of Random Variables and Random Processes
    • Providence, RI, USA: AMS
    • V. V. Buldygin and Y. V. Kozachenko, Metric Characterization of Random Variables and Random Processes (Translations of Mathematical Monographs), vol. 188. Providence, RI, USA: AMS, 2000.
    • (2000) Translations of Mathematical Monographs , vol.188
    • Buldygin, V.V.1    Kozachenko, Y.V.2
  • 38
    • 34250435546 scopus 로고
    • Stochastic optimization problems with nondifferentiable cost functionals
    • D. P. Bertsekas, "Stochastic optimization problems with nondifferentiable cost functionals," J. Optim. Theory Appl., vol. 12, no. 2, pp. 218-231, 1973.
    • (1973) J. Optim. Theory Appl. , vol.12 , Issue.2 , pp. 218-231
    • Bertsekas, D.P.1


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.