  We prove that two-layer (Leaky)ReLU networks initialized by e.g. the widely
used method proposed by He et al. (2015) and trained using gradient descent on
a least-squares loss are not universally consistent. Specifically, we describe
a large class of one-dimensional data-generating distributions for which, with
high probability, gradient descent only finds a bad local minimum of the
optimization landscape, since it is unable to move the biases far away from
their initialization at zero. It turns out that in these cases, the found
network essentially performs linear regression even if the target function is
non-linear. We further provide numerical evidence that this happens in
practical situations, for some multi-dimensional distributions and that
stochastic gradient descent exhibits similar behavior. We also provide
empirical results on how the choice of initialization and optimizer can
influence this behavior.
