  Neural networks with REctified Linear Unit (ReLU) activation functions
(a.k.a. ReLU networks) have achieved great empirical success in various
domains. Nonetheless, existing results for learning ReLU networks either pose
assumptions on the underlying data distribution being e.g. Gaussian, or require
the network size and/or training size to be sufficiently large. In this
context, the problem of learning a two-layer ReLU network is approached in a
binary classification setting, where the data are linearly separable and a
hinge loss criterion is adopted. Leveraging the power of random noise
perturbation, this paper presents a novel stochastic gradient descent (SGD)
algorithm, which can \emph{provably} train any single-hidden-layer ReLU network
to attain global optimality, despite the presence of infinitely many bad local
minima, maxima, and saddle points in general. This result is the first of its
kind, requiring no assumptions on the data distribution, training/network size,
or initialization. Convergence of the resultant iterative algorithm to a global
minimum is analyzed by establishing both an upper bound and a lower bound on
the number of non-zero updates to be performed. Moreover, generalization
guarantees are developed for ReLU networks trained with the novel SGD
leveraging classic compression bounds. These guarantees highlight a key
difference (at least in the worst case) between reliably learning a ReLU
network as well as a leaky ReLU network in terms of sample complexity.
Numerical tests using both synthetic data and real images validate the
effectiveness of the algorithm and the practical merits of the theory.
