Title: Linear Convergence of SGD on Overparametrized Shallow Neural Networks. Abstract: Despite the non-convex landscape, first-order methods can be shown to reach global minima when training overparameterized neural networks, where the number of parameters far exceed the number of training data. In this work, we prove linear convergence of stochastic gradient descent when training a two-layer neural network with smooth activations. While the existing theory either requires a high degree of overparameterization or non-standard initialization and training strategies, e.g., training only a single layer, we show that a subquadratic scaling on the width is sufficient under standard initialization and training both layers simultaneously if the minibatch size is sufficiently large and it also grows with the number of training examples. Via the batch size, our results interpolate between the state-of-the-art subquadratic results for gradient descent and the quadratic results in the worst case.