  The diffusion approximation of stochastic gradient descent (SGD) in current
literature is only valid on a finite time interval. In this paper, we establish
the uniform-in-time diffusion approximation of SGD, by only assuming that the
expected loss is strongly convex and some other mild conditions, without
assuming the convexity of each random loss function. The main technique is to
establish the exponential decay rates of the derivatives of the solution to the
backward Kolmogorov equation. The uniform-in-time approximation allows us to
study asymptotic behaviors of SGD via the continuous stochastic differential
equation (SDE) even when the random objective function $f(\cdot;\xi)$ is not
strongly convex.
