SlowGuess/NeurIPS_2023_OCR huggingface dataset:
where  $\omega_{2} := \frac{2}{(r_{2} - L_{y})\tau_{1}}\left(\frac{r_{1}(1 - \beta)}{\beta} + \frac{r_{1}^{2}}{r_{1} - L_{x}}\right)^{\frac{1}{\theta_{1}}}$  and  $\omega_{3} := \frac{4r_{1}\operatorname{diam}(\mathcal{X})}{r_{2} - L_{y}}\left(\frac{1 - \beta}{\beta} + \sigma_{2}\right)$ . Here,  $\operatorname{diam}(\mathcal{X})$  denotes the diameter of the set  $\mathcal{X}$ .
Equipped with these results, we could also derive similar complexity results for (non)convex-nonconcave minimax problems as Theorem 1.
I Proof of Theorem 2
In general, analyzing the local convergence rate of sequences could be challenging. Fortunately, when  $\theta \in (0,\frac{1}{2} ]$ , the homogeneous sufficient descent property holds as follows:
$$
\begin{array}{l} \Phi (x ^ {t}, y ^ {t}, z ^ {t}, v ^ {t}) - \Phi (x ^ {t + 1}, y ^ {t + 1}, z ^ {t + 1}, v ^ {t + 1}) \\ \geq \frac {r _ {1}}{4 0} \| x ^ {t + 1} - x ^ {t} \| ^ {2} + \frac {r _ {2}}{2 0} \| y ^ {t} - y _ {+} ^ {t} (z ^ {t}, v ^ {t}) \| ^ {2} + \frac {r _ {1}}{4 \beta} \| z ^ {t} - z ^ {t + 1} \| ^ {2} + \frac {6 r _ {2}}{2 5 \mu} \| v _ {+} ^ {t} (z ^ {t + 1}) - v ^ {t} \| ^ {2}. \\ \end{array}
$$
Consequently, we can use the unified analysis framework in [3]. Next, we prove the "Safeguard" property as required there.
Proposition 7 ("Safeguard" property) There exists a series of parameters  $\nu_{1},\nu_{2},\nu_{3},\nu_{4} > 0$  such that for all  $t\in \{0,1,\ldots ,T - 1\}$ , the following holds:
$$
\begin{array}{l} \operatorname {d i s t} (\mathbf {0}, \partial \Phi (x ^ {t + 1}, y ^ {t + 1}, z ^ {t + 1}, v ^ {t + 1})) \\ \leq \nu_ {1} \| x ^ {t} - x ^ {t + 1} \| + \nu_ {2} \| y ^ {t} - y _ {+} ^ {t} (z ^ {t}, v ^ {t}) \| + \nu_ {3} \| v ^ {t} - v _ {+} ^ {t} (z ^ {t + 1}) \| + \nu_ {4} \| z ^ {t} - z ^ {t + 1} \|. \\ \end{array}
$$
Proof We observe from the update of  $x^{t + 1} = \operatorname*{argmin}_{x\in \mathcal{X}}\frac{1}{2}\| x - (x^t -c\nabla_xF(x^t,y^t,z^t,v^t))\|^2$  and  $y^{t + 1} = \operatorname*{argmin}_{y\in \mathcal{Y}}\frac{1}{2}\| y - (y^t +\alpha \nabla_yF(x^{t + 1},y^t,z^t,v^t))\|^2$  that
$$
\begin{array}{l} \mathbf {0} \in x ^ {t + 1} - x ^ {t} + c \nabla_ {x} F (x ^ {t}, y ^ {t}, z ^ {t}, v ^ {t}) + \partial \mathbf {1} _ {\mathcal {X}} (x ^ {t + 1}), \\ \mathbf {0} \in y ^ {t + 1} - y ^ {t} - \alpha \nabla_ {y} F \left(x ^ {t + 1}, y ^ {t}, z ^ {t}, v ^ {t}\right) + \partial \mathbf {1} _ {\mathcal {Y}} \left(y ^ {t + 1}\right). \tag {25} \\ \end{array}
$$
$$
\begin{array}{l} \partial \Phi (x ^ {t + 1}, y ^ {t + 1}, z ^ {t + 1}, v ^ {t + 1}) \\ = \left\{\nabla_ {x} F \left(x ^ {t + 1}, y ^ {t + 1}, z ^ {t + 1}, v ^ {t + 1}\right) + \partial \mathbf {1} _ {\mathcal {X}} \left(x ^ {t + 1}\right) \right\} \times \\ \{\nabla_ {y} F (x ^ {t + 1}, y ^ {t + 1}, z ^ {t + 1}, v ^ {t + 1}) - 2 \nabla_ {y} d (y ^ {t + 1}, z ^ {t + 1}, v ^ {t + 1}) + \partial \mathbf {1} _ {\mathcal {Y}} (y ^ {t + 1}) \} \times \\ \{\nabla_ {z} F (x ^ {t + 1}, y ^ {t + 1}, z ^ {t + 1}, v ^ {t + 1}) - 2 \nabla_ {z} d (y ^ {t + 1}, z ^ {t + 1}, v ^ {t + 1}) + 2 \nabla_ {z} q (z ^ {t + 1}) \} \times \\ \{\nabla_ {v} F (x ^ {t + 1}, y ^ {t + 1}, z ^ {t + 1}, v ^ {t + 1}) - 2 \nabla_ {v} d (y ^ {t + 1}, z ^ {t + 1}, v ^ {t + 1}) \}. \\ \end{array}
$$
This, together with (25), gives
$$
