Symmetry & critical points for a model shallow neural network

We consider the optimization problem associated with fitting two-layer ReLU networks with $k$ neurons over $k$-dimensional input, where labels are assumed to be generated by a target network. We leverage the rich symmetry exhibited by such models to identify various families of critical points and express them as infinite series in $1/\sqrt{k}$. These expressions are then used to derive estimates for several related quantities which imply that not all spurious minima are alike. For example, we show that while the loss function at certain types of spurious minima decays to zero as $O(k^{-1})$, in other cases the loss converges to a strictly positive constant. The methods used depend on symmetry breaking, bifurcation, and algebraic geometry, notably Artin's implicit function theorem.