  While over-parameterization is widely believed to be crucial for the success
of optimization for the neural networks, most existing theories on
over-parameterization do not fully explain the reason -- they either work in
the Neural Tangent Kernel regime where neurons don't move much, or require an
enormous number of neurons. In practice, when the data is generated using a
teacher neural network, even mildly over-parameterized neural networks can
achieve 0 loss and recover the directions of teacher neurons. In this paper we
develop a local convergence theory for mildly over-parameterized two-layer
neural net. We show that as long as the loss is already lower than a threshold
(polynomial in relevant parameters), all student neurons in an
over-parameterized two-layer neural network will converge to one of teacher
neurons, and the loss will go to 0. Our result holds for any number of student
neurons as long as it is at least as large as the number of teacher neurons,
and our convergence rate is independent of the number of student neurons. A key
component of our analysis is the new characterization of local optimization
landscape -- we show the gradient satisfies a special case of Lojasiewicz
property which is different from local strong convexity or PL conditions used
in previous work.
