Escaping limit cycles: Global convergence for constrained nonconvex-nonconcave minimax problems

Abstract: This paper introduces a new extragradient-type algorithm for a class of nonconvex-nonconcave minimax problems. It is well-known that finding a local solution for general minimax problems is computationally intractable. This observation has recently motivated the study of structures sufficient for convergence of first order methods in the more general setting of variational inequalities when the so-called weak Minty variational inequality (MVI) holds. This problem class captures non-trivial structures as we demonstrate with examples, for which a large family of existing algorithms provably converge to limit cycles. Our results require a less restrictive parameter range in the weak MVI compared to what is previously known, thus extending the applicability of our scheme. The proposed algorithm is applicable to constrained and regularized problems, and involves an adaptive stepsize allowing for potentially larger stepsizes. Our scheme also converges globally even in settings where the underlying operator exhibits limit cycles.

Reviews and author responses:

 I also keep my score 8. 
The questions proposed by the reviewer wk1L who gave 5 points are solved well by the authors from my perspective. 

 I have gone through other reviewers' comments and authors' detailed response, and I decided to keep my score. The clarity of the paper has been improved after revision, and I recommend accepting the paper in its current form.

 Thanks for the clarification, I will keep my score.

 The convergence result in Corollary B.1 for CurvatureEG+ requires $\rho>-\gamma_{k} / 2$ for all $k$. Since $\gamma_k$ could potentially be larger than $1/L$ when picked adaptively, CurvatureEG+ may have convergence for problems where EG+ does not. The difficulty with formally showing that it converges for Forsaken specifically, is that despite $\gamma_k$ being larger than $1/L$ in practice, we cannot guarantee that it would be large enough. In addition, we would need to compute $\rho$ for this example. In contrast with the other examples we provide, computing $\rho$ is challenging for Forsaken, since $\rho$ relies on the critical point, which has a non-trivial solution. That is why we rely on simulations for Forsaken instead.

Thank you for pointing out the additional improvement. We will make sure to include them.

 Hello, authors,

Thank you for your elaborate reply and updates.

Just a few comments:

About concern (1): So the theoretical range of $\rho$ in both your CurvatureEG+ and EG+ in Diakonikolas et al. (2021) do not contain the Forsaken example, but your CurvatureEG+ can converge in the simulation of Forsaken example while EG+ cannot, right? 

About concern (4): In the caption of Figure 2, you could change the second sentence into $\rho=-1/8L$ or "the region below the dated line indicates where $\rho>-1/8L$" in the accepted version.  

About concern (7): In Figure 3, the arrows are very dense and you could make them sparser so that the direction will be clearer in the accepted version. 

 
We thank the reviewer for the thorough feedback. We address each comment one by one below.



