# Respsonse to Reviewer DVEg

We deeply appreciate the reviewer's time and valuable comments. Glad to hear the acknowledgement of the generality of our proposed framework and its ease to use.

- Re: have assumptions similiar to our assumption on the 3rd-order derivative of the potential been introduced before?

The exact form of this assumption is new. However, it is related to, for example, the Hessian Lipschitz condition, i.e., $\|\nabla^2 f(\boldsymbol{x}) - \nabla^2f(\boldsymbol{y})\| \leq \tilde{L} \|\boldsymbol{x} - \boldsymbol{y}\|$, used in, e.g., the classical works of (Durmus et al., 2019b) and (Ma et al., 2021). Hessian Lipschitz condition + the standard smoothness assumption -> our assumption (i.e., bounded $\|\nabla (\Delta f)(\boldsymbol{x}) \|$).

We appreciate this question as it indeed helps clarify our novelty and contribution. We will revise accordingly.

- Re: randomized midpoint.

Randomized midpoint is a highly creative discretization of underdamped Langevin and its resulting $\mathcal{O}(d^\frac{1}{3})$ dimension dependence is remarkable. Although randomized midpoint was cited in the original submission, we indeed should add more discussions around line249; will amend. Due to page limit, we chose to present our general framework (mean sq. analysis) and then used LMC discretization of overdamped Langevin as an example application; however, calculation based on our framework seemed to give $\mathcal{O}(d^\frac{1}{3})$ dependence for randomized midpoint as well (due to its 2nd-order strong error), which means our framework is indeed working.

- Re: clarify in what sense the analysis and algorithm is optimal or use the word "optimal" sparingly.

We fully agree. The current results have not given a tight dimension dependence of LMC for the class of smooth and strongly convex potentials, but only for a subclass that also satisfy the 3rd derivative condition. This limitation was stated in the original submission, e.g., line 244 - 248, but we will make it more visible. Fortunately, this extra assumption is not making our subclass much smaller than those considered by the frontier of the literature (kindly see reply to the 1st point above; see also Remark in line 214-219). More precisely, prior to this work, the best known mixing time result for LMC is $\tilde{\mathcal{O}}(\frac{d}{\epsilon^2})$ in W2 distance without any extra assumption, and $\tilde{\mathcal{O}}(\frac{d}{\epsilon})$ with extra assumption such as Hessian Lipschitz, as summarized in Table 1. Our result improves the mixing time bound to $\tilde{\mathcal{O}}(\frac{\sqrt{d}}{\epsilon})$ after replacing the Hessian Lipschitz assumption by a similar one. We thank the reviewer as this comment prompts us to clarify more in a revision, and hopefully now we explained our novelty, contribution, and position in the literature a little better.

# Additional Response to Reviewer DVEg

[comment]: # (Thank you very much for additional discussion. We are glad hear that our earlier responses addressed most of concerns and strengthened generality of our analysis framework.)

[comment]: # (We appreciate the discussion on condition number dependence. Altough dimension dependence is our main focus in this work, condition number dependence can also be of great theoretical and practical value. We fully agree that the current presentation of the framework, when directly applied to the creative randomized midpoint discretization of ULD, may not necessarily yield a comparable condition number dependence as that in Shen and Lee, 2019. However, we feel that it might be possible to customize the current framework to leverage the special properties of randomized midpoint algorithm to obtaitn a similar result, for example, since the local order of bias of RMA is $p_1=4$, and local order of standard deviation is $p_2=2$, the additional gap between $p_1$ and $p_2$, i.e. $p_1 - p_2 - \frac{1}{2} = \frac{3}{2}$ can be used to remove the dependence on $C_1, D_1$ in Eq.8.)

Thank you very much for the reply and additional discussion. We are glad to hear that our earlier responses addressed most of the concerns and strengthened the generality of our analysis framework.

The condition number dependence of randomized midpoint is a very interesting additional question. Despite that our submission chose LMC and its dimension dependence as its focused application, we thank the reviewer for this great scientific problem. We need more time to do the full analysis but believe it is actually nontrivial, especially if one wants a tight condition number dependence. At least for LMC which is our focus application, the condition number dependence has to be tracked very carefully throughout each lemma (on two different types of local error, etc.) and then merged, and in the end we obtained a condition number dependence that is not always worse nor better than the existing results (it is just a different kind of dependence; see Table 1 for more details). Therefore, without detailed calculations for randomized midpoint, we apologize that we cannot confirm the reviewer's suggested bound at this moment, nor do we have any a priori belief that our framework will produce a dependence comparable or better to the remarkable analysis (and results) of [Shen and Lee, 2019].

