, listed below, that should not be difficult to fix in the final version of the paper. - in the comparison with the prior work, the authors do not present the latest results. Therefore, the comparison embellishes the contributions of this paper. Indeed, in table 1, in the row corresponding to the Langevin diffusion, the authors should present the result of "Analysis of Langevin Monte Carlo via convex optimization", which improves on [13,17]. In addition, for the Underdamped Langevin Diffusion, in the table,in the abstract and in the text, it would be more appropriate to compare with [15], which provides an improved bound (kappa^1.5 instead of kappa^2). - The authors do not keep track of numerical constants. I would very much appreciate to see results with precise values of constants, as it is done in many recent references cited by the authors. SPECIFIC REMARKS/TYPOS - Line 9: compare with [15] instead of [10] - Line 14: applies -> apply - Line 16: problems -> problem - Lines 32-33: this lines oversell the contribution (which is not needed). If one uses the methodology of evaluating a method adopted by the authors, that I find fully justified, the current best algorithm is independent of d. By the way, the sentence on lines 32-33 contradicts the content of table 1. - Lines 44-45: when mentioning ``the current fastest algorithm'', it might be helpful to cite [15] as well. - Lines 106-107 - same remark - Line 131: depend -> depends - Line 132: I guess Omega(d^1.5) should be replaced by Omega(d^2) - Line 136: I suggest to use another letter than f, which refers to a precise function (the potential) - Lines 170-171: [18,15] -> [18,13] - Lines 180-181: the meaning of "accurate" is not clear here. My suggestion is to remove any comment about accuracy here and to keep only the comment on unbiasedness. - Line 191: please emphasize that the distribution is Gaussian conditionnally to alpha. - Lemma 2: it should be clearly mentioned that these claims are true when u=1/L - Lemma 2: I suggest to replace v_n(0) and x_n(0) by v_n and x_n - Line 201: same remark - Section A: everything is conditional to alpha - Lemma 6: mention that u=1/L - Line 400: Schwarz is mispelled
- Line 191: please emphasize that the distribution is Gaussian conditionnally to alpha.