Strengths
The presentation is clear. Based on my knowledge on both shuffling and minimax optimization, I found that the algorithms and complexity results are reasonable. The experiments provide sufficient details for reproducibility.
Weaknesses
The contribution is in general combinational, with slightly more techniques such as smoothing and shuffling-based function evaluations. Also, comparison with previous stochastic algorithms for minimax optimization and shuffling for nonconvex optimization is lacking, as shown in question 1 below.
Questions
(1) You could compare your complexity results (Theorem 2) with previous stochastic gradient-based minimax optimization works (e.g. [a]) and shuffling for nonconvex optimization (e.g. [b]).
[a] Lin, T., Jin, C., \& Jordan, M. (2020, November). On gradient descent ascent for nonconvex-concave minimax problems. In International Conference on Machine Learning (pp. 6083-6093). PMLR.
[b] Nguyen, L. M., Tran-Dinh, Q., Phan, D. T., Nguyen, P. H., \& Van Dijk, M. (2021). A unified convergence analysis for shuffling-type gradient methods. Journal of Machine Learning Research, 22(207), 1-44.
(2) At the bottom of page 1, for problem (NL), it looks possible to remove matrix $K$ and use $F_i:\mathbb{R}^p\to \mathbb{R}^q$, since $\langle F_i(w),Ku\rangle=\langle K^{\top}F_i(w),u\rangle$ and we could replace $K^{\top}F_i(w)\in \mathbb{R}^q$ with $F_i(w)\in \mathbb{R}^q$.
(3) In Assumption 2, do you mean for any $v\in {\rm dom}(h)$, $||v||_2\le M_h$?
(4) In "Condition (6) can be relaxed to the form $\frac{1}{n} \sum_{i=1}^n||\nabla F_i(w)-\nabla F(w)||^2 \leq \sigma_J^2+\Theta_J||\nabla \Phi_0(w)||^2$", should $\Phi_0$ be $F$?
(5) After Assumption 5, ``If $f=0$, then $\mathcal{G} _ {\eta}(w)\equiv\nabla\Phi_{\gamma}(w)$''. Should it be $\nabla\Phi_0(w)$ instead of $\nabla\Phi_{\gamma}(w)$, since the definition of $\mathcal{G} _ {\eta}(w)$ in eq. (18) involves $\nabla\Phi_0(w)$ not $\nabla\Phi _ {\gamma}(w)$? Or you might consider defining $\mathcal{G} _ {\eta,\gamma}(w)$ with $\nabla\Phi_0(w)$ replaced by $\nabla\Phi _ {\gamma}(w)$, such that $\epsilon$-stationary point of (3) and that of (10) in Lemma 2 correspond to $\mathcal{G} _ {\eta,0}(w)$ and $\mathcal{G} _ {\eta,\gamma}(w)$ respectively.
(6) I found that Eq. (21) (Option 1) needs to save {$F_{\pi^{(t)}(j)(w_0^{(t)})}$}$_{j=1}^n$ to avoid repeated evaluation of these $n$ values. Yes?
(7) Line 255: "fix number" $\rightarrow$ "fixed number".
(8) At the end of Theorem 2, do you mean to change one of the $\nabla_w\mathcal{H}_i$ into $\nabla_u\mathcal{H}_i$? It seems that the complexity of semi-shuffling is bettter than full-shuffling. Then what's the advantage of full-shuffling?
Limitations
The checklist mentions the limitation that this work only focuses on nonconvex-linear and nonconvex-strongly convex minimax optimization, and says "We do not yet know if our paper has an immediate broader impact. However, since our problems and our algorithms are sufficiently general, we hope they will create broader impacts." I agree with both of them.