  We formulate and study a general family of (continuous-time) stochastic
dynamics for accelerated first-order minimization of smooth convex functions.
Building on an averaging formulation of accelerated mirror descent, we propose
a stochastic variant in which the gradient is contaminated by noise, and study
the resulting stochastic differential equation. We prove a bound on the rate of
change of an energy function associated with the problem, then use it to derive
estimates of convergence rates of the function values, (a.s. and in
expectation) both for persistent and asymptotically vanishing noise. We discuss
the interaction between the parameters of the dynamics (learning rate and
averaging weights) and the covariation of the noise process, and show, in
particular, how the asymptotic rate of covariation affects the choice of
parameters and, ultimately, the convergence rate.