1. **Does Forsaken satisfy weak MVI?** Forsaken does not satisfy the weak MVI for $\rho > -1/2L$, not even when restricting the definitions to the constraint set $\|z\|_\infty \leq 3/2$. So the example does not fall into the parameter range $\rho \in (-1/2L, \infty)$, but nonetheless CurvatureEG+ still converges in the simulations. The details can be found in Appendix C.4 of the updated manuscript.
2. **Convergence rate** Thank you for the comment. As the reviewer noted this would affect the constant. If the parameters are restricted to the setting of Diakonikolas et al. (2021), that is, $\gamma=1/L$ and $\rho = -1/8L$, then it can be shown that the constant obtained is three times smaller than that of Diakonikolas et al. (2021). Due to page limitation we omit this in the paper. 
3. **Why use $\|H z^{k}-H z^{k}\|$?** The reason goes back to the proof of Theorem 3.1 and basically to the fact that H is $(1-\gamma L)$-strongly monotone, and thus when $\gamma = 1/L$ is permitted then $\|H\bar z- Hz\|> c\|\bar z - z\|$ no longer holds for some $c>0$ (see Lemma A.3(ii)). In fact, in this setting it is only possible to establish that the limit points belong to Zer T.  Nevertheless, as shown in the proof of Theorem 3.1(ii) if the stepsize $\gamma$ stays away from $1/L$, then the more familiar term $\|\bar z - z\|$ is obtained. 
4. **“Positive results” in Figure 2** That is indeed correct. We have replaced “positive result” with “convergence result”.
5. **Lower bound of Algorithm 1 instead?** The lower bounds are derived to contextualize existing algorithms. We would not be able to characterize the role of the stepsize for EG+ if we only considered Algorithm 1. With that said, it is an interesting result in its own right but not one we pursue here. The kind of argument we use easily generalizes to any sequence of $\alpha_k$. The difficulty lies in treating an arbitrary choice of $\gamma_k$.
6. **Definition of origin** By origin we mean the point $(0,0)$. We have clarified this throughout in the revised text.
7. **Why does AdaptiveEG+ have limit cycles?** Left subfigure of Figure 3 illustrates a case where $\rho < -1/(2L)$ for which we cannot guarantee convergence of AdaptiveEG+. Note that the two examples in Figure 3a and Figure 3b are distinct. We have updated Figure 3b and the caption to make this clearer.
8. **Labels for Figure 3-4** We thank the reviewer for paying so close attention. We have changed the y label.
9. **Show specific stepsizes in experimental section** We have updated the manuscript to reflect the suggestion.
10. **What does “this” refer to** We thank the reviewer for pointing out this ambiguity. It is indeed only CEG+ and we have updated the manuscript.
11. **Convergence of CurvatureEG+ is assumed not proved** Thank you for the comment. We have added the convergence results as a Corollary in the appendix.

We address the additional minor comments below.



1. **nonconvex-nonconcave minimax paper** We have added the reference to the related work.
2. **Recommended parameter choices for Algorithm 1?** For Theorem 3.1 to yield the fastest convergence rate, one should pick $\gamma_k$ as large as possible, i.e. $\gamma_k=1/L$, while $\delta_k$ should be as close to $\rho$ as possible in order to maximize $\kappa$ as it appears in Thm 3.1. However, if the rate is not of concern, and the aim is instead to guarantee convergence for the largest class possible under unknown $\rho$, then choose $\delta_k$ close to $-\gamma_k/2$.
3. **Figure 1 curves** The curves are indeed multiple trajectories from different initial points while the black ends indicate the initialization. We have included a description to clarify.
4. **Missing definitions** The missing definitions have now been added to the appendix. 
5. **Typos** The remaining typos have been corrected.

 
We thank the reviewer for the feedback. We address the specific concerns below.



* **Relationship with lower bounds for linear-span algorithms** Our lower bound is indeed different. We want to explicitly capture the dependency on the stepsize since it dictates the $\rho$. This is sufficient to show tightness of EG+ convergence rate. It should also be noted that the lower bound is of a different flavor than the bounds that usually considers linear-span algorithms. It is not concerned with giving a complexity result, but rather showing impossibility of convergence for the given algorithmic family.
* **Will the parameter choice of proposed algorithm bring any benefits in terms of the complexity (even in the constant)** As the reviewer noted, the parameter choice would result in potentially smaller constants compared to those in the literature. Without additional assumption the rate appears to be tight for this class of algorithms. However, this does not exclude a potentially better complexity under the weak MVI assumption for a different algorithmic class. 


 
We thank the reviewer for the comment.

We would like to clarify why we consider the algorithms contained in the paper. Our starting point is, as the reviewer mentions, the existing algorithm EG+ of (Diakonikolas et al. 2021). We show that not only can the range of the weak MVI constant be extended, but constrained problems can also be tackled. Our basic scheme is presented in Algorithm 1 which unlike EG+ involves an adaptive rule for the selection of stepsize $\alpha$. This leads to potentially larger stepsize. Indeed as we show in Section 3.1 by conservatively lower bounding $\alpha_k$ a (smaller) nonadaptive stepsize (denoted by $\bar \alpha_k$) may be obtained and in this setting we recover the known algorithms such as EG+ and Tseng's FBF. Finally, in Section 4 we explore a backtracking linesearch for the other stepsize $\gamma_k$. We then present the stochastic variant. However, in the stochastic setting it is not possible to establish convergence for the adaptive stepsize choice $\alpha$ as in Algorithm 1, and in fact diminishing stepsize is necessary. The aim of the paper is to say as much as possible in each setting.

