High-dimensional limit of one-pass SGD on least squares

We give a description of the high-dimensional limit of one-pass single-batch stochastic gradient descent (SGD) on a least squares problem. This limit is taken with non-vanishing step-size, and with proportionally related number of samples to problem-dimensionality. The limit is described in terms of a stochastic differential equation in high dimensions, which is shown to approximate the state evolution of SGD. As a corollary, the statistical risk is shown to be approximated by the solution of a convolution-type Volterra equation with vanishing errors as dimensionality tends to infinity. The sense of convergence is the weakest that shows that statistical risks of the two processes coincide. This is distinguished from existing analyses by the type of high-dimensional limit given as well as generality of the covariance structure of the samples.