We have submitted *Almost Sure Convergence of Dropout Algorithms for Neural Networks*, and it is currently under review. This is joint work between Albert Senen-Cerda and myself. A preprint is available on arXiv.

## Dropout algorithms, neural networks

Our manuscript investigates mathematically the convergence properties of a class of well-known and often used training algorithms in neural networks. The techniques used lie in the domains of stochastic approximation, and nonconvex optimization. The mathematical analysis of neural networks is highly interesting, topical, and challenging.

## Abstract

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.

## Preprint

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