We indeed overload for instance EG+ to refer to both the general scheme and to $\gamma=1/L$ and $\bar{\alpha}=1/2$ specifically depending on context. This, in particular, is in order to compare with the convergence results of (Diakonikolas et al. 2021). We have clarified further, in particular in the description of Figure 3b.

Concerning adaptive stepsizes, in general we stick to the convention that $\alpha_k$ denotes an adaptive stepsize while $\bar{\alpha}_k$ is constant. If there is ambiguity, we refer to the main theorems and lemmas, which we have made sure are self-contained, such that they refer to the exact assumptions and algorithms that they address.

Concerning applying the methods to real world problems, we agree that it is an important direction, but it is outside the scope of this work. We refer to the overview of problems that fall under weak MVI in Section 2.

Below we address the additional comments.



* **Missing definitions** We thank the reviewer for the comment. The missing definitions have now been added in the appendix A.
* **Typos** We thank the reviewer for pointing them out. We have fixed them in the revised text.
* **Naming Alg. 1** Names are included for the algorithms in the updated manuscript.
* **Convergence thm of CurvatureEG+** Thank you for the comment. We have added the convergence results as a Corollary in the appendix.
* **CEG+ on Forsaken** Both CEG+ and AdaptiveEG+ are not guaranteed to converge to the critical point since $\rho < - \frac{1}{2L}$. The updated manuscript now contains appendix C.4 showing this claim.
* **Does CurvatureEG+ escape unstable critical points?** In the almost bilinear toy example of Hsieh et al '21 that the reviewer mentions, CurvatureEG+ is not attracted to the attracting limit cycle of the underlying operator F but instead converges to the (unstable) critical point. In order for CurvatureEG+ to converge elsewhere we would need the zeros of F to contain more elements.
* **Show oracle calls for CurvatureEG+** We agree that comparing the methods in terms of the number of iterates can be misleading and we have updated Figure 3b to show the trajectories instead. However, even oracle calls or computational time on these toy examples can be misleading as CurvatureEG+ has very few backtracking calls and within a few iterations it converges to zero calls. We defer this type of empirical comparison for real world settings which we leave for future work. 
* **Comparing the convergence rate with [1]** Convergence rate analysis under assumptions such as Lipschitz continuity of the Jacobian (Assumption 2 in [1]) is not pursued in our work and is considered for future work.  \
 \
 [1] Last Iterate is Slower than Averaged Iterate in Smooth Convex-Concave Saddle Point Problems, Golowich et. al, 2020
 


 We thank the reviewer for the feedback and address the main concern below.

**Global Lipschitz continuity** For the bilinear Example 5 the assumption is satisfied. In the case of Example 4 and 5, we consider a constrained problem and the Lipschitz modulus of $F$ is computed within the constraint sets. Note that the Lipschitz continuity assumption is only needed to hold along the generated sequence at the points bar $z^k$ and $z^k$ (see (B.3)). By definition, $\bar{z}^k$ always stays within the bounded constraint set (see Step 1.1 of the algorithm). In these examples $z^k$ too stays within the constraint set, although in general it may not. We have now added additional clarifications for these examples in the paper. 

Note also that it is possible to relax the global Lipschitz continuity assumption to local Lipschitz continuity. This is not pursued in this paper and is considered for future work.  

We address the additional minor comments below.



* **Less tuning parameters** We have removed the claim.
* **Explicit interpretation of the projection onto a certain hyperplane** We agree with the reviewer, but due to page limitation we keep the interpretation in the appendix. Instead, we make it clear in the text that it can be found in the main proof. 
* **Add names of algorithms before the experimental section** The updated manuscript now includes the names in the definition of the algorithms.

