Acceleration and Averaging in Stochastic Mirror Descent Dynamics
This provides theoretical analysis for stochastic optimization algorithms, though it appears to be an incremental extension of existing accelerated mirror descent methods to stochastic continuous-time settings.
The authors tackled the problem of accelerated first-order minimization of smooth convex functions under stochastic gradient noise by proposing a continuous-time stochastic variant of accelerated mirror descent. They proved convergence rate bounds for function values under both persistent and vanishing noise, showing how noise covariation affects parameter choices and convergence rates.
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.