On the ISS Property of the Gradient Flow for Single Hidden-Layer Neural
  Networks with Linear Activations
Arthur Castello B. de Oliveira Milad Siami Eduardo D. Sontag
Recent research in neural networks and machine learning suggests that using
many more parameters than strictly required by the initial complexity of a
regression problem can result in more accurate or faster-converging models --
contrary to classical statistical belief. This phenomenon, sometimes known as
``benign overfitting'', raises questions regarding in what other ways might
overparameterization affect the properties of a learning problem. In this work,
we investigate the effects of overfitting on the robustness of gradient-descent
training when subject to uncertainty on the gradient estimation. This
uncertainty arises naturally if the gradient is estimated from noisy data or
directly measured. Our object of study is a linear neural network with a
single, arbitrarily wide, hidden layer and an arbitrary number of inputs and
outputs. In this paper we solve the problem for the case where the input and
output of our neural-network are one-dimensional, deriving sufficient
conditions for robustness of our system based on necessary and sufficient
conditions for convergence in the undisturbed case. We then show that the
general overparametrized formulation introduces a set of spurious equilibria
which lay outside the set where the loss function is minimized, and discuss
directions of future work that might extend our current results for more
general formulations.