Understanding the role of (stochastic) gradient descent (SGD) in the training and generalisation of deep neural networks (DNNs) with ReLU activation has been the object study in the recent past. In this paper, we make use of deep gated networks (DGNs) as a framework to obtain insights about DNNs with ReLU activation. In DGNs, a single neuronal unit has two components namely the pre-activation input (equal to the inner product the weights of the layer and the previous layer outputs), and a gating value which belongs to $[0,1]$ and the output of the neuronal unit is equal to the multiplication of pre-activation input and the gating value. The standard DNN with ReLU activation, is a special case of the DGNs, wherein the gating value is $1/0$ based on whether or not the pre-activation input is positive or negative. We theoretically analyse and experiment with several variants of DGNs, each variant suited to understand a particular aspect of either training or generalisation in DNNs with ReLU activation. Our theory throws light on two questions namely i) why increasing depth till a point helps in training and ii) why increasing depth beyond a point hurts training? We also present experimental evidence to show that gate adaptation, i.e., the change of gating value through the course of training is key for generalisation.

ReLU neural-networks have been in the focus of many recent theoretical works, trying to explain their empirical success. Nonetheless, there is still a gap between current theoretical results and empirical observations, even in the case of shallow (one hidden-layer) networks. For example, in the task of memorizing a random sample of size $m$ and dimension $d$, the best theoretical result requires the size of the network to be $\tilde{\Omega}(\frac{m^2}{d})$, while empirically a network of size slightly larger than $\frac{m}{d}$ is sufficient. To bridge this gap, we turn to study a simplified model for ReLU networks. We observe that a ReLU neuron is a product of a linear function with a gate (the latter determines whether the neuron is active or not), where both share a jointly trained weight vector. In this spirit, we introduce the Gated Linear Unit (GaLU), which simply decouples the linearity from the gating by assigning different vectors for each role. We show that GaLU networks allow us to get optimization and generalization results that are much stronger than those available for ReLU networks. Specifically, we show a memorization result for networks of size $\tilde{\Omega}(\frac{m}{d})$, and improved generalization bounds. Finally, we show that in some scenarios, GaLU networks behave similarly to ReLU networks, hence proving to be a good choice of a simplified model.