We thank Reviewer U6Tv for the thoughtful comments. Please find our responses below.
**1. The reason for comparing to CSGM [18]**
We chose to compare to CSGM [18] since [18] is asymptotically optimal and compares favorably to several previous algorithms, as demonstrated in the experiments in [18]. By showing that our proposed scheme compares favorably to [18], we show that our scheme compares favorably to those previous works as well.
**2. Regarding comparison to other techniques**
Regarding Feldman and Talwar [30]: [30] mentioned applying [30, Theorem 3.4] to Gaussian mechanism. The main obstacle in comparing [30, Theorem 3.4] to our result is that [30] relies on a computational hardness assumption on the pseudorandom number generator, and it is unclear how many bits of random seed (communication) are necessary to guarantee computational indistinguishability. Also, [30] does not prove that their scheme has a communication cost close to the theoretical minimum.
Regarding Bassily and Smith [5]: [5] does not exactly preserves the distribution of the simulated mechanism (there is a 50% chance that the data is dropped). Additional (likely non-trivial) analyses are necessary to characterize its central-DP guarantee for mean estimation, to be compared to our scheme in Figure 1.
[65] is also non-exact, making their central-DP guarantees unclear.
**3. About whether "approximate simulation would imply biased estimates"**
We emphasize that unbiasedness is a mathematical property that requires proof. If an approximate method cannot be proved to be unbiased, then it should not be considered to be unbiased, regardless of how close to zero the bias appears to be in experiments (the bias might become large outside of the cases experimented). For some specific approximate methods, it may be possible to add a debiasing step with a proof of unbiasedness (e.g., [65]), though the feasibility of such a step depends on the specific task and the mathematical tractability of the output distribution. The advantage of exactness is that it can readily imply unbiasedness, and no additional steps are necessary.
In sum, approximate method does not necessarily imply biasedness (e.g., [65]), but an approximate method without a proof of unbiasedness should not be considered to be unbiased. Therefore, unbiasedness is an advantage of our exact method over other approximate methods without proofs of unbiasedness. This is a theoretical advantage that does not require experiments to show.
**4. Regarding distortion for a fixed $(\\varepsilon, \\delta)$ and approximate techniques**
Indeed, if $(\\varepsilon, \\delta)$ is fixed, then we have to reduce the $\\varepsilon, \\delta$ of the simulated mechanism. However, this reduction can be small according to Theorem 4.8. Also, the distortion introduced by lowering $\\varepsilon, \\delta$ is considerably different from the distortion introduced by the methods in [30,65,71]. If we exactly simulate the Gaussian mechanism with a lower $\\varepsilon, \\delta$, the noise is still exactly Gaussian (with a larger variance), and can be added to other Gaussian noises nicely. This "summable" noise is the key to providing central DP guarantees (in addition to local DP). However, if we simulate the Gaussian mechanism using the approximate techniques in [30,65,71], the noise introduced has a mathematically intractable distribution, which will be an obstacle in obtaining theoretic guarantees for downstream tasks (e.g., for distributed mean estimation, we do not know the overall noise distribution and its central-DP properties after summing all the data). The main benefits of exactness are mathematical tractability and ease of proving guarantees for downstream tasks.
**5. The benefits of simple theoretical guarantees**
Also, we believe that the theoretical contributions of our work, namely the privacy-communication trade-off given in Theorems 4.3-4.8 given in simple and clean expressions, with exact preservation of the output distribution, is noteworthy in itself. To the best of our knowledge, our work is the first method for compressing DP mechanisms that has a bound on its compression size universally close to the I(X;Z) lower bound (i.e., the bound depends on the simulated mechanism only through I(X;Z) as in Theorem 4.3, and hence is always almost-optimal regardless of the situation). Even though we agree that more experiments can be beneficial, experiments are not strictly necessary to demonstrate the almost-optimality of our method, when we have a mathematical proof of its universal almost-optimality for the compression size.
We believe these simple guarantees (in terms of simple quantities like I(X;Z)) and exact distribution preservation can make the proposed method a useful general technique for designing more specific DP mechanisms in the future.
We hope that we have adequately addressed the concerns and questions, and kindly invite the reviewer to consider updating the score.