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We analyze the dynamics of streaming stochastic gradient descent (SGD) in thehigh-dimensional limit when applied to generalized linear models andmulti-index models (e.g. logistic regression, phase retrieval) with generaldata-covariance. In particular, we demonstrate a deterministic equivalent ofSGD in the form of a system of ordinary differential equations that describes awide class of statistics, such as the risk and other measures ofsub-optimality. This equivalence holds with overwhelming probability when themodel parameter count grows proportionally to the number of data. Thisframework allows us to obtain learning rate thresholds for stability of SGD aswell as convergence guarantees. In addition to the deterministic equivalent, weintroduce 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 numericalsimulations which give an excellent match to the theory.

Hitting the High-Dimensional Notes: An ODE for SGD learning dynamics on GLMs and multi-index models