This paper constructs methods named CurvatureEG+ (and Adaptive EG+ and CEG+), built upon a recently proposed EG+ [Diakonikolas et al., 2021], a variant of EG, that works under the weak MVI condition. Most importantly, the CurvatureEG+ (and Adaptive EG+/CEG+) works for a range (of the weak MVI) larger than that of EG+. The corresponding nonconvex-nonconcave setting includes non-trivial problems illustrated in the paper, where the proposed method converges, while other existing methods reach limit cycles. Unlike EG+, the CurvatureEG+ can handle both constrained and composite cases. A stochastic variant is also studied. Although the experiments consider toyish problems, they seem interesting. **Strengths**
- A larger range of $\rho$ of the weak MVI: The Adaptive EG+ and (a line-search-version) CurvatureEG+ work for a wider range of $\rho$, $\rho>-\frac{1}{2}$ or $\rho>-\frac{\gamma_k}{2}$ respectively, compared to that of EG+, $\rho>-\frac{1}{8}$. (This bound of EG+ could be weakened, but the authors show that this cannot be better than $\rho>-\frac{1}{4}$.) One might be view this improvement negligible (which I did when I glanced through the paper), but the wider region includes, e.g., the difficult "forsaken" example in [Hsieh et al., 2021] and the corresponding experimental results are interesting to read. For the stochastic case, the proposed SEG+ works for a same range of $\rho$, while a stochastic version of EG+ in [Diakonikolas et al., 2021] is shown to work for a more restrictive setting.
- Unified analysis of a larger class of algorithms: This work first used a projection onto a certain hyperplane technique with an adaptive step size. Then, the authors show that it includes a non-adaptive variant, named CEG+, which covers both EG+ and FBF [Tseng, 2000]. (This also allows a larger step size for EG+.) Although the projection on to a hyperplane is not new (while probably new to the machine learning community), such unifying analysis seems theoretically inspiring.
- Stochastic variant: A constant step size for the exploration step (while requiring diminishing step size for the other step) seems new and worth further investigation.
- Experiments: At a first glance, the weak MVI condition looked artificial, but the examples (while toyish) well complemented the importance of such new condition introduced recently.

**Weaknesses**
- One might view the convergence guarantee improvement of CurvatureEG+ over that of EG+ not significant.
- To be precise, none of the examples satisfy Assumption 1 (specifically the Lipschitz continuity) globally. 
- The authors claim that they have less tuning parameters, compared to the method in [Lee & Kim, 2021a], but Algorithm 1 seems to have relatively many tuning parameters. I think this can be resolved, for example, by providing standard choices such as $\lambda_k=1$, $\gamma_k = \frac{1}{L}$, $\delta_k = \rho$, requiring the knowledge of $\rho$ and $L$. This however still does not seem to have less tuning parameters than that in [Lee & Kim, 2021a].

**Other comments**
- Although readers can check the appendix to see an explicit interpretation of the projection onto a certain hyperplane in Algorithm 1, I personally suggest at least further introduce it in the main page if there is a space.
- I think it is worth mentioning related convergence results on the projection onto a certain hyperplane. What is new in that perspective seems missing in this paper. 
- The names of the proposed methods are given in the experiment section, but I suggest mentioning them in the main sections where they first appear for the readers who do not have time to read from the beginning to the end in details.
- After (.2): $\sigma_k$ -> $\delta_k$
- Before (.4): $\bar{z}$ -> $\bar{z}^k$ Built upon a recent study on generalizing EG for nonconvex-nonconcave setting, this paper provides a nice unified framework of EG using the projection onto a hyperplane. Although the improvement in terms of the convergence guarantee is not significant, this paper is theoretically inspiring, and the toyish experiments are informative, which I believe has potential for further research both theoretically and practically.


This paper focuses on variants of ExtraGradient (EG) by considering different options for the two step-sizes of the method: one for the extrapolation step and the second for the iterate update. 
Building on the analysis of (Diakonikolas et al. 2021),  the paper relaxes the setup therein by allowing a larger range of values for the $\rho$ parameter (see Fig. 2) of weak Minty Variational Inequality (MVI) which parameter controls the degree of nonconvexity.

