Non-asymptotic convergence analysis for the Unadjusted Langevin Algorithm

Understand

In this paper, we study a method to sample from a target distribution $\pi$ over $\mathbb{R}^d$ having a positive density with respect to the Lebesgue measure, known up to a normalisation factor.

• This method is based on the Euler discretization of the overdamped Langevin stochastic differential equation associated with $\pi$.
• For both constant and decreasing step sizes in the Euler discretization, we obtain non-asymptotic bounds for the convergence to the target distribution $\pi$ in total variation distance.
• A particular attention is paid to the dependency on the dimension $d$, to demonstrate the applicability of this method in the high dimensional setting.