  In this paper, we consider the joint task of simultaneously optimizing (i)
the weights of a deep neural network, (ii) the number of neurons for each
hidden layer, and (iii) the subset of active input features (i.e., feature
selection). While these problems are generally dealt with separately, we
present a simple regularized formulation allowing to solve all three of them in
parallel, using standard optimization routines. Specifically, we extend the
group Lasso penalty (originated in the linear regression literature) in order
to impose group-level sparsity on the network's connections, where each group
is defined as the set of outgoing weights from a unit. Depending on the
specific case, the weights can be related to an input variable, to a hidden
neuron, or to a bias unit, thus performing simultaneously all the
aforementioned tasks in order to obtain a compact network. We perform an
extensive experimental evaluation, by comparing with classical weight decay and
Lasso penalties. We show that a sparse version of the group Lasso penalty is
able to achieve competitive performances, while at the same time resulting in
extremely compact networks with a smaller number of input features. We evaluate
both on a toy dataset for handwritten digit recognition, and on multiple
realistic large-scale classification problems.
