Theoretical guarantees for stochastic gradient sampling methods via Gaussian convolution inequalities
For researchers in MCMC and sampling methods, this work offers tighter theoretical guarantees for stochastic gradient sampling, addressing a previously open problem.
The paper provides first-order bounds on the bias of stochastic gradient kinetic Langevin dynamics in Wasserstein distances, resolving an open problem on invariant measure accuracy. The bounds sharpen existing non-asymptotic guarantees for stochastic-gradient MCMC methods.
We derive first-order (in the stepsize) bounds on the bias in Wasserstein distances of the invariant measure of stochastic gradient kinetic Langevin dynamics with minimal assumptions on the stochastic gradient noise. These bounds sharpen existing non-asymptotic guarantees for stochastic-gradient MCMC methods and provide a quantitative resolution of a previously open problem on invariant measure accuracy. The main technical ingredients are new Gaussian convolution inequalities controlling the Wasserstein-$p$ distance between a Gaussian convolved with a mean-zero perturbation and the Gaussian itself. We anticipate that these inequalities will be of independent interest beyond the present application.