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In spite of the accomplishments of deep learning based algorithms in numerous applications and very broad corresponding research interest, at the moment there is still no rigorous understanding of the reasons why such algorithms produce useful results in certain situations.
“Deep Neural Network Approximation Theory”
Philipp Grohs, Dmytro Perekrestenko, Dennis Elbrächter and Helmut Bölcskei · 1901
Earlier work this paper cites.
Martin Hutzenthaler, Arnulf Jentzen, Thomas Kruse and Tuan Nguyen · 1901
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“Error bounds for approximations with deep ReLU neural networks in W s , p W^{s,p} norms”
Ingo Gühring, Gitta Kutyniok and Philipp Petersen · 1902
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“Non-asymptotic Analysis of Biased Stochastic Approximation Scheme”
Belhal Karimi, Blazej Miasojedow, Eric Moulines and Hoi-To Wai · 1902
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“Stochastic Gradient Descent for Nonconvex Learning without Bounded Gradient Assumptions”
Yunwen Lei, Ting Hu, Guiying Li and Ke Tang · 1902
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“A Priori Estimates of the Population Risk for Residual Networks”
Weinan E, Chao Ma and Qingcan Wang · 1903
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Christoph Reisinger and Yufei Zhang · 1903
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“Convergence rates for the stochastic gradient descent method for non-convex objective functions”
Benjamin Fehrman, Benjamin Gess and Arnulf Jentzen · 1904
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“A Theoretical Analysis of Deep Neural Networks and Parametric PDEs”
Gitta Kutyniok, Philipp Petersen, Mones Raslan and Reinhold Schneider · 1904
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“Approximation in L p ( μ ) L^{p}(\mu) with deep ReLU neural networks”
Felix Voigtlaender and Philipp Petersen · 1904
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“On stochastic gradient Langevin dynamics with dependent data streams: the fully non-convex case”
Ngoc Chau, Éric Moulines, Miklos Rásonyi, Sotirios Sabanis and Ying Zhang · 1905
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“Approximation spaces of deep neural networks”
Rémi Gribonval, Gitta Kutyniok, Morten Nielsen and Felix Voigtlaender · 1905
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“Deep Network Approximation Characterized by Number of Neurons”
Zuowei Shen, Haizhao Yang and Shijun Zhang · 1906
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“Space-time error estimates for deep neural network approximations for differential equations”
Philipp Grohs, Fabian Hornung, Arnulf Jentzen and Philipp Zimmermann · 1908
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“Deep neural network approximations for Monte Carlo algorithms”
Philipp Grohs, Arnulf Jentzen and Diyora Salimova · 1908
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“Full error analysis for the training of deep neural networks”
Christian Beck, Arnulf Jentzen and Benno Kuckuck · 1910
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“Uniform error estimates for artificial neural network approximations for heat equations”
Lukas Gonon, Philipp Grohs, Arnulf Jentzen, David Kofler and David Šiška · 1911
Earlier work this paper cites.
“Efficient approximation of high-dimensional functions with deep neural networks”
Patrick Cheridito, Arnulf Jentzen and Florian Rossmannek · 1912
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“Central limit theorems for stochastic gradient descent with averaging for stable manifolds”
Steffen Dereich and Sebastian Kassing · 1912
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“Note on the gamma function”
J.. Wendel · 1948
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“A stochastic approximation method”
Herbert Robbins and Sutton Monro · 1951
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“Dynamic programming”
Richard Bellman · 1957
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“Some elementary inequalities relating to the gamma and incomplete gamma function”
Walter Gautschi · 1959
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“Approximation by superpositions of a sigmoidal function”
G. Cybenko · 1989
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“On the approximate realization of continuous mappings by neural networks”
Ken-Ichi Funahashi · 1989
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“Multilayer feedforward networks are universal approximators”
Kurt Hornik, Maxwell Stinchcombe and Halbert White · 1989
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“Layered Neural Networks with Gaussian Hidden Units as Universal Approximations”
Eric. Hartman, James. Keeler and Jacek. Kowalski · 1990
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“Universal approximation of an unknown mapping and its derivatives using multilayer feedforward networks”
Kurt Hornik, Maxwell Stinchcombe and Halbert White · 1990
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“Approximation theory and feedforward networks”
Edward. Blum and Leong Li · 1991
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“Approximation capabilities of multilayer feedforward networks”
Kurt Hornik · 1991
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“Universal Approximation Using Radial-Basis-Function Networks”
J. Park and I.. Sandberg · 1991
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“Universal approximation bounds for superpositions of a sigmoidal function”
Andrew. Barron · 1993
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“Some new results on neural network approximation”
Kurt Hornik · 1993
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“Multilayer feedforward networks with a nonpolynomial activation function can approximate any function”
Moshe Leshno, Vladimir. Lin, Allan Pinkus and Shimon Schocken · 1993
Cited alongside, same era.
“Approximation and estimation bounds for artificial neural networks”
Andrew. Barron · 1994
Cited alongside, same era.
“Neural networks for localized approximation”
C.. Chui, Xin Li and H.. Mhaskar · 1994
Cited alongside, same era.
“Aspects of the numerical analysis of neural networks”
S.. Ellacott · 1994
Cited alongside, same era.
“Approximation capability to functions of several variables, nonlinear functionals, and operators by radial basis function neural networks”
Tianping Chen and Hong Chen · 1995
Cited alongside, same era.
