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In this paper, we investigate a continuous time version of the StochasticLangevin Monte Carlo method, introduced in [WT11], that incorporates astochastic sampling step inside the traditional over-damped Langevin diffusion.This method is popular in machine learning for sampling posterior distribution.We will pay specific attention in our work to the computational cost in termsof $n$ (the number of observations that produces the posterior distribution),and $d$ (the dimension of the ambient space where the parameter of interest isliving). We derive our analysis in the weakly convex framework, which isparameterized with the help of the Kurdyka-\L ojasiewicz (KL) inequality, thatpermits to handle a vanishing curvature settings, which is far less restrictivewhen compared to the simple strongly convex case. We establish that the finalhorizon of simulation to obtain an $\varepsilon$ approximation (in terms ofentropy) is of the order $( d \log(n)^2 )^{(1+r)^2} [\log^2(\varepsilon^{-1}) +n^2 d^{2(1+r)} \log^{4(1+r)}(n) ]$ with a Poissonian subsampling of parameter$\left(n ( d \log^2(n))^{1+r}\right)^{-1}$, where the parameter $r$ is involvedin the KL inequality and varies between $0$ (strongly convex case) and $1$(limiting Laplace situation).

Stochastic Langevin Monte Carlo for (weakly) log-concave posterior distributions