Paper: Semi-Anchored Gradient Methods for Nonconvex-Nonconcave Minimax Problems (ICLR-2024)

Reviewer comment (Experiments):
It is advisable to compare the proposed algorithm with recent papers on nonconvex-nonconcave minimax problems that are based on various regularity conditions, such as dominant conditions and the PL inequality, in order to demonstrate the competitiveness of the proposed approach.

Author response:
Regarding the dominant and PL conditions mentioned in the weaknesses section, the fair classification problem is linear in the max-player, so it is obvious that the dominant and PL conditions do not hold. (The toy example also does not satisfy those conditions.) Nevertheless, we have already ran the GDmax (without regularization) that is shown to work under the PL condition in the max-player (but not under the weak MVI), which was slower than the SA-GDmax in our fair classification experiment.

Reviewer comment (Theory):
In Theorems 5 and 6, the use of gradient computational cost may not be ideal. The $\mathcal{O}(\log(1/\epsilon))$ cost pertains to the iteration cost of the proximal gradient descent method, where the computational cost of the proximal operator is neglected.

Author response:
You are correct that we omitted the computational cost of the proximal operator. We clarified in revision that we are considering the prox-friendly function, which makes the computational cost of the proximal operation negligible.

Reviewer comment (Experiments):
What is the numerical performance of the algorithm for nonconvex-nonconcave problems without the weak MVI condition?

Author response:
The fair classification experiment is the problem possibly without the weak MVI condition, which is exactly the setting the reviewer is asking for. We clarified this in the paper.

Reviewer comment (Novelty):
All remaining results in the paper instead show rates for the _tangent residual_ (as soon as either inexactness appears or the hyperplane projection is used). If we are interested in the tangent residual in the first place, then a  $\mathcal O(1/k)$ rate can be achieved by an _explicit_ scheme  _without_ (inexact) max-oracles by a primal-dual extragradient scheme.

Author response:
- (W2) We agree that it would have been the best if we were able to come up with an explicit method that exactly reduces to the PDHG for the bilinear problem and that is comparable to the extragradient in terms of the convergence rate under the weak MVI. This, however, was not a simple task, which we hoped to resolve by starting from the Bregman proximal point perspective. Nevertheless, we still believe our finding has a merit, and we are sharing this work in the hope that this becomes a foundation of developing a decent PDHG-type method for general minimax problems.

Reviewer comment (Theory):
The reviewer is surprised that bounded domain is needed in Thm. 5, and asks if it is not possible to use that inexact proximal point is (approximately) nonexpansive up to an error you can control (through the approximate subsolver of the max-oracle).

Author response:
- (Q1) We really appreciate your feedback, and as you expected, we were able to simply remove the bounded domain condition in Theorem 5 of the SA-MGDA, by adopting the flow of the proof of Theorem 6.

Reviewer comment (Reproducibility):
In numerical experiments, it is unclear whether the authors implemented their SA-GDmax or the more practical SA-MGDA algorithm. If it is SA-GDmax as in (7), clarification is needed on how the authors conducted the v substep precisely.

Author response:
We implemented the exact SA-GDmax, as we have an efficient max-oracle for both experiments. In particular, the max step of the fair classification problem is exactly maximizing a quadratic function over a simplex, or equivalently, an orthogonal projection problem to a simplex, given the value of the loss function for each category. This can be efficiently done in $O(d_v\log(d_v))$, where $d_v$ is the dimension of $v$ (arXiv: 1309.1541). Note that the orthogonal projection to a simplex is also needed for other methods.

Reviewer comment (Experiments):
The authors did not compare with many of the methods in Table 1, and they should consider adding more numerical comparisons.

Author response:
Among extragradient-type methods, we chose CEG+ that is known to work under the weakest condition we consider. We already have ran EG and EG+ and those worked poorly.

Reviewer comment (Writing):
Some typos and unclear points are listed in the questions below.

Author response:
- (Q4-6,Q10-11) Thanks for the corrections and the suggestions. We have revised accordingly.

