  We propose a new framework, inspired by random matrix theory, for analyzing
the dynamics of stochastic gradient descent (SGD) when both number of samples
and dimensions are large. This framework applies to any fixed stepsize and the
finite sum setting. Using this new framework, we show that the dynamics of SGD
on a least squares problem with random data become deterministic in the large
sample and dimensional limit. Furthermore, the limiting dynamics are governed
by a Volterra integral equation. This model predicts that SGD undergoes a phase
transition at an explicitly given critical stepsize that ultimately affects its
convergence rate, which we also verify experimentally. Finally, when input data
is isotropic, we provide explicit expressions for the dynamics and average-case
convergence rates (i.e., the complexity of an algorithm averaged over all
possible inputs). These rates show significant improvement over the worst-case
complexities.
