  We investigate the test risk of continuous-time stochastic gradient flow
dynamics in learning theory. Using a path integral formulation we provide, in
the regime of a small learning rate, a general formula for computing the
difference between test risk curves of pure gradient and stochastic gradient
flows. We apply the general theory to a simple model of weak features, which
displays the double descent phenomenon, and explicitly compute the corrections
brought about by the added stochastic term in the dynamics, as a function of
time and model parameters. The analytical results are compared to simulations
of discrete-time stochastic gradient descent and show good agreement.