Reviewer comment (Writing):
The drawbacks of extragradient and advantages of PDHG could be briefly mentioned in the abstract and the beginning of the Introduction, instead of stating "there is still room for improvement". Also, in the abstract, is the "worst-case convergence rate" lacking in extragradient methods? If yes, this could be mentioned in the abstract.

Author response:
- (Q4-6,Q10-11) Thanks for the corrections and the suggestions. We have revised accordingly.

Reviewer comment (Novelty):
What is your advantage over the works "Fast extra gradient methods for smooth structured nonconvex-nonconcave minimax problems" and "Stable Nonconvex-Nonconcave Training via Linear Interpolation"? You may cite the latter.

Author response:
- (Q3) They are both interesting, as the former discusses the acceleration, and the latter studies last iterate convergence for the first-time in nonconvex-nonconave problems. However, they both assume the (negative-)comonotonicity that is stronger than the weak MVI that we assume, and how one can extend their results to the weak MVI seem interesting but unclear at the moment. We cited the latter one, next to the former.

Reviewer comment (Writing):
In the abstract sentence "This was studied on a general convex-concave problem, but it has not been found useful in a more general nonconvex-nonconcave minimax problem. In this paper, we demonstrate its natural extension to a structured nonconvex-nonconcave minimax problem", the words "it" and "its" are far away from PDHG and could be replaced by PDHG for clarity.

Author response:
- (Q4-6,Q10-11) Thanks for the corrections and the suggestions. We have revised accordingly.

Reviewer comment (Writing):
At the end of the second paragraph of the Introduction, the phrase "a new nonlinear variant of the PDHG, named semi-anchored (SA) gradient method" could be clearer.

Author response:
- (Q4-6,Q10-11) Thanks for the corrections and the suggestions. We have revised accordingly.

Reviewer comment (Writing):
On page 4, in the sentence "the GDmax minimizes the equivalent minimization problem", the word "minimizes" could be changed to "solves".

Author response:
- (Q4-6,Q10-11) Thanks for the corrections and the suggestions. We have revised accordingly.

Reviewer comment (Theory):
How to compute \( R(x) \)? You could explain or cite this in your paper. Can \( R(x) \) be exactly solved? If not, it is recommended to include such an error in the convergence results.

Author response:
- (Q7) Computing $R(x)$ exactly and inexactly corresponds to SA-GDmax and SA-MGDA, respectively. In particular, the analysis for the inexact computation of $R(x)$ is provided in Theorems 5 and 6 for SA-MGDA (with and without projection). The explicit and detailed analysis of the inexactness of $R(x)$ can be found in Appendix D where we present the proofs of Theorems 5 and 6.

Reviewer comment (Writing):
In Section 4.3, you stated "This has several advantages over the standard BPP, which will be detailed later." Later, only one advantage of a larger range of \( \rho \) is found. Are there any other advantages?

Author response:
- (Q8) As mentioned in the general comment, the projection technique was originally introduced to relax the bounded domain assumption in Theorem 5 of the SA-MGDA, which is now removed. So, now there is no advantage other than having a larger range of $\rho$.

Reviewer comment (Writing):
In Section 5.1, can \( \widehat{L} \) be replaced with the previously defined \( \gamma \)?

Author response:
- (Q9) You are right that they can be interchanged in the current form. To clarify this, we moved the definition of $\hat{L}$ earlier in front of Assumption A2, and leave $\gamma$ only for the definition of weak monotonicity.

Reviewer comment (Theory):
Right after "it resembles GDmax", you could indicate that we can also obtain SA-GDmax with projection using BPP with projection (4) using \( h \) in (5).

Author response:
- (Q4-6,Q10-11) Thanks for the corrections and the suggestions. We have revised accordingly.

Reviewer comment (Theory):
In the toy example, what is the function \( \phi \)? Should it be \( +\frac{L^2\rho}{4}u^2 \) and \( -\frac{L^2\rho}{4}v^2 \) to correspond to \( +f(u) \) and \( -g(v) \) in problem (1)?

Author response:
- (Q12) The considered function is $\phi$, as written in the paper, and we forgot to inform $f(u)=g(v)=0$, which we added in the revision.