  We consider the problem of learning a one-hidden-layer neural network with
non-overlapping convolutional layer and ReLU activation, i.e., $f(\mathbf{Z},
\mathbf{w}, \mathbf{a}) = \sum_j a_j\sigma(\mathbf{w}^T\mathbf{Z}_j)$, in which
both the convolutional weights $\mathbf{w}$ and the output weights $\mathbf{a}$
are parameters to be learned. When the labels are the outputs from a teacher
network of the same architecture with fixed weights $(\mathbf{w}^*,
\mathbf{a}^*)$, we prove that with Gaussian input $\mathbf{Z}$, there is a
spurious local minimizer. Surprisingly, in the presence of the spurious local
minimizer, gradient descent with weight normalization from randomly initialized
weights can still be proven to recover the true parameters with constant
probability, which can be boosted to probability $1$ with multiple restarts. We
also show that with constant probability, the same procedure could also
converge to the spurious local minimum, showing that the local minimum plays a
non-trivial role in the dynamics of gradient descent. Furthermore, a
quantitative analysis shows that the gradient descent dynamics has two phases:
it starts off slow, but converges much faster after several iterations.
