  We analyze the dynamics of large batch stochastic gradient descent with
momentum (SGD+M) on the least squares problem when both the number of samples
and dimensions are large. In this setting, we show that the dynamics of SGD+M
converge to a deterministic discrete Volterra equation as dimension increases,
which we analyze. We identify a stability measurement, the implicit
conditioning ratio (ICR), which regulates the ability of SGD+M to accelerate
the algorithm. When the batch size exceeds this ICR, SGD+M converges linearly
at a rate of $\mathcal{O}(1/\sqrt{\kappa})$, matching optimal full-batch
momentum (in particular performing as well as a full-batch but with a fraction
of the size). For batch sizes smaller than the ICR, in contrast, SGD+M has
rates that scale like a multiple of the single batch SGD rate. We give explicit
choices for the learning rate and momentum parameter in terms of the Hessian
spectra that achieve this performance.
