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Register a new userOn the effect of the activation function on the distribution of hidden nodes in a deep network
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Added by
Hanie Sedghi
Publication date
2019
fields
Informatics Engineering
and research's language is
English
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We analyze the joint probability distribution on the lengths of the vectors of hidden variables in different layers of a fully connected deep network, when the weights and biases are chosen randomly according to Gaussian distributions, and the input is in ${ -1, 1}^N$. We show that, if the activation function $phi$ satisfies a minimal set of assumptions, satisfied by all activation functions that we know that are used in practice, then, as the width of the network gets large, the `length process converges in probability to a length map that is determined as a simple function of the variances of the random weights and biases, and the activation function $phi$. We also show that this convergence may fail for $phi$ that violate our assumptions.
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Recent work on the representation of functions on sets has considered the use of summation in a latent space to enforce permutation invariance. In particular, it has been conjectured that the dimension of this latent space may remain fixed as the cardinality of the sets under consideration increases. However, we demonstrate that the analysis leading to this conjecture requires mappings which are highly discontinuous and argue that this is only of limited practical use. Motivated by this observation, we prove that an implementation of this model via continuous mappings (as provided by e.g. neural networks or Gaussian processes) actually imposes a constraint on the dimensionality of the latent space. Practical universal function representation for set inputs can only be achieved with a latent dimension at least the size of the maximum number of input elements.
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Dropout is a simple but effective technique for learning in neural networks and other settings. A sound theoretical understanding of dropout is needed to determine when dropout should be applied and how to use it most effectively. In this paper we continue the exploration of dropout as a regularizer pioneered by Wager, et.al. We focus on linear classification where a convex proxy to the misclassification loss (i.e. the logistic loss used in logistic regression) is minimized. We show: (a) when the dropout-regularized criterion has a unique minimizer, (b) when the dropout-regularization penalty goes to infinity with the weights, and when it remains bounded, (c) that the dropout regularization can be non-monotonic as individual weights increase from 0, and (d) that the dropout regularization penalty may not be convex. This last point is particularly surprising because the combination of dropout regularization with any convex loss proxy is always a convex function. In order to contrast dropout regularization with $L_2$ regularization, we formalize the notion of when different sources are more compatible with different regularizers. We then exhibit distributions that are provably more compatible with dropout regularization than $L_2$ regularization, and vice versa. These sources provide additional insight into how the inductive biases of dropout and $L_2$ regularization differ. We provide some similar results for $L_1$ regularization.
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