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This paper proposes a thorough theoretical analysis of Stochastic Gradient Descent (SGD) with non-increasing step sizes.
Yuanyuan Feng, Tingran Gao, Lei Li, Jian-Guo Liu, and Yulong Lu · 1902
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A stochastic approximation method
Herbert Robbins and Sutton Monro · 1951
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Some properties of diffusion processes depending on a parameter
Ju. N. Blagovescenskii and M. I. Freidlin · 1961
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Analysis of recursive stochastic algorithms
Lennart Ljung · 1977
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Stochastic approximation methods for constrained and unconstrained systems , volume 26 of Applied Mathematical Sciences
Harold J. Kushner and Dean S. Clark · 1978
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On the decomposition of solutions of stochastic differential equations
Hiroshi Kunita · 1981
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Problem complexity and method efficiency in optimization
Arkadi S. Nemirovsky and David B. Yudin · 1983
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A method for solving the convex programming problem with convergence rate O ( 1 / k 2 ) O(1/k^{2})
Yurii E. Nesterov · 1983
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Applications of a Kushner and Clark lemma to general classes of stochastic algorithms
Michel Métivier and Pierre Priouret · 1984
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Théorèmes de convergence presque sure pour une classe d’algorithmes stochastiques à pas décroissant
Michel Métivier and Pierre Priouret · 1987
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Numerical integration of stochastic differential equations , volume 313 of Mathematics and its Applications
Grigori N. Milstein · 1988
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Adaptive algorithms and stochastic approximations , volume 22 of Applications of Mathematics (New York)
Albert Benveniste, Michel Métivier, and Pierre Priouret · 1990
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Expansion of the global error for numerical schemes solving stochastic differential equations
Denis Talay and Luciano Tubaro · 1990
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Brownian motion and stochastic calculus , volume 113 of Graduate Texts in Mathematics
Ioannis Karatzas and Steven E. Shreve · 1991
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Diffusions, Markov processes, and martingales. Vol. 2
Chris Rogers and David Williams · 1994
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A dynamical system approach to stochastic approximations
Michel Benaim · 1996
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Nonlinear programming
Dimitri P Bertsekas · 1997
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Inequalities for differential and integral equations , volume 197 of Mathematics in Science and Engineering
Baburao G. Pachpatte · 1998
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Introductory lectures on convex optimization , volume 87 of Applied Optimization
Yurii E. Nesterov · 2004
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Solving large scale linear prediction problems using stochastic gradient descent algorithms
Tong Zhang · 2004
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Infinite dimensional analysis
Charalambos D. Aliprantis and Kim C. Border · 2006
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Multidimensional diffusion processes
Daniel W Stroock and SR Srinivasa Varadhan · 2007
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Gradient flows in metric spaces and in the space of probability measures
Luigi Ambrosio, Nicola Gigli, and Giuseppe Savaré · 2008
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Robust stochastic approximation approach to stochastic programming
Arkadi Nemirovski, Anatoli Juditsky, Guanghui Lan, and Alexander Shapiro · 2009
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Optimal transport , volume 338 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]
Accelerated mirror descent in continuous and discrete time
Walid Krichene, Alexandre Bayen, and Peter L Bartlett · 2015
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Linear convergence of gradient and proximal-gradient methods under the polyak-lojasiewicz condition
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A differential equation for modeling nesterov’s accelerated gradient method: Theory and insights
Weijie Su, Stephen P. Boyd, and Emmanuel J. Candès · 2016
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From error bounds to the complexity of first-order descent methods for convex functions
Jérôme Bolte, Trong Phong Nguyen, Juan Peypouquet, and Bruce W Suter · 2017
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Qianxiao Li, Cheng Tai, and Weinan E · 2017
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Cédric Villani · 2009
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Proximal alternating minimization and projection methods for nonconvex problems: An approach based on the kurdyka-łojasiewicz inequality
Hédy Attouch, Jérôme Bolte, Patrick Redont, and Antoine Soubeyran · 2010
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Information-theoretic lower bounds on the oracle complexity of stochastic convex optimization
Alekh Agarwal, Peter L. Bartlett, Pradeep Ravikumar, and Martin J. Wainwright · 2011
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Non-asymptotic analysis of stochastic approximation algorithms for machine learning
Francis R. Bach and Eric Moulines · 2011
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Numerical Solution of Stochastic Differential Equations
Peter E. Kloeden and Eckhard Platen · 2011
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Hogwild: A lock-free approach to parallelizing stochastic gradient descent
Benjamin Recht, Christopher Ré, Stephen J. Wright, and Feng Niu · 2011
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Pegasos: primal estimated sub-gradient solver for SVM
Shai Shalev-Shwartz, Yoram Singer, Nathan Srebro, and Andrew Cotter · 2011
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Asymptotic bias of stochastic gradient search
V. B. Tadić and A. Doucet · 2017
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Optimal convergence rates for nesterov acceleration
Jean Franccois Aujol, Aude Rondepierre, and Charles Dossal · 2018
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Gradient descent learns linear dynamical systems
Moritz Hardt, Tengyu Ma, and Benjamin Recht · 2018
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An alternative view: When does SGD escape local minima?
Robert Kleinberg, Yuanzhi Li, and Yang Yuan · 2018
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Understanding the acceleration phenomenon via high-resolution differential equations
Bin Shi, Simon S. Du, Michael I. Jordan, and Weijie J. Su · 2018
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Convergence rates of an inertial gradient descent algorithm under growth and flatness conditions
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Tight analyses for non-smooth stochastic gradient descent
Nicholas J. A. Harvey, Christopher Liaw, Yaniv Plan, and Sikander Randhawa · 2019
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Stochastic modified equations and dynamics of stochastic gradient algorithms I: mathematical foundations
Qianxiao Li, Cheng Tai, and Weinan E · 2019
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Continuous-time models for stochastic optimization algorithms
Antonio Orvieto and Aurélien Lucchi · 2019
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Stochastic first-order methods: non-asymptotic and computer-aided analyses via potential functions
Adrien Taylor and Francis Bach · 2019
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Stagewise training accelerates convergence of testing error over SGD
Zhuoning Yuan, Yan Yan, Rong Jin, and Tianbao Yang · 2019
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Stein’s method for normal approximation in Wasserstein distances with application to the multivariate central limit theorem
Thomas Bonis · 2020
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