In particular, it provides the following contributions:
*(i)* Primarily it defines Alg.1 where the step size for the iterate update is adaptive and shows the main convergence result (Thm  3.1.) that under some assumptions (Asm 1), Alg. 1 converges on weak MVI problem.
*(ii)* It then considers a non-adaptive variant of Alg. 1, dubbed CEG+, which can be seen as a generalization of the EG+ method of (Diakonikolas et al. 2021), and complements the result of the latter by showing that the convergence result is tight on weak MVI;
*(iii)* extends CEG+ by using an adaptive scheme for the first step size (for the extrapolation) which uses Lipschitz constant backtracking, and Alg.1 for the latter step size (for the iterate update), dubbed CurvatureEG+, which method is shown to empirically converge on some toy-examples on which CEG+ does not;
*(iv)* finally, for the stochastic setup the authors consider one of the step-sizes to be diminishing and the other can be constant, and show that this variant converges on weak MVI problems.
 # Strengths
- The paper has several theoretical contributions referring to different options for the two step-sizes of EG (see summary above), and importantly, it relaxes the setup of prior work--in particular of (Diakonikolas et al. 2021).
- The technical contributions are relevant for the community, given past negative results and the current state of existing provable convergence results. 
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# Weaknesses
- Inconsistency of methods the authors provide theoretical/empirical results for.  This paper mixes several different options for the step sizes, and the convergence for the stochastic counterparts of the considered full-batch methods is not shown. In particular, the stochastic method requires one of the step sizes to be diminishing, and the considered full-batch methods require different assumptions on the step sizes. 
- Lack of empirical evaluation to verify if the results are helpful for challenging real-world problems, or alternatively a discussion of the importance of the provided results in practice. 
- In terms of novelty, the paper largely builds on (Diakonikolas et al. 2021), and combines it with recent developments of operator splitting techniques (Latafat & Patrinos, 2017; Giselsson, 2021) to provide the relaxation of the problem.
- Due to shifting between the different options, the writing is somewhat confusing and hard to follow (see examples below). 

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# Minor Comments and Suggestions
- While the writeup seems technically precise, it is not self-contained. In particular, there are missing definitions. As these are the basis for understanding the presented results the definitions should be provided. Examples of non-defined terms: monotone operators, maximally monotone, cohypomonotonicity, etc., and even defining L-Lipschitz is common. 
- Some typos: 
   - in contribution 3: our result are $\rightarrow$ is
   - related works: approaches has been $\rightarrow$  have been
   - Hsieh et al .. considers  $\rightarrow$ consider
   - last sentence Sec. 1: they only considering $\rightarrow$ consider
- Consider naming Alg.1 
- In my opinion, the title could be misleading, it could be interpreted as a novel technique to escape limit cycles for general non-convex contained problems. It does not emphasize that this work specifically focuses on (variants of) extragradient and that the global convergence holds solely for the weak MVI setup. 
- It is hard to follow which proposed method the listed contributions refer to. It might be helpful to define the method in a general form before this listing so that you can be more precise about which step size is adaptive/constant/diminishing etc for each of the contributions.
- adding more detailed related work section (if lack of space in Appendix) to list relevant works on convergence results (e.g. the reference below)

----
# Questions
- Is it possible to provide a separate convergence theorem for CurvatureEG+ on weak MVI, ideally for a larger set of problems than Thm 3.1. as the results indicate?
- How does CEG+ perform on Forsaken (Fig.1)?
- Does CurvatureEG+ escape unstable critical points? In particular, how does it perform on the almost bilinear toy example of Hsieh et al '21? 
- Since CurvatureEG+ uses an iterative way to find the step size, it is important that the  authors provide a comparison with EG on a problem where EG converges with the x-axis being the computation time
- Could you comment on the convergence rate you get relative to existing results of EG on monotone variational inequalities, e.g. [1]

[1] Last Iterate is Slower than Averaged Iterate in Smooth Convex-Concave Saddle Point Problems, Golowich et. al, 2020 The paper provides several contributions for different variants of the extragradient method, which in general are of interest to the community. 
It lacks stochastic counterpart convergence guarantees of the full-batch results, and it considers several variants of the method rather than focusing mainly on the method with best convergence properties and to consistently discuss its convergence in full-batch setting, then stochastic, and finally verifying these results and ideally discussing/showing if these are relevant for challenging real-world setups.
In summary, the paper is somewhat rushed, and although it is in a good and valuable direction, at this stage, it requires major modifications (see comments above) and additional stochastic counterpart results.

