While classic work in convex-concave min-max optimization relies on averageiterate convergence results, the emergence of nonconvex applications such as training Generative Adversarial Networks has led to renewed interest in last-iterate convergence guarantees. Proving last-iterate convergence is challenging because many natural algorithms, such as Simultaneous Gradient Descent/Ascent, provably diverge or cycle even in simple convex-concave min-max settings, and previous work on global last-iterate convergence rates has been limited to the bilinear and convex-strongly concave settings. In this work, we show that the HAMILTONIAN GRADIENT DESCENT (HGD) algorithm achieves linear convergence in a variety of more general settings, including convex-concave problems that satisfy a novel "sufficiently bilinear" condition. We also prove convergence rates for stochastic HGD and for some parameter settings of the Consensus Optimization algorithm of Mescheder et al. (2017).
Summary:
The paper, considers methods for solving smooth unconstrained min-max optimization problems.  In particular, the authors prove that the Hamiltonian Gradient Descent (HGD) algorithm converges with linear convergence rate to the min-max solution. One of the main contributions of this work is that the proposed analysis is focusing on last iterate convergence guarantees for the HGD. This result, as the authors claim can be particularly useful in the future for analyzing more general settings (nonconvex-nonconcave min-max problems).
In addition, two preliminary convergence theorems were provided for two extensions of HGD: (i) a stochastic variant of HGD and (ii)  Consensus Optimization Algorithm (CO) (by establishing connections of CO and HGD).
Main Comments:
The paper is well written and the main contributions are clear. I believe that the idea of the paper is interesting and the convergence analysis seems correct, however i have some concerns regarding  the presentation and the combination of different assumptions used in the theory. 
1) I think definition 2.5 of Higher order Lipschitz is very strong assumption to have. What exactly means? Essentially the authors upper bounded any difficult term appear in the theorems. Is it possible to avoid having something so strong? Please elaborate.
2) In assumption 3.1 is not clear what $L_H$ is. This quantity never mentioned before. Reading the Lemmas of Section 4 (Lemma 4.4) you can see that it is the smoothness parameter of function $H$. Thus, is not necessary to have it there (not important for the definition).
3) What is the main difference on the combination of assumptions on Theorems 3.2, 3.2 and 3.4. Which one is stronger. Is there a reason for the existence of Theorem 3.3?
4) All the results heavily depend on the PL condition. I think having this in mind, showing the convergence of Theorems 3.2-3.4 is somehow trivial. In particular, one can propose several combinations of assumptions in order for the function H to satisfy the PL condition. Can we avoid having the PL condition? The authors need to elaborate more on this.
5) In Theorem 5.2, the term 1/sqrt(2) is missing from the final bound.
Minor Suggestions:
In first paragraph of page 5 where the authors divide the existing literature into the three particular cases, I am suggesting to add the refereed papers inside each one of this cases (which papers assumed function g bilinear , which papers strongly convex-concave etc.)
I understand that the main contribution of the work is the theoretical analysis of the proposed method but would like to see some numerical evaluation in the main paper. There are some preliminary results in the appendix but it will be useful for the reader if there are are some plots showing the benefit of the method in comparison with existing methods that guarantee convergence (which method is faster?). In the current experiments there is a comparison only with CO algorithm and SGDA.
In general i find the paper interesting, with nice ideas and I believe that will be appreciated from researchers that are interested on smooth games and their connections to machine learning applications. 
I suggest weak accept but I am open to reconsider in case that my above concerns are answered.
**********after rebuttal********
I would like to thank the authors for their reply and for the further clarification. 
I will keep my score the same but I highly encourage the authors to add some clarification related to my last comment on the globally bounded gradient. 
In their response they mentioned that the analysis only requires that  H is smooth and that $\|\xi(x^{(0)})\|$ is sufficient bound. This needs to be clear in the paper (add clear arguments and related references). 
In addition, in their response they highlight the non-increasing nature of function H over the course of the algorithm which is important for their argument. Having this in mind note that the theoretical results on stochastic variant presented in the paper are wrong. In SGD,  function H does not necessarily decrease over the course of the algorithm. The authors either need to remove these results or restate them in a different way in order to satisfy the assumed conditions.