Nevertheless, here is some spectulation just for scientific discussion purposes: we feel that it might be possible to customize our current framework to leverage the special properties of randomized midpoint to obtain a similar result; for example, since the local order of bias of randomized midpoint is $p_1=4$, and the local order of standard deviation is $p_2=2$, the additional gap between $p_1$ and $p_2$, i.e. $p_1 - p_2 - \frac{1}{2} = \frac{3}{2}$ can be used to remove the dependence on $C_1, D_1$ in Eq.8 of our submission.

In all cases, we hope the expert reviewer could allow us to focus on the general framework + application to LMC in this short conference submission. The suggestion however helps develop future research and is deeply appreciated.

# Response to Reviewer LgVW
We deeply appreciate the reviewer's time and valuable comments.

- Re: Theorem 8 of Durmus and Moulines (2019b) already gave an $\tilde{\mathcal{O}}(\sqrt{d})$ bound.

We are deeply thankful for the opportunity of clarification. The result (Theorem 8) in Durmus and Moulines' milestone paper actually says that the mixing time of LMC is $\tilde{\mathcal{O}}(d)$ in 2-Wasserstein distance, NOT in 2-Wasserstein squared. Table 1 of **their** paper (not ours) summarized this.

For more details, note that the right hand side of Theorem 8 in Durmus and Moulines (2019b) consists of two terms, the first term is $\mathcal{O}(d)$, yet the second term (i.e. $u_n^{(3)}(\gamma)$) is $\mathcal{O}(d^2)$ (there is a leading $d$ in Eq. (11), and also another factor of $d$ near the end of the first line of Eq. (11); typeset here refers to their final arXiv version). Therefore, the right hand side of Theorem 8 is actually $\mathcal{O}(d^2)$, and after taking the square root, one ends up with $\mathcal{O}(d)$ mixing time in $W_2$. It may be easier to see this fact from Corollary 9 in Durmus and Moulines (2019b), which serves as a constant step size version of Theorem 8 and is more comparable to Theorem 4.1 of our paper (which also uses constant step size).

Once this confusion is cleared up, we see that prior to this paper, the best known mixing time bound of LMC in $W_2$ is $\tilde{\mathcal{O}}(d)$, whereas a discretization of ULD (Cheng et al. 2018b; Dalalyan and Riou-Durand, 2020) can achieve $\tilde{\mathcal{O}}(\sqrt{d})$ mixing time bound in $W_2$. Hence we have been thinking whether ULD algorithm truly has better dimension dependence than LMC algorithm.

It seems that the interpretation of Theorem 8 in Durmus and Moulines (2019b) is a major concern in the review and may have negatively affected the rating. We hope the above clarification could turn the doubts away, and we would deeply appreciate it if the reviewer could kindly re-evaluate the contribution of our work.

- Re: Verifiability of A2 (3rd order derivative condition)

This is a great suggestion. It helps us show that A2 is in fact not a very strong assumption: after direct (although lengthy) calculations, we found that potentials considered in both examples in Sec. 5 satisfy A2. We will revise accordingly, and the comment is very much appreciated.

# Addtional Response to Reviewer LgVW

Thanks very much for the additional discussions. We are glad that the confusion with Theorem 8 in Durmus and Moulines 2019b has cleared up.

Regarding Theorem 5 in their paper, it leads to an $\tilde{\mathcal{O}}(\frac{d}{\epsilon^2})$ mixing time bound, which is worse than that produced by Theorem 8 which required an additional regularity assumption. This is summarized in Table 1 of **their** paper (not ours).

To explain the details, we agree that the right hand side of the main result of Theorem 5 is $\mathcal{O}(d)$, and after square rooting one gets an $\mathcal{O}(\sqrt{d})$ bound of $W_2$. However, this bound on $W_2$ does **not** directly translate into the final dimension dependence of *mixing time*, because one needs to further solve for a lower bound of the number of iterations needed for the algorithm to reach $\epsilon$-neighborhood of the target measure in $W_2$ topology.

