  We consider the optimization problem associated with fitting two-layer ReLU
networks with respect to the squared loss, where labels are assumed to be
generated by a target network. Focusing first on standard Gaussian inputs, we
show that the structure of spurious local minima detected by stochastic
gradient descent (SGD) is, in a well-defined sense, the \emph{least loss of
symmetry} with respect to the target weights. A closer look at the analysis
indicates that this principle of least symmetry breaking may apply to a broader
range of settings. Motivated by this, we conduct a series of experiments which
corroborate this hypothesis for different classes of non-isotropic non-product
distributions, smooth activation functions and networks with a few layers.
