  We analyze the dynamics of streaming stochastic gradient descent (SGD) in the
high-dimensional limit when applied to generalized linear models and
multi-index models (e.g. logistic regression, phase retrieval) with general
data-covariance. In particular, we demonstrate a deterministic equivalent of
SGD in the form of a system of ordinary differential equations that describes a
wide class of statistics, such as the risk and other measures of
sub-optimality. This equivalence holds with overwhelming probability when the
model parameter count grows proportionally to the number of data. This
framework allows us to obtain learning rate thresholds for stability of SGD as
well as convergence guarantees. In addition to the deterministic equivalent, we
introduce an SDE with a simplified diffusion coefficient (homogenized SGD)
which allows us to analyze the dynamics of general statistics of SGD iterates.
Finally, we illustrate this theory on some standard examples and show numerical
simulations which give an excellent match to the theory.