More precisely, let us investigate the non-asymptotic bound
$$
W_2(\pi_n, \pi) \leq (1 - mh)^{n} W_2(\pi_0, \pi) + C \sqrt{d} h^p
$$ where $C$ is a constant independent of dimension $d$ and step size $h$. The limiting sampling error, i.e., $W_2(\pi_\infty,\pi)$, will be $\mathcal{O}(\sqrt{d})$ as discussed above. But this is not enough for an $\sqrt{d}$ dependence in the *mixing time* bound. In fact, when $p=\frac{1}{2}$, one obtains $\tilde{\mathcal{O}}(\frac{d}{\epsilon^2})$ mixing time in $W_2$, which is Durmus & Moulines 2019b Theorem 5's result (see also e.g., Theorem 1 in Dylalyan 2017a). However, if it can be shown that $p=1$, one will obtain $\tilde{\mathcal{O}}(\frac{\sqrt{d}}{\epsilon})$ mixing time in $W_2$ (not $W_2^2$), and that is Theorem 4.1 in this work.

# Response to Reviewer 8SkQ
We deeply appreciate the reviewer's time and valuable comments. We are glad to hear the acknowledgement that this work is interesting, novel and important.

Potential applications outside the sampling regime, such as optimization, sound very interesting, and we are thankful for the suggestion.

# Response to Reviwer j2p5
We deeply appreciate the reviewer's time and valuable comments.

- Re: framework only applies to contractive SDEs which essentially requires the target distribution to be strongly log-concave

This is a fair statement. It is indeed desirable to weaken the assumption on the target measure, e.g., requiring isoperimetric inequalities such as Poincare's inequality and log Sobolev inequality, or growth conditions on the potential, etc. However, most existing techniques for these nonconvex problems need case-by-case analyses and this is a little complementary to our goal of having a user-friendly generic framework.

For example, a common way to utilize isoperimetric inequalities for analyzing non-log-concave samplers is to construct an objective functional/Lyapunov function that leverages on the special structure of the underlying stochastic process. A well-known example is the choice of KL divergence as the Lyapunov function for the geometric ergodicity of overdamped Langevin under log Sobolev assumption, but KL divergence no longer works (directly) if the dynamics is underdamped Langevin instead, and the Lyapunov function needs to be modified in a nontrivial way (e.g., Ma et al., 2021). Another powerful class of approaches, based on coupling, require the design of coupling (e.g., synchronous, reflection, maximum) and a smart partition of state space, both of which are tailored to the specific dynamics. In short, there seems to be a trade-off between the restrictiveness of the assumptions one puts on target measures, and the range of stochastic processes an approach can quantitatively analyze. As this work tries to establish a easy-to-use framework for a generic class of SDE-based sampling algorithms, we chose to work in the standard strongly convex and smooth setup. Contractivity then follows as the expert pointed out (sometimes after some additional work, such as in the case of underdamped Langevin (Dalalyan and Riou-Durand, 2020)). As there are still many open and important problems in the area, the convexity assumption, albeit strong, is still widely used in literature, e.g., LMC (Dalalyan 2017; Cheng and Bartlett 2018; Durmus and Moulines 2019a & 2019b), Underdamped Langevin (Cheng et al. 2018, Dalalyan and Riou-Durand, 2020), MALA (Dwivedi et al. 2019, Chewi et al. 2020) and Hamiltonian Monte Carlo (Mangoubi and Smith 2017, Chen and Vempala, 2019).

Meanwhile, we believe there is at least one way to extend the framework and replace contractivity by a weaker requirement. Progress is along the way, but we're hoping the reviewer could kindly allow us to take step by step.

- Re: no comparison between our 3rd-order derivative condition and the Hessian Lipschitz condition in the literature

This helpful comment is greatly appreciated. We believe discussing these two assumptions will significantly strengthen this work and it will be included in a revision. In short, neither is always a stronger requirement than the other. More precisely, there are two levels of discussions, first being on the satisfaction of the condition, and second being on the detail of the condition, such as the constant.

For the former, we note:
1. A2 is no stronger than Hessian Lipschitz condition, if not weaker. Consider for example $f(x)=x^4$, and $f$ satisfies A2 but is not Hessian Lipschitz.
2. Hessian Lipschitz + Smoothness --> A2.

For the latter (detail of the constant), we fully agree that it matters a lot whether the constant $G$ in our condition and/or the Hessian Lipschitz constant (denoted by $\tilde{L}$) are implicitly dimension dependent. The great comment on the 'tensorized' special cases is very much appreciated. In fact, there are two situations, namely (1) what the review suggested, namely $f(x)=\sum_i f_i(x_i)$, i.e., independent but not identically distributed, and (2) the i.i.d. case, in which $f(x)=\sum_i f(x_i)$.

