First, we would like to thank the reviewer for a quick response!
We still believe that there is no mathematical gap in our assertions, as the sentence _This result gives no hope for improving the communication complexity (2) in the worst case_ is not a strong mathematical statement. What we intended to say is that _This result **indicates** it is not possible to improve the communication complexity (2) in the worst case_, which means that while we cannot be certain the bound is impossible to break with *any technique*, it is definitely not possible with approaches like the Rand$K$ compressor due the lower bound theorem.
Nevertheless, we agree that this can lead to misunderstandings. Therefore, we propose to rewrite the contributions section in the following way (changes are in **bold**):
1. We start by proving the impossibility of devising a method where the server communicates with the workers using unbiased compressors $\mathbb{U}(\omega)$ (or biased compressors from Section B) and achieves an iteration rate faster than $\Omega\left(\frac{(\omega + 1) L \delta^0}{\varepsilon}\right)$ (Theorem 3.1) under Assumptions 1.1, 1.2, and 1.6.
~~**This result gives no hope for improving the communication complexity (2) in the worst case**~~.
**This result provides a lower bound for any method that applies such compressors to vectors sent from the server to the workers in every iteration. Moreover, we prove a more general iteration lower bound of
$\Omega\left(\frac{(\omega + 1)L \delta^0}{\varepsilon}\right)$ for all methods where the server zeroes out a coordinate with probability $1 / (\omega + 1)$ (see Remark 3.2).**
2. In view of this results, it is clear that an extra assumption is needed to break the lower bound $\Omega\left(\frac{(\omega + 1) L \delta^0}{\varepsilon}\right).$ In response, we introduce a novel assumption termed "Functional $(L_A, L_B)$ Inequality" (see Assumption 4.2). We prove that this assumption is relatively weak and holds, for instance, under the local smoothness of the functions $f_i$ (see Assumption 1.5).
3. We develop a new method for downlink compression, MARINA-P, and show that, under our new assumption, **along with Assumptions 1.1, 1.2 and 1.6, it can achieve the iteration rate of**
\begin{align*}
\textstyle O\left(\frac{\delta^0 L}{\varepsilon} + \frac{\delta^0 L_A (\omega + 1)}{\varepsilon} + \frac{\delta^0 L_B (\omega + 1)}{\varepsilon \sqrt{n}}\right).
\end{align*}
**(see Theorem D.1 with $p = 1 / (\omega + 1)$ + Lemma A.5)**.
Notably, when $L_A$ is small and $n \gg 1,$ this **iteration** complexity is provably superior to **$\Theta(\frac{\delta^0 L (\omega + 1)}{\varepsilon})$** and the complexities of the previous compressed methods. In this context, $L_A$ serves as a measure of the similarity between the functions $f_i$, and can be bounded by the ``variance'' of the Hessians of the functions $f_i$ (see Theorem 4.8). Thus, MARINA-P is the first method whose **iteration** complexity can provably improve with the number of workers $n$.
4. **Moreover, MARINA-P can achieve the s2w communication complexity of**
\begin{align*}
\textstyle O\left(\frac{d \delta^0 L}{n \varepsilon} + \frac{d \delta^0 L_A}{\varepsilon}\right).
\end{align*}
**When $L_A$ is small and $n \gg 1,$ this communication complexity is provably superior to (2) and the communication complexities of the previous compressed methods.**
5. Our theoretical improvements can be combined with techniques enhancing the w2s communication complexities. In particular, by combining MARINA-P with MARINA (Gorbunov et al., 2021) and adding the crucial momentum step, we develop a new method, M3, that guarantees a total communication complexity (s2w + w2s) of
\begin{align*}
\textstyle O\left(\frac{d \delta^0 L_{\max}}{n^{1/3} \varepsilon} + \frac{d \delta^0 L_A}{\varepsilon}\right).
\end{align*}
When $n > 1$ and in the close-to-homogeneous regime, i.e., when $L_A$ is small, this **communication** complexity is better than (2) and the complexities of the previous bidirectionally compressed methods.
6. Our theoretical results are supported by numerical experiments (see Section F).
Additionally, we will change Lines 122-123 to "Let us first investigate the possibility of improving the iteration complexity $O\left(\frac{(\omega + 1)\delta^0 L}{\varepsilon}\right)$ under Assumptions 1.1, 1.2, and 1.6."
These extra 4-5 sentences fully resolve any potential misunderstandings and clarify our contributions. These are very minor adjustments that can be easily incorporated into the camera-ready version of the paper.
We believe that we have resolved the concerns and will happily answer any additional questions.