Thank you for your time and comments. Below, we try to address your concerns:

Weaknesses

• There is a vast literature: The vast literature on this subject justifies the relevance of our work. We believe that our contribution is both significant and timely, as it unifies various approaches within this extensive body of literature, explaining the connections between them, all while achieving improved convergence rates.

• Rates are expected: We respectfully disagree. Please observe that proving these rates through a continuous-time analysis is a non-trivial task, and it was not clear whether it was possible. It is important to note that the rates presented in the earlier works [1,2] are known to be suboptimal in discrete and continuous-time settings, whereas in many cases we achieve the optimal rates. To further highlight this, we have enhanced the discussion about comparisons with prior work in Section 5 with Table 2 in our revision to compare our rates with existing results.

• Improvements are in the constants: We agree that our rates are linear, just as in the prior work. However, please note that our improvements are in the exponents and not the constants. This forms a significant enhancement in comparison with the prior work. For example, Theorem 4.1 proves a convergence rate of order $C_{GM}(1-\sqrt{ 1/\kappa})^k)$, and the best previous result in this setting was $5L||x_0-x^*||^2_2(1+\sqrt{ 1/\kappa}/9)^{-k}$. Suppose $L=10,\mu=1,\kappa=10$ and $k=50$, then $C_{GM}(1-\sqrt{ 1/\kappa})^k \approx 2.35\times 10^{-7}\times ||x_0-x^*||_2^2$ while $5L||x_0-x^*||_2^2((1+\sqrt{ 1/\kappa}/9)^{-k}) \approx 8.8939||x_0-x^*||_2^2$. Similarly, for QHM, we prove a rate of order $O((1-\sqrt{\frac{1}{3\kappa}})^k)$ while the previous best rate was $O((1+\frac{1}{40\sqrt{\kappa}})^{-k})$. If $\kappa=10$ and $k=50$, then $(1-\sqrt{\frac{1}{3\kappa}})^k \approx 4.19 \cdot 10^{-5}$ while $(1+\frac{1}{40\sqrt{\kappa}})^{-k} \approx 0.6745$. Finally, we would like to draw your attention to the ability of (GM2-ODE) to recover other high-resolution ODEs.

• Leading to other ODEs As shown in Table 1, (GM2-ODE) and its SIE discretization recover the TM method and its ODE through different choices of coefficients. In addition, the current existing rates on (HR_TM) in the paper are not comparable to the algorithm's convergence rate $(\mathcal O\left((1-\sqrt{\tfrac{\mu}{L}}\right)^{2k}$ which should correspond to $\mathcal{O}\left(e^{-2\sqrt{\mu}t}\right)$ in continuous time). This observation suggests a new high-resolution ODE for the TM method. Setting the same coefficients that recover the TM method in (GM2-ODE) reads $\ddot{X}_t+\sqrt{\mu}\left(\frac{3-\sqrt{\tfrac{\mu}{L}}}{1-\sqrt{\tfrac{\mu}{L}}}\right)\dot X_t +\frac{1}{\sqrt{L}}\nabla^2 f(X_t)\dot X_t+\left(\frac{2}{1-\sqrt{\tfrac{\mu}{L}}}+\sqrt{\frac{\mu}{L}}\right)\nabla f(X_t)=0.$ Surprisingly, if the step-size $\tfrac{1}{\sqrt{L}}\rightarrow 0$ the recent ODE reduces to$\ddot{X}_t+3\sqrt{\mu}\dot X_t+2\nabla f(X_t)=0$ which is the low-resolution ODE corresponding to the TM method [3]. We will include this discussion in the revision of our paper. .

Thank you for your positive evaluation and comments on our paper.

Thank you for your time, careful reading, and valuable comments. Our responses to the specific concerns and comments are below:

Novelty and Significance

• Use of IQCs: It is true that we leveraged IQCs, but please note we use it only in one of the three Lyapunov functions we propose in this paper. We introduced two more Lyapunov functions which cannot be derived from the IQC theorem: One is in continuous time (Theorem 3.2), and the other one is in discrete time (equation (7)). The latter is the first Lyapunov function that uniformly captures the discrete time behavior of the algorithms for different choices of parameters.