For (1), the Hessian Lipschitz condition is actually not always dimension-free and does not always have better dimension-(in)dependence than our condition. An example is the following strongly-convex and smooth potential function $f(\boldsymbol{x}) = \sum_{i=1}^d \left( \frac{i^2 + 1}{2} x_i^2 + i^4 \cos (\frac{x_i}{i}) \right)$, for which the Hessian Lipschitz constant $\tilde{L}$ scales as $\mathcal{O}(d)$ whereas the constant $G$ in A2 is $G = \mathcal{O}(1)$.

For (2), however, $\tilde{L}$ will be dimension free but $G$ may not be. Nonetheless, because all the dimensions decouple, our claimed $\tilde{\mathcal{O}}(\sqrt{d})$ results can be proved in a much easier way in this special case, which is why we wondered whether we really needed assumption A2 if we were smarter. But we fully agree with the review that this is just a spectulation at this point and will make sure this is clearly stated. One reason we asked ourselves whether LMC (overdamped Langevin) really has a different dimension dependence from KLMC (kinetic Langevin) is: as one lets $\gamma\to\infty$ in kinetic Langevin, the process provably converges (weakly) to overdamped Langevin (after time dilation), but dimension doesn't explicitly appear in the proof of weak convergence, and no assumption like A2 is needed either. These are of course even more spectulative but hopefully they explain why we thought maybe A2 can be weakened one day without bringing the result back to $\tilde{\mathcal{O}}(d)$.

Despite of (1) or the more complicated correlated cases, the Hessian Lipschitz condition still seems to be popular in the contemporary literature (e.g., Durmus & Moulines, 2019b; Ma et al., 2021), where it is often implicitly assumed to be dimension-free. Therefore, we hope we could also assume our alternative constant $G$ to be dimension-free for now as well. Of course as the review pointed out, we need to make sure this is clarified upfront, so that results can be precisely stated without creating confusion or overclaim, and this will be done in a revision. Meanwhile, we feel that the dimension dependence issue is universal to any assumption that brings in new constants (e.g. smoothmess, strong convexity, etc.), not unique to assumption A2. We will take the reviewer's suggestion and further investigate $G$ and/or A2.

- Re: experiments used a poor proxy for the Wasserstein distance.

We agree that 'difference in means' is a coarse proxy; however, it has two outstanding advantages -- being computationally cheap, and being a lower bound of $W_2$ and hence also obeying the error bound in Eq. (10). Entropy-regularized $W_2$ is a great approximation of $W_2$. Unfortunately, despite its reduced cost, it still needs $\tilde{\mathcal{O}}(\frac{n^2}{\epsilon^4})$ runtime (Altschuler et al, 2017), where $n$ is the number of samples, and $\epsilon$ is the tolerance parameter. Since our experiments were designed to validate our statistical bound, the number of samples can be as large as $n=10^8$, $\epsilon$ needs to be of order $\epsilon = 10^{-2}$, and we need to record the distance in every iteration for each one of the $10^4$ independent Markov chains. Therefore, entropy-regularized Wasserstein distance is computationally infeasible to us given the scale of our experiments.

- Re: modify the abstract.

We will revise it accordingly to make it clearer. Thanks.

- Re: Thm 4.3

In our paper, $m$ is set to 1, yet $L$ is free as long as it is no less than 4. We are sorry for the confusion and will update the notation to make it clearer. Also, great thanks to the alternative way to deduce Eq. (27).

# Additional Response to Reviewer j2p5

Thanks very much for the additional discussions. We are glad to hear that the identification of a third-order condition which allows for $d^{1/2}$ dimension dependence is interesting (and the general framework is as well). Your reconsideration is deeply appreciated.

We agree that the example is special. We are still thinking about ways to soften A2, if not to remove it.

The comments on two possible routes of dealing with implicit dimension dependencies in parameters are valuable and insightful. We definitely will state the assumption and its possible ramification clearly upfront. This is actually deeply appreciated as we're suggesting a significant result and we also would like to be precise but not overclaiming.

