Asymptotic convergence rate of Dropout on shallow linear neural networks
This work provides theoretical insights into the convergence behavior of Dropout for researchers and practitioners using regularization techniques in neural networks, specifically for shallow linear architectures.
This paper analyzes the convergence rate of gradient flows for Dropout and Dropconnect in shallow linear neural networks, which can be viewed as matrix factorization with a specific regularizer. The authors provide a local convergence proof and a bound on the convergence rate that depends on data, dropout probability, and network width, which qualitatively matches numerical simulations.
We analyze the convergence rate of gradient flows on objective functions induced by Dropout and Dropconnect, when applying them to shallow linear Neural Networks (NNs) - which can also be viewed as doing matrix factorization using a particular regularizer. Dropout algorithms such as these are thus regularization techniques that use 0,1-valued random variables to filter weights during training in order to avoid coadaptation of features. By leveraging a recent result on nonconvex optimization and conducting a careful analysis of the set of minimizers as well as the Hessian of the loss function, we are able to obtain (i) a local convergence proof of the gradient flow and (ii) a bound on the convergence rate that depends on the data, the dropout probability, and the width of the NN. Finally, we compare this theoretical bound to numerical simulations, which are in qualitative agreement with the convergence bound and match it when starting sufficiently close to a minimizer.