**Abstract: **Despite the development of many new optimization techniques, constant

step-size stochastic gradient descent (SGD) remains the tool of choice

in modern machine learning. It performs unreasonably well in practice,

both in convex settings where there are poor guarantees, and also in

non-convex settings, where there are often no guarantees. In this

talk, we will give an argument as to why it works for low-rank models,

illustrating the argument for the particular case of phase retrieval.

Phase retrieval is the problem of solving systems of rank-1 quadratic

equations, and can be solved by optimizing a non-convex, non-smooth

objective. In order to obtain guarantees, however, a spectral

initialization method is first used in order to get a constant error

estimate. We will show how SGD works even from arbitrary

initialization. The key insight is that the low-rank nature of the

signal allows us to study the dynamics as a Markov process with a

two-dimensional phase space, and in this phase space, the effective

step size decreases as the dimension increases. Our analysis also

introduces and makes use of several new ideas from stochastic process

theory, which we believe will be of use much more broadly.