Regarding the discussion on overdamped and kinetic Langevin, we fully agree that discretization needs to be considered and it is actually the core of the problem. Every word the reviewer said is true, and a lot of complications can arise. Still, we keep on asking ourselves how can dimension enter the discretization; if the stepsize is chosen to be the traditional o(1/L) value, perhaps the dimension actually does not appear, unless it is lurking inside L? In any case, this paragraph is purely for brainstorming purposes; it is not science yet but just spectulative.

# Message to the area chairs (didn't use because reviewers increased their scores)
Dear Chairs,

We wonder if you could kindly advise us what to do given the following situation: we have 3 positive reviews and 1 negative one; the negative review was incorrect and the reviewer acknowledged this, however without changing the rating.

More precisely, Reviewer LgVW first rated 4; after our 1st round of response, s/he acknowledged that her/his main concern was incorrect, and then s/he asked another question, which is again incorrect and clarified in our 2nd round of response. However, after 12 days, the reviewer hasn't responded and the rating remained to be 4. We're afraid that s/he forgot.

A similar but less critical observation is, Reviewer DVEg gave an initial rating of 6, and after discussions s/he wrote "the detailed response (which) resolves most of my concerns" and "this strengthens the generality of their analysis framework". The reviewer then asked another question, which is a great one and appreciated despite of being not directly related to our paper. We again replied in details; unfortunately, there was no more response, which we can understand, but the rating was never changed.

Since the deadline given to the reviewers is approaching, may we ask if there is anything that can be done to remind the reviewers?

In any case, thank you very much for your time and consideration.

Best wishes,

Paper4350 Authors

# Response to urgent question from AC

(Sorry it took us a weekend to reply as we carefully went through the important paper of [Li et al., 2019])

We sincerely thank the AC for this reference. It is really appreciated because otherwise we would have missed this very important reference, which is unacceptable to ourselves, damaging our valued scholarship, and unfair to the authors of that great work.

As the AC pointed out, we thought we had two contributions in the initial submission, one being the mean-square analysis framework, and the other being improving the known dimension dependence of LMC from $d$ to $\sqrt{d}$ under an additional assumption. The second remains valid and we still think it is an interesting result, but of course what we’re discussing now is the first:

Foremost of all, we completely agree that [Thm 1, Li et al., 2019] and our section 3 share significant resemblance. As both are based on [Milstein and Tretyakov, 2013] (numerical analysis of SDEs), both their goal (transferring local integration error to global sampling error) and the proof techniques are similar, and of course we’re the one who needs to revise.

Meanwhile, two observations pertinent to our work are worth mentioning: (i) our requirement on the local integration error is weaker; (ii) since our focus, namely the dimension dependence of LMC, is not necessarily identical to that of [Li et al. 2019], our Theorem 3.3 is actually more suitable than [Thm 1, Li et al] for tracking dimension dependence.

More precisely (for (i)), note our constant $C_1$ and $C_2$ correspond to their $\lambda_1$ and $\lambda_2$ via $C_1=\sqrt{\lambda_2}$ and $C_2=\sqrt{\lambda_1}$, but we also have additional $D_1$ and $D_2$ terms which weaken the needed conditions on local accuracy (they need $D_1=D_2=0$, i.e., uniform local errors, while we can work with non-uniform local errors).
In general, local error tends to depend on initial values, i.e. $D_1 \neq 0, D_2 \neq 0$. Consider the simplest example of 1D standard Gaussian being the target distribution, then we have
$E\| E[x_{k+1} - \bar{x}_{k+1}|\mathcal{F}_k]\| = (e^{-h} - 1 + h) E\|\bar{x}_k\| = (\frac{h^2}{2} + o(h^2)) E\|\bar{x}_k\| \leq (\frac{h^2}{2} + o(h^2)) \sqrt{E\|\bar{x}_k\|^2}$, where $x_{k+1}$ and $\bar{x}_{k+1}$ are respectively the exact solution of Langevin dynamics and the one-step-iterate of LMC from initial value $\bar{x}_k$. One can see that the local error does depend on $\bar{x}_k$ and is not uniform, at least not directly. Our weaker local error condition (i.e. $D_1, D_2 \neq 0$) reflects this, and the proof of the final sampling error bound consequently used significantly more work.

