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This paper shows that a perturbed form of gradient descent converges to a second-order stationary point in a number iterations which depends only poly-logarithmically on dimension (i.e., it is almost "dimension-free").
Gradient methods for the minimisation of functionals
Boris T Polyak · 1963
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Learning representations by back-propagating errors
David E Rumelhart, Geoffrey E Hinton, and Ronald J Williams · 1988
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Introductory lectures on convex programming volume i: Basic course
Yu Nesterov · 1998
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Cubic regularization of newton method and its global performance
Yurii Nesterov and Boris T Polyak · 2006
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On decompositional algorithms for uniform sampling from n-spheres and n-balls
Radoslav Harman and Vladimír Lacko · 2010
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Phase retrieval using alternating minimization
Praneeth Netrapalli, Prateek Jain, and Sujay Sanghavi · 2013
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The loss surface of multilayer networks
Anna Choromanska, Mikael Henaff, Michael Mathieu, Gérard Ben Arous, and Yann LeCun · 2014
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A trust region algorithm with a worst-case iteration complexity of \ \backslash mathcal { \{ O } \} ( \ \backslash epsilonˆ { \{ -3/2 } \} ) for nonconvex optimization
Frank E Curtis, Daniel P Robinson, and Mohammadreza Samadi · 2014
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Identifying and attacking the saddle point problem in high-dimensional non-convex optimization
Yann N Dauphin, Razvan Pascanu, Caglar Gulcehre, Kyunghyun Cho, Surya Ganguli, and Yoshua Bengio · 2014
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Convex optimization: Algorithms and complexity
Sébastien Bubeck et al · 2015
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Phase retrieval via wirtinger flow: Theory and algorithms
Emmanuel J Candes, Xiaodong Li, and Mahdi Soltanolkotabi · 2015
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Escaping from saddle points—online stochastic gradient for tensor decomposition
Rong Ge, Furong Huang, Chi Jin, and Yang Yuan · 2015
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Computing matrix squareroot via non convex local search
Prateek Jain, Chi Jin, Sham M Kakade, and Praneeth Netrapalli · 2015
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Finding approximate local minima for nonconvex optimization in linear time
Naman Agarwal, Zeyuan Allen-Zhu, Brian Bullins, Elad Hazan, and Tengyu Ma · 2016
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Linear convergence of gradient and proximal-gradient methods under the Polyak-Lojasiewicz condition
Hamed Karimi, Julie Nutini, and Mark Schmidt · 2016
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Deep learning without poor local minima
Kenji Kawaguchi · 2016
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Gradient descent only converges to minimizers
Jason D Lee, Max Simchowitz, Michael I Jordan, and Benjamin Recht · 2016
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The power of normalization: Faster evasion of saddle points
Kfir Y Levy · 2016
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Non-square matrix sensing without spurious local minima via the burer-monteiro approach
Dohyung Park, Anastasios Kyrillidis, Constantine Caramanis, and Sujay Sanghavi · 2016
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Srinadh Bhojanapalli, Behnam Neyshabur, and Nathan Srebro · 2016
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Gradient descent efficiently finds the cubic-regularized non-convex newton step
Yair Carmon and John C Duchi · 2016
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Accelerated methods for non-convex optimization
Yair Carmon, John C Duchi, Oliver Hinder, and Aaron Sidford · 2016
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Matrix completion has no spurious local minimum
Rong Ge, Jason D Lee, and Tengyu Ma · 2016
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Complete dictionary recovery over the sphere i: Overview and the geometric picture
Ju Sun, Qing Qu, and John Wright
Cited in the paper.
A geometric analysis of phase retrieval
Ju Sun, Qing Qu, and John Wright · 2016
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Guaranteed matrix completion via non-convex factorization
Ruoyu Sun and Zhi-Quan Luo · 2016
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Qinqing Zheng and John Lafferty · 2016
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