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In this paper, we provide non-asymptotic upper bounds on the error ofsampling from a target density using three schemes of discretized Langevindiffusions. The first scheme is the Langevin Monte Carlo (LMC) algorithm, theEuler discretization of the Langevin diffusion. The second and the thirdschemes are, respectively, the kinetic Langevin Monte Carlo (KLMC) fordifferentiable potentials and the kinetic Langevin Monte Carlo fortwice-differentiable potentials (KLMC2). The main focus is on the targetdensities that are smooth and log-concave on $\mathbb R^p$, but not necessarilystrongly log-concave. Bounds on the computational complexity are obtained undertwo types of smoothness assumption: the potential has a Lipschitz-continuousgradient and the potential has a Lipschitz-continuous Hessian matrix. The errorof sampling is measured by Wasserstein-$q$ distances. We advocate for the useof a new dimension-adapted scaling in the definition of the computationalcomplexity, when Wasserstein-$q$ distances are considered. The obtained resultsshow that the number of iterations to achieve a scaled-error smaller than aprescribed value depends only polynomially in the dimension.

Bounding the error of discretized Langevin algorithms for non-strongly log-concave targets