For (ii), note that even the extra dependence on $D_1$ and $D_2$ aside, the specific forms of our bounds in Theorem 3.3 do have different expressions from those in [Thm 1, Li et al]. In fact, bounds and constants in [Thm 1, Li et al] were probably not designed for tightly tracking the dimension dependence, as the focus of that great paper was more on $\epsilon$ dependence; consequently, [Thm 1, Li et al] only led to an $\tilde{O}(d)$ bound of LMC (see Example 1 of their paper), whereas our Theorem 3.3 led to $\tilde{O}(\sqrt{d})$. Working out and tracking all the dimension dependence is the hardest part of our work, and for that (see Sec.4) we did rely on the specific form of our bound in Sec.3.

<!-- Please allow us to first explain that even if $D_1$ and $D_2$ are indeed
zero, the two results are not entirely the same: (i) for example, our
error is linear in $C_1$ and $C_2$, while that for [Thm 1, Li et al] is
the square root of a different linear combination of $\lambda_1=C_2^2$
and $\lambda_2=C_1^2$. <**Ruilin: This may not be a significant difference as one can use inequality $\sqrt{a^2 + b^2} \leq |a| + |b|$ to make things linear, and hence I feel this is not a strong argument** > (ii) Bounds and constants in [Thm 1, Li et al. 2019]
are not designed for tightly tracking the dimension dependence, as the
focus of that great paper was more on $\epsilon$ dependence;
consequently, [Li et al. 2019] only gave an $O(d)$ dependence of LMC (see Example 1 of their paper), whereas
our Theorem 3.3 gave $O(\sqrt(d))$. In addition, (iii) in general, local error tends to depend on initial values, i.e. $D_1 \neq 0, D_2 \neq 0$. Consider for example a 1D standard Gaussian as target distribution, and one runs one iteration of LMC from $\bar{x}_k$ with step size $h$ in $k$-th itetaion, then $E\|x_{k+1} - \bar{x}_{k+1}\|^2 = \frac{1}{4} h^4 E\|\bar{x}_k\|^2 + \frac{2}{3}h^3 - \frac{1}{4}h^4 + o(h^4)$ clearly depends on the initial value $\bar{x}_k$, where $x_{k+1}$ and $\bar{x}_{k+1}$ are the exact solution of Langevin dynamcis and the one-step-iterate of LMC algorithm from initial value $\bar{x}_k$ respectively. The dependence on initial values is captured by our weakened assumption (i.e. $D_1, D_2 \neq 0$). -->

Having said all these, we would completely agree that our Section 3 is not worth a publication if it was not preparational for Section 4 (LMC dimension dependence). The very important paper of [Li et al] already made the first contribution despite of our technical differences, and we pledge that what’s old and new will be fairly clarified in our revision. Now we think our contribution is mostly the second.

As the AC mentioned, a major revision of Section 3 is imperative, but this is actually easy because we will just say (in Sec.3) that here is a variation of the important results in [Li et al. 2019]. Now we know the important reference of [Li et al. 2019] (again, our bad, and thank you), stating things in a fair and factual way is the minimum we need to do, and it will be done in our revision (we will not be stupid enough to lie on OpenReview).

We hope the AC and everyone involved in this process could consider our explanations. In any case, thank you for your time and helping us improve our paper significantly.

<!-- < For discussion only >
We have
$$
x_1 = e^{-h} x_0 + \sqrt{2}\int_0^h e^{-(h-s)} dB_s \\
\bar{x}_1 = (1 - h) x_0 + \sqrt{2}\int_0^h 1 dB_s
$$
and hence
$$
\mathbb{E}\|x_1 - \bar{x}_1\|^2 = (e^{-h} - （1-h）)^2\mathbb{E}\|x_0\|^2 + 2\int_0^h (e^{-(h-s)} - 1)^2 ds = \frac{1}{4} h^4 E\|\bar{x}_k\|^2 + \frac{2}{3}h^3 - \frac{1}{4} h^4 + o(h^4)
$$
Please correct me if I am wrong.
-->

# more response to reviewer j2p5

Thanks for your update. May we try to convince again that the needed revision is not large-scale? As our results in Sec.3 are technically *not* the identical to those in [Li et al. 2019], we just need to change some descriptions and add acknowledgments of their importance. If the concern was not being able to provide further feedback to the authors in the form of follow-up reviews, (i) what we can do is to provide a revised version, asap, before the decision deadline (we just need to be told that this could be a possibility); (ii) may we note that if it is killed and resubmitted to another venue, there will be no follow-up reviews either because reviewers will likely be different?

Is there any possibility that the reviewer could reconsider a score of 4, even assuming Sec.3 had zero contribution but just based on the contribution of Sec.4?