Learning ReLU Networks on Linearly Separable Data: Algorithm, Optimality, and Generalization

Provably escape local minima by means of noise-injected SGD in training single-hidden-layer ReLU networks!

Learning ReLU Networks on Linearly Separable Data: Algorithm, Optimality, and Generalization.

Authors: G. Wang, G. B. Giannakis, and J. Chen

Neural networks with REctified Linear Unit (ReLU) activations have achieved great empirical success in various domains. However, existing results for learning ReLU networks either pose assumptions on the underlying data distribution being e.g. Gaussian, or require the network size andor 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, this paper presents a novel stochastic gradient descent (SGD) algorithm, which can 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, trainingnetwork 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. 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.