Paper: Langevin Monte Carlo for strongly log-concave distributions: Randomized midpoint revisited (ICLR-2024)

Reviewer comment (Evaluation):
Some of the results are potentially still not optimal as the authors also suggest.

Author response:
While our bound offers the best-known dependence on the condition number for Kinetic Langevin Monte Carlo, it's essential to note that the optimal dependence remains an open question.

Reviewer comment (Experiments):
It would also be nice to see how the improved bounds can be applied to some more concrete problems even for theoretical results.

Author response:
The primary focus of our work is to present a thorough and refined theoretical analysis of midpoint randomization applied to Langevin Monte Carlo. While we acknowledge that demonstrating a more concrete and tangible real-world application could enhance the practical significance of our findings, delving into the applied aspects of midpoint randomization presents challenges. Factors such as the calculation of Wasserstein distance in high dimension, measurement of the distributions of the samples and the true underlying distribution, estimation of smoothness and strong convexity parameters, etc., may require extensive efforts. Consequently, we regard this avenue as a direction for future research.

Reviewer comment (Writing):
Minor comment - Page 2, line 8: “strongly” -> “strong”.

Author response:
We thank the reviewer for pointing out the typos, and we corrected them in the revision of the paper.

Reviewer comment (Writing):
Minor comment - Page 2, line 22: “at” -> “a”.

Author response:
We thank the reviewer for pointing out the typos, and we corrected them in the revision of the paper.

Reviewer comment (Writing):
Minor comment - Page 2, “notation” paragraph, line 4: “semi-definite positive” -> “positive semi-definite”.

Author response:
We thank the reviewer for pointing out the typos, and we corrected them in the revision of the paper.

Reviewer comment (Writing):
Minor comment - Page 6, line 5: “designatex” -> “designate”.

Author response:
We thank the reviewer for pointing out the typos, and we corrected them in the revision of the paper.

Reviewer comment (Experiments):
Do you have an example application of the improved analysis to get better results for a specific problem?

Author response:
The primary focus of our work is to present a thorough and refined theoretical analysis of midpoint randomization applied to Langevin Monte Carlo. While we acknowledge that demonstrating a more concrete and tangible real-world application could enhance the practical significance of our findings, delving into the applied aspects of midpoint randomization presents challenges. Factors such as the calculation of Wasserstein distance in high dimension, measurement of the distributions of the samples and the true underlying distribution, estimation of smoothness and strong convexity parameters, etc., may require extensive efforts. Consequently, we regard this avenue as a direction for future research.