  The stochastic gradient Langevin Dynamics is one of the most fundamental
algorithms to solve sampling problems and non-convex optimization appearing in
several machine learning applications. Especially, its variance reduced
versions have nowadays gained particular attention. In this paper, we study two
variants of this kind, namely, the Stochastic Variance Reduced Gradient
Langevin Dynamics and the Stochastic Recursive Gradient Langevin Dynamics. We
prove their convergence to the objective distribution in terms of KL-divergence
under the sole assumptions of smoothness and Log-Sobolev inequality which are
weaker conditions than those used in prior works for these algorithms. With the
batch size and the inner loop length set to $\sqrt{n}$, the gradient complexity
to achieve an $\epsilon$-precision is
$\tilde{O}((n+dn^{1/2}\epsilon^{-1})\gamma^2 L^2\alpha^{-2})$, which is an
improvement from any previous analyses. We also show some essential
applications of our result to non-convex optimization.