“Degree of approximation by neural and translation networks with a single hidden layer”
“Error bounds for approximations with deep ReLU networks”
Dmitry Yarotsky · 2017
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“Solving stochastic differential equations and Kolmogorov equations by means of deep learning”
Christian Beck, Sebastian Becker, Philipp Grohs, Nor Jaafari and Arnulf Jentzen · 2018
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Julius Berner, Philipp Grohs and Arnulf Jentzen · 2018
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“Gradient Descent Provably Optimizes Over-parameterized Neural Networks”
Simon. Du, Xiyu Zhai, Barnabás Poczós and Aarti Singh · 2018
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“Exponential convergence of the deep neural network approximation for analytic functions”
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H.. Mhaskar and Charles. Micchelli · 1995
Cited alongside, same era.
“Neural Networks for Optimal Approximation of Smooth and Analytic Functions”
H.. Mhaskar · 1996
Cited alongside, same era.
“Approximation by feed-forward neural networks”
Ronald. DeVore, Konstantin. Oskolkov and Pencho. Petrushev · 1997
Cited alongside, same era.
“Ridgelets: theory and applications”, 1998
Emmanuel. Candès · 1998
Cited alongside, same era.
“Approximation of functions and their derivatives: A neural network implementation with applications”
T. Nguyen-Thien and T. Tran-Cong · 1999
Cited alongside, same era.
“Approximation theory of the MLP model in neural networks”
Allan Pinkus · 1999
Cited alongside, same era.
“Lower Bounds on the Complexity of Approximating Continuous Functions by Sigmoidal Neural Networks”
Michael Schmitt · 2000
Cited alongside, same era.
Weinan E and Qingcan Wang · 2018
Later among the works it cites.
“DNN Expression Rate Analysis of High-dimensional PDEs: Application to Option Pricing”
Dennis Elbrächter, Philipp Grohs, Arnulf Jentzen and Christoph Schwab · 2018
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Philipp Grohs, Fabian Hornung, Arnulf Jentzen and Philippe von Wurstemberger · 2018
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“Approximation capability of two hidden layer feedforward neural networks with fixed weights”
Namig. Guliyev and Vugar. Ismailov · 2018
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“On the approximation by single hidden layer feedforward neural networks with fixed weights”
Namig. Guliyev and Vugar. Ismailov · 2018
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“Convergence of the Deep BSDE Method for Coupled FBSDEs”
Jiequn Han and Jihao Long · 2018
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“Strong error analysis for stochastic gradient descent optimization algorithms”
Arnulf Jentzen, Benno Kuckuck, Ariel Neufeld and Philippe von Wurstemberger · 2018
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Arnulf Jentzen, Diyora Salimova and Timo Welti · 2018
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“The universal approximation power of finite-width deep ReLU networks”
Dmytro Perekrestenko, Philipp Grohs, Dennis Elbrächter and Helmut Bölcskei · 2018
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“Topological properties of the set of functions generated by neural networks of fixed size”
Philipp Petersen, Mones Raslan and Felix Voigtlaender · 2018
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“Equivalence of approximation by convolutional neural networks and fully-connected networks”
Philipp Petersen and Felix Voigtlaender · 2018
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“Optimal approximation of piecewise smooth functions using deep ReLU neural networks”
Philipp Petersen and Felix Voigtlaender · 2018
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“DGM: A deep learning algorithm for solving partial differential equations”
Justin Sirignano and Konstantinos Spiliopoulos · 2018
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“Universal approximations of invariant maps by neural networks”
Dmitry Yarotsky · 2018
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“Fine-Grained Analysis of Optimization and Generalization for Overparameterized Two-Layer Neural Networks”
Sanjeev Arora, Simon Du, Wei Hu, Zhiyuan Li and Ruosong Wang · 2019
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“Machine Learning Approximation Algorithms for High-Dimensional Fully Nonlinear Partial Differential Equations and Second-order Backward Stochastic Differential Equations”
Christian Beck, Weinan E and Arnulf Jentzen · 2019
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“Optimal approximation with sparsely connected deep neural networks”
Helmut Bölcskei, Philipp Grohs, Gitta Kutyniok and Philipp Petersen · 2019
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“General multilevel adaptations for stochastic approximation algorithms of Robbins-Monro and Polyak-Ruppert type”
Steffen Dereich and Thomas Müller-Gronbach · 2019
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“Gradient Descent Finds Global Minima of Deep Neural Networks”
Simon Du, Jason Lee, Haochuan Li, Liwei Wang and Xiyu Zhai · 2019
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“A priori estimates of the population risk for two-layer neural networks”
Weinan E, Chao Ma and Lei Wu · 2019
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“New error bounds for deep ReLU networks using sparse grids”
Hadrien Montanelli and Qiang Du · 2019
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“Deep learning in high dimension: neural network expression rates for generalized polynomial chaos expansions in UQ”
Christoph Schwab and Jakob Zech · 2019
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“Exponential Convergence Time of Gradient Descent for One-Dimensional Deep Linear Neural Networks”
Ohad Shamir · 2019
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“Nonlinear approximation via compositions”
Zuowei Shen, Haizhao Yang and Shijun Zhang · 2019
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“Fast Convergence of Natural Gradient Descent for Over-Parameterized Neural Networks”
Guodong Zhang, James Martens and Roger Grosse · 2019
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“Gradient descent optimizes over-parameterized deep ReLU networks”
Difan Zou, Yuan Cao, Dongruo Zhou and Quanquan Gu · 2019
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“A comparative analysis of optimization and generalization properties of two-layer neural network and random feature models under gradient descent dynamics”
Weinan E, Chao Ma and Lei Wu · 2020
Closest in time.
“Lower error bounds for the stochastic gradient descent optimization algorithm: Sharp convergence rates for slowly and fast decaying learning rates”
Arnulf Jentzen and Philippe von Wurstemberger · 2020
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