This paper proposed CEG+ and CurvatureEG+, which extend extragradient method to a proximal variant (regarding the operator $A$), and apply it in nonconvex-nonconcave minimax optimization with Weak MVI condition in both deterministic and stochastic cases, and proved their complexities, which has the same order as those in literature (e.g., Diaonikolas et al., 2021). The authors also studied the lower bound in the simpler case when $A\equiv 0$, showing a difference compared to EG+ in (Diaonikolas et al., 2021).

In the deterministic case, it proposed an adaptive stepsize strategy to allow larger range of the MVI parameter $\rho$, and further a curvature-based strategy to avoid the lower bound requirement of $\rho$. The authors also executes several experiments, showing that CurvatureEG+ can avoid cycling in the experiments. The paper focuses on general minimax problems, which is a very important problem (even though with the special Weak MVI structure). The paper is clearly written and easy to read. 

The results of curvature-based adaptive stepsize and lower bound look interesting, which can reveal a significant difference compared to EG+, and rationalize the extension of the regime of $\rho$.

My main concern is that:
1. The lower bound requires EG+ to have a fixed stepsize $\gamma$, but as far as I know, lower bound analysis generally has no requirement on the stepsize (in fact they are using linear-span algorithms, so it can be arbitrary stepsize). So to some extent, now I view the lower bound result too specified and therefore a little restrictive, can the author comment more about it?
2. Even though with the craft stepsize strategy, the final complexity result is still similar to those in existing literature, will the parameter choice of proposed algorithm bring any benefits in terms of the complexity (even in the constant)? Or do you think the lower bound for the whole class of weak MVI problems is already achieved? Can the author comment more about it? Thank you.

Generally I view the results in the paper pretty interesting. This paper provides approach to avoid lower bound of Weak MVI parameter and provide a lower bound perspective on the fundamental difference compared with existing literature.

This work extends the extragradient algorithm in Diakonikolas et al. (2021) from unconstrainted and unregularized inclusion problems that satisfy weak Minty inequality (MVI) to their constrained and regularized counterparts. Compared with the original extragradient algorithm, the extended algorithm is proved to converge with a larger range of stepsize choices and MVI-related constant $\rho$ (implying a larger set of applicable problems) in both deterministic and stochastic inclusion problems. The range of $\rho$ is also proved tight by providing a lower bound of $\rho$. The extended algorithm also generalizes the celebrated forward-backward-forward (FBF) algorithm in Tseng (2000). Finally, an improvement of this extended algorithm is proposed using Lipschitz constant backtracking.  Pros:

(1) This paper studies the constrained/regularized nonconvex-nonconcave minimax optimization and its generalization called weak MVI problem, which are popular, important, general and hard problems. 

(2) There are abundant extensions of the original extragradient algorithm. 

(3) Both theorems and experiments convincingly demonstrate that there are inclusion problems where the original extragradient algorithm diverges while the proposed algorithm with adaptive stepsizes converges. 

Concerns: 

(1) In the introduction, you said "However, given the range of $\rho$ in Diakonikolas et al. (2021), the new class is still too small to include even the simplest counterexample of Hsieh et al. (2021) for the general Robbins-Monro schemes." Does your setting contain this simplest counterexample? If yes, I think it better to elaborate this counterexample and demonstrate that it belongs to your setting but not that in Diakonikolas et al. (2021), possibly in the appendix. If not, then I think it better not to mention this counterexample. 

(2) In Section 3.1, your maximum stepsize $\overline{\alpha}_k$ is $3/2$ of Diakonikolas et al. (2021). Is your convergence rate also faster than or comparable with that paper? A similar comparison of convergence rate might also be made between your Theorem 3.5 and Theorem 4.5(i) in Diakonikolas et al., 2021.  

(3) To my understanding, $\|\overline{z}^k-z^k\|$ is a generalization of proximal gradient, a popular measure for nonsmooth nonconvex minimization problems. Why do you use the convergence measure $\|H\overline{z}^k-Hz^k\|$ instead of $\|\overline{z}^k-z^k\|$?