\begin{array}{l} \operatorname {d i s t} (\mathbf {0}, \partial \Phi (x ^ {t + 1}, y ^ {t + 1}, z ^ {t + 1}, v ^ {t + 1})) \\ \leq \left\| \frac {x ^ {t} - x ^ {t + 1}}{c} - \nabla_ {x} F (x ^ {t}, y ^ {t}, z ^ {t}, v ^ {t}) + \nabla_ {x} F (x ^ {t + 1}, y ^ {t + 1}, z ^ {t + 1}, v ^ {t + 1}) \right\| + \\ \left\| \frac {y ^ {t} - y ^ {t + 1}}{\alpha} + \nabla_ {y} F \left(x ^ {t + 1}, y ^ {t}, z ^ {t}, v ^ {t}\right) + \nabla_ {y} F \left(x ^ {t + 1}, y ^ {t + 1}, z ^ {t + 1}, v ^ {t + 1}\right) - 2 \nabla_ {y} d \left(y ^ {t + 1}, z ^ {t + 1}, v ^ {t + 1}\right) \right\| + \\ \left\| r _ {1} \left(z ^ {t + 1} - x ^ {t + 1}\right) - 2 r _ {1} \left(z ^ {t + 1} - x \left(y ^ {t + 1}, z ^ {t + 1}, v ^ {t + 1}\right)\right) + 2 r _ {1} \left(z ^ {t + 1} - x \left(z ^ {t + 1}, v \left(z ^ {t + 1}\right)\right)\right) \right\| + \\ \left\| r _ {2} \left(y ^ {t + 1} - v ^ {t + 1}\right) \right\|. \tag {26} \\ \end{array}
$$
Under Assumption 1 and utilizing the error bounds in Lemmas 2 and 4, we can simplify (26). When  $z^t$  reaches the local region that satisfies  $\beta \| x(z^t, v^{t+1}) - z^t \| \leq 1$ , we can show that the right-hand side is actually upper bounded by the linear combination of the desired four terms:
$$
\begin{array}{l} \operatorname {d i s t} (\mathbf {0}, \partial \Phi (x ^ {t + 1}, y ^ {t + 1}, z ^ {t + 1}, v ^ {t + 1})) \\ \leq \left(\frac {1}{c} + L _ {x} + r _ {1}\right) \| x ^ {t} - x ^ {t + 1} \| + L _ {x} \| y ^ {t} - y ^ {t + 1} \| + r _ {1} \| z ^ {t} - z ^ {t + 1} \| + \left(\frac {1}{\alpha} + L _ {y} + r _ {2}\right) \| y ^ {t} - y ^ {t + 1} \| + \\ 2 L _ {x} \left\| x ^ {t + 1} - x \left(y ^ {t + 1}, z ^ {t + 1}, v ^ {t + 1}\right) \right\| + r _ {2} \left\| v ^ {t} - v ^ {t + 1} \right\| + r _ {1} \left\| z ^ {t + 1} - x ^ {t + 1} \right\| + \\ 2 r _ {1} \| x \left(y ^ {t + 1}, z ^ {t + 1}, v ^ {t + 1}\right) - x \left(z ^ {t + 1}, v \left(z ^ {t + 1}\right)\right) \| + r _ {2} \| y ^ {t + 1} - v ^ {t + 1} \| \\ \leq \nu_ {1} \| x ^ {t} - x ^ {t + 1} \| + \nu_ {2} \| y ^ {t} - y _ {+} ^ {t} (z ^ {t}, v ^ {t}) \| + \nu_ {3} \| z ^ {t} - z _ {+} ^ {t} (v ^ {t + 1}) \| + \nu_ {4} \| v ^ {t} - v ^ {t + 1} \|. \\ \end{array}
$$
Here,  $\nu_{1} = \frac{1}{c} + L_{x} + r_{1} + 2L_{x}\sigma_{6} + 2r_{1}\sigma_{1}L_{y}\alpha\sigma_{6} + (L_{x} + \frac{1}{\alpha} + L_{y} + 2r_{2} + 2L_{x}\sigma_{1})L_{y}\alpha\sigma_{6},$ $\nu_{2} = L_{x} + \frac{1}{\alpha} + L_{y} + r_{2} + 2L_{x}\sigma_{1} + r_{1}(2\sigma_{1} + 1)(\sigma_{8} + 1), \nu_{3} = \frac{r_{1}}{\beta} + 2L_{x}\sigma_{2} + 2r_{1}\sigma_{1}\sigma_{3} + r_{2}\sigma_{3},$  and  $\nu_{4} = \frac{r_{2}}{\mu} + 2r_{1}\sigma_{1}\sigma_{5} + 2r_{1}\sqrt{\omega_{1}}$ . Moreover, since  $f(\cdot, y)$  and  $f(x, \cdot)$  is semi-algebraic [10, Theorem 3], using results in [10, Example 2], we find that  $\Phi(x, y, z, v) = F(x, y, z, v) - 2d(y, z, v) + 2q(z)$  is also semi-algebraic in  $x, y, z, v$ . Based on Assumption 1,  $\Phi(x, y, z, v)$  is continuous in  $x, y, z, v$ . Building on the unified convergence analysis framework in [4], we can conclude that  $\{(x^{t}, y^{t}, z^{t}, v^{t})\}$  converges to one stationary point  $(x^{*}, y^{*}, z^{*}, v^{*})$ . On top of Lemma 8 and  $\lim_{t \to \infty} \| x^{t} - x^{t+1} \| = 0$ ,  $\lim_{t \to \infty} \| y^{t} - y^{t+1} \| = 0$ , we have  $\{(x^{t}, y^{t})\}$  actually converges to a GS. For  $z^{t}$ , we have
$$
\begin{array}{l} \left\| z ^ {t} - x ^ {*} \left(z ^ {t}\right) \right\