  The step function is one of the simplest and most natural activation
functions for deep neural networks (DNNs). As it counts 1 for positive
variables and 0 for others, its intrinsic characteristics (e.g., discontinuity
and no viable information of subgradients) impede its development for several
decades. Even if there is an impressive body of work on designing DNNs with
continuous activation functions that can be deemed as surrogates of the step
function, it is still in the possession of some advantageous properties, such
as complete robustness to outliers and being capable of attaining the best
learning-theoretic guarantee of predictive accuracy. Hence, in this paper, we
aim to train DNNs with the step function used as an activation function (dubbed
as 0/1 DNNs). We first reformulate 0/1 DNNs as an unconstrained optimization
problem and then solve it by a block coordinate descend (BCD) method. Moreover,
we acquire closed-form solutions for sub-problems of BCD as well as its
convergence properties. Furthermore, we also integrate
$\ell_{2,0}$-regularization into 0/1 DNN to accelerate the training process and
compress the network scale. As a result, the proposed algorithm has a high
performance on classifying MNIST and Fashion-MNIST datasets. As a result, the
proposed algorithm has a desirable performance on classifying MNIST,
FashionMNIST, Cifar10, and Cifar100 datasets.
