How does stochastic gradient descent converge to a continuous stochastic differential equation?

SGD converges in the continuous limit to a stochastic differential equation (SDE) resembling Langevin dynamics.

When the learning rate becomes infinitesimally small and batch sampling introduces randomness, SGD dynamics approximate a continuous-time stochastic process. The discrete updates become a drift term (negative gradient of loss) plus a diffusion term (gradient noise). This SDE interpretation explains why SGD can escape sharp minima and why noise is not purely harmful but structurally important to exploration of the loss landscape.