### Abstract

Original language | English |
---|---|

Article number | 2002.02247v1 |

Number of pages | 20 |

Journal | arXiv |

Publication status | Published - 6 Feb 2020 |

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### Bibliographical note

20 pages, 2 figures### Keywords

- math.OC
- cs.LG
- math.PR

### Cite this

*arXiv*, [2002.02247v1].

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**Almost sure convergence of dropout algorithms for neural networks.** / Senen-Cerda, Albert; Sanders, Jaron.

Research output: Contribution to journal › Article › Academic

TY - JOUR

T1 - Almost sure convergence of dropout algorithms for neural networks

AU - Senen-Cerda, Albert

AU - Sanders, Jaron

N1 - 20 pages, 2 figures

PY - 2020/2/6

Y1 - 2020/2/6

N2 - We investigate the convergence and convergence rate of stochastic training algorithms for Neural Networks (NNs) that, over the years, have spawned from Dropout (Hinton et al., 2012). Modeling that neurons in the brain may not fire, dropout algorithms consist in practice of multiplying the weight matrices of a NN component-wise by independently drawn random matrices with $\{0,1\}$-valued entries during each iteration of the Feedforward-Backpropagation algorithm. This paper presents a probability theoretical proof that for any NN topology and differentiable polynomially bounded activation functions, if we project the NN's weights into a compact set and use a dropout algorithm, then the weights converge to a unique stationary set of a projected system of Ordinary Differential Equations (ODEs). We also establish an upper bound on the rate of convergence of Gradient Descent (GD) on the limiting ODEs of dropout algorithms for arborescences (a class of trees) of arbitrary depth and with linear activation functions.

AB - We investigate the convergence and convergence rate of stochastic training algorithms for Neural Networks (NNs) that, over the years, have spawned from Dropout (Hinton et al., 2012). Modeling that neurons in the brain may not fire, dropout algorithms consist in practice of multiplying the weight matrices of a NN component-wise by independently drawn random matrices with $\{0,1\}$-valued entries during each iteration of the Feedforward-Backpropagation algorithm. This paper presents a probability theoretical proof that for any NN topology and differentiable polynomially bounded activation functions, if we project the NN's weights into a compact set and use a dropout algorithm, then the weights converge to a unique stationary set of a projected system of Ordinary Differential Equations (ODEs). We also establish an upper bound on the rate of convergence of Gradient Descent (GD) on the limiting ODEs of dropout algorithms for arborescences (a class of trees) of arbitrary depth and with linear activation functions.

KW - math.OC

KW - cs.LG

KW - math.PR

M3 - Article

JO - arXiv

JF - arXiv

M1 - 2002.02247v1

ER -