• Use of well-known techniques: It is true that we use well-established techniques like Lyapunov analysis and Grönwall lemma. Nevertheless, we leverage these tools to achieve a family of Lyapunov functions whose form is not seen before in the literature. Crucially, our analysis is NOT a trivial extension of another work, the form of the unified Lyapunov function requires different steps in the analysis. For example, we use a limit analysis after (55) in the appendix which is different from all of the prior works. This is just one example of the many analytical and algebraic complexities that arise from the generality of the Lyapunov function we addressed in our analysis.

• Improvements are in the constants: We agree that the rates are linear similar to the state-of-the-art. However, it is important to note that the improvements are not in the constants but in the exponents of the linear rate, which significantly tightens the bounds. For example, Theorem 4.1 proves a convergence rate of order $C_{GM}(1-\sqrt{ 1/\kappa})^k)$, and the best previous result in this setting was $5L||x_0-x^*||^2_2(1+\sqrt{ 1/\kappa}/9)^{-k})$. Suppose $L=10,\mu=1,\kappa=10$ and $k=50$, then $C_{GM}(1-\sqrt{1/\kappa})^k \approx 2.35\times 10^{-7}\times ||x_0-x^*||^2_2$ while $5L\|x_0-x^*\|^2_2((1+\sqrt{ 1/\kappa}/9)^{-k}) \approx 8.8939||x_0-x^*||_2^2$. Similarly, for QHM, we prove a rate of order $O((1-\sqrt{\frac{1}{3\kappa}})^k)$ while the previous best rate was $O((1+\frac{1}{40\sqrt{\kappa}})^{-k})$. If $\kappa=10$ and $k=50$, then $(1-\sqrt{\frac{1}{3\kappa}})^k \approx 4.19 \cdot 10^{-5}$ while $(1+\frac{1}{40\sqrt{\kappa}})^{-k} \approx 0.6745$. To highlight our improvements, we have now added Table 2 to showcase the faster convergence rates both in continuous and discrete-time.

• Comparison with GM-ODE: We discussed the differences between GM-ODE and GM2-ODE in detail in pages 8-9 in our submission. The main drawback of GM-ODE is its inconsistency in terms of ODE and algorithm recovery. Moreover, GM2-ODE is more capable of recovering algorithms and leads to better convergence guarantees. Please note that the Lyapunov function corresponding to GM-ODE does not achieve the optimal rates. Also note that discretization of GM-ODE using SIE method does not exactly recover the NAG algorithm. Finally, we would like to draw your attention to the ability of GM2-ODE to recover other high-resolution ODEs. Our unified model (GM^2-ODE) has the following two attractive features:

1. Not only does it unify the ODEs in continuous time, but also existing algorithms through SIE discretization. Surprisingly, all these algorithms are found to be well-known methods (NAG,HB,TMM, etc) and they are all recovered through the same routine.
2. The corresponding unifying Lyapunov function leads to better convergence rates than the previous results obtained by continuous time analysis. To further emphasize this, we added Table 2 to compare our rates with existing results, which shows precisely the improvements.

• Less condition on the convergence of the QHM algorithm: Please note that the previous rate on QHM requires $L/\mu\geq 9$ [6], while our result in corollary 4.1.1 is free from such condition.
• Does GM2-ODE lead to new methods?: As shown in Table1, (GM2-ODE) and its SIE discretization recover the TM method and its ODE through different choices of coefficients. In addition, the current existing rates on (HR_TM) in the paper are not comparable to the algorithm's convergence rate ($\mathcal O\left((1-\sqrt{\tfrac{\mu}{L}}\right)^{2k}$ which should correspond to $\mathcal{O}\left(e^{-2\sqrt{\mu}t}\right)$ in continuous time). This observation suggests a new high-resolution ODE for the TM method. Setting the same coefficients that recover the TM method in (GM2-ODE) reads

$\ddot{X}_t+\sqrt{\mu}\left(\frac{3-\sqrt{\tfrac{\mu}{L}}}{1-\sqrt{\tfrac{\mu}{L}}}\right)\dot X_t +\frac{1}{\sqrt{L}}\nabla^2 f(X_t)\dot X_t+\left(\frac{2}{1-\sqrt{\tfrac{\mu}{L}}}+\sqrt{\frac{\mu}{L}}\right)\nabla f(X_t)=0.$

Surprisingly, if the step-size $\tfrac{1}{\sqrt{L}}\rightarrow 0$ the above ODE reduces to $\ddot{X}_t+3\sqrt{\mu}\dot X_t+2\nabla f(X_t)=0$ which is the low-resolution ODE corresponding to the TM method [3]. This discussion will be added to the paper.