(4) In Figure 2, should the region below the dashed line indicates where $\rho>-1/8L$ and the dashed line indicates where $\rho=-1/8L$? What does "positive result" mean? Diakonikolas et al. 2021 only obtains convergence results at the black square point while you obtain convergence results at the blue round point, right? 

(5) In Theorem 3.4, is it better to derive a lower bound of Algorithm 1 instead of EG+, i.e., the lower bound where Algorithm 1 may diverge? 

(6) What is the definition of "origin" in Example 4? 

(7) In the left subfigure of Figure 3, Adaptive EG+ seems to converge to the limit cycle instead of a stationary point. How do you explain? 

(8) In Figures 3 and 4, does "lower bound example" mean Example 5? Example 5 seems to have the maximally monotone operator $A\ne 0$. In this case, does $\|H\overline{z}^k-Hz^k\|= \|F\overline{z}^k\|$ still hold? If not, I think the y label should perhaps be $\|H\overline{z}^k-Hz^k\|$ instead of $\|F\overline{z}^k\|$. 

(9) In the experiment, for the algorithms with adaptive stepsizes, I think it better to give the specific stepsize values, such as $\gamma_k=1/k$, etc. 

(10) In the experiment, you said "this is the direct generalization to the constraint setting of the EG+ scheme studied Diakonikolas et al. (2021)." Does "this" mean CEG+ only, or both CEG and CEG+? 

(11) The convergence of CurvatureEG+ is assumed not proved in Lemma 4.1. Is it feasible to prove?

Minor comments:
(1) You might cite the following paper about nonconvex-nonconcave minimax optimization, and discuss about your advantage over it. 

[1] Yang, J., Kiyavash, N., \& He, N. (2020). Global convergence and variance-reduced optimization for a class of nonconvex-nonconcave minimax problems. ArXiv:2002.09621.

(2) In Theorem 3.1 and 3.4, is it feasible to recommend adaptive stepsizes that lead to the fastest possible convergence rate? 

(3) In Figure 1, I think it is better to explain the blue and red curves in either legend or caption. They are multiple trajectories from different initial points, right? Why do some red curves on the right subfigure have small black ends?

(4) I think it better to define "maximally monotone" and "cohypomonotonicity" or cite the papers that have the definition, as some readers like me see these concepts for the first time. 

(5) You might add "(MVI)" right after "Weak Minty inequality" in Assumption 1.

(6) There's a typo in "VIs provides a convenient abstraction for a range of problems", you could use "provide". 

(7) There's a typo in "Recal that the resolvent of a maximally monotone operator", you could use "Recall".

(8) In "by making the extrapolation step adaptive as well" in Section 3, you might explain which of equation (EG+) is "extrapolation step", or maybe use "making $\gamma_k$ adaptive as well"?

(9) Right below Corollary 3.2, should $\overline{z}^k-F\overline{z}^k-Hz^k$ be $\overline{z}^k-\gamma F\overline{z}^k-Hz^k$? 

(10) In the experiment, I think it better to add "(cf. Appendix A.1)" right after "Example 5".  Due to the aforementioned pros, I think this is an excellent paper and would like to see it accepted. Some points need to be clarified as aforementioned in the concerns and minor comments. 

Meta-review: The paper considers the saddle point problem of finding non-convex/non-concave minimax solutions. Building onEG+ of Diakonikolas et al., 2021 that works under weak MVI conditions, the work presents a new algorithm CurvatureEG+ that works for a larger range of weak MVI condition compared to previous work and also works for the constrained and composite cases. The authors show cases where this algorithm converges while the previous algorithms can be shown to reach limit cycles. Overall, this theoretical work seems strong. Most reviewers seem to agree that the contribution is good enough for publication. Compared to EG+ the additional contribution is to expand the range of weak MVI condition. While this seems like a slight improvement, looking beyond just the final convergence rate, the paper has some nice insights that provides a unifying view that captures past algorithms (like EG+ as special case). I recommend acceptance.

Decision: Accept (Spotlight)