Non-asymptotic Error Bounds in W2-Distance with Sqrt(d) Dimension Dependence and First Order Convergence for Langevin Monte Carlo beyond Log-Concavity

Efficient sampling from complex and high dimensional target distributions turns out to be a fundamental task in diverse disciplines such as scientific computing, statistics and machine learning. In this paper, we propose a new kind of randomized splitting Langevin Monte Carlo (RSLMC) algorithm for sampling from high dimensional distributions without logconcavity. Compared with the existing randomized Langevin Monte Carlo (RLMC), the newly proposed RSLMC algorithm requires less evaluations of gradients and is thus computationally cheaper. Under the gradient Lipschitz condition and the log-Sobolev inequality, we prove a uniform-in-time error bound in W 2 -distance of order O( √ dh) for both RLMC and RSLMC sampling algorithms, which matches the best one in the literature under the logconcavity condition. Moreover, when the gradient of the potential U is non-globally Lipschitz with superlinear growth, new modified R(S)LMC algorithms are introduced and analyzed, with non-asymptotic error bounds established. Numerical examples are finally reported to corroborate the theoretical findings.