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Register a new userCharacteristics of Monte Carlo Dropout in Wide Neural Networks
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Added by
Maram Akila
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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Monte Carlo (MC) dropout is one of the state-of-the-art approaches for uncertainty estimation in neural networks (NNs). It has been interpreted as approximately performing Bayesian inference. Based on previous work on the approximation of Gaussian processes by wide and deep neural networks with random weights, we study the limiting distribution of wide untrained NNs under dropout more rigorously and prove that they as well converge to Gaussian processes for fixed sets of weights and biases. We sketch an argument that this property might also hold for infinitely wide feed-forward networks that are trained with (full-batch) gradient descent. The theory is contrasted by an empirical analysis in which we find correlations and non-Gaussian behaviour for the pre-activations of finite width NNs. We therefore investigate how (strongly) correlated pre-activations can induce non-Gaussian behavior in NNs with strongly correlated weights.
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We prove two universal approximation theorems for a range of dropout neural networks. These are feed-forward neural networks in which each edge is given a random ${0,1}$-valued filter, that have two modes of operation: in the first each edge output is multiplied by its random filter, resulting in a random output, while in the second each edge output is multiplied by the expectation of its filter, leading to a deterministic output. It is common to use the random mode during training and the deterministic mode during testing and prediction. Both theorems are of the following form: Given a function to approximate and a threshold $varepsilon>0$, there exists a dropout network that is $varepsilon$-close in probability and in $L^q$. The first theorem applies to dropout networks in the random mode. It assumes little on the activation function, applies to a wide class of networks, and can even be applied to approximation schemes other than neural networks. The core is an algebraic property that shows that deterministic networks can be exactly matched in expectation by random networks. The second theorem makes stronger assumptions and gives a stronger result. Given a function to approximate, it provides existence of a network that approximates in both modes simultaneously. Proof components are a recursive replacement of edges by independent copies, and a special first-layer replacement that couples the resulting larger network to the input. The functions to be approximated are assumed to be elements of general normed spaces, and the approximations are measured in the corresponding norms. The networks are constructed explicitly. Because of the different methods of proof, the two results give independent insight into the approximation properties of random dropout networks. With this, we establish that dropout neural networks broadly satisfy a universal-approximation property.
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.
Overparametrized neural networks, where the number of active parameters is larger than the sample size, prove remarkably effective in modern deep learning practice. From the classical perspective, however, much fewer parameters are sufficient for optimal estimation and prediction, whereas overparametrization can be harmful even in the presence of explicit regularization. To reconcile this conflict, we present a generalization theory for overparametrized ReLU networks by incorporating an explicit regularizer based on the scaled variation norm. Interestingly, this regularizer is equivalent to the ridge from the angle of gradient-based optimization, but is similar to the group lasso in terms of controlling model complexity. By exploiting this ridge-lasso duality, we show that overparametrization is generally harmless to two-layer ReLU networks. In particular, the overparametrized estimators are minimax optimal up to a logarithmic factor. By contrast, we show that overparametrized random feature models suffer from the curse of dimensionality and thus are suboptimal.
Due to complex experimental settings, missing values are common in biomedical data. To handle this issue, many methods have been proposed, from ignoring incomplete instances to various data imputation approaches. With the recent rise of deep neural networks, the field of missing data imputation has oriented towards modelling of the data distribution. This paper presents an approach based on Monte Carlo dropout within (Variational) Autoencoders which offers not only very good adaptation to the distribution of the data but also allows generation of new data, adapted to each specific instance. The evaluation shows that the imputation error and predictive similarity can be improved with the proposed approach.
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