  Overparameterization refers to the important phenomenon where the width of a
neural network is chosen such that learning algorithms can provably attain zero
loss in nonconvex training. The existing theory establishes such global
convergence using various initialization strategies, training modifications,
and width scalings. In particular, the state-of-the-art results require the
width to scale quadratically with the number of training data under standard
initialization strategies used in practice for best generalization performance.
In contrast, the most recent results obtain linear scaling either with
requiring initializations that lead to the "lazy-training", or training only a
single layer. In this work, we provide an analytical framework that allows us
to adopt standard initialization strategies, possibly avoid lazy training, and
train all layers simultaneously in basic shallow neural networks while
attaining a desirable subquadratic scaling on the network width. We achieve the
desiderata via Polyak-Lojasiewicz condition, smoothness, and standard
assumptions on data, and use tools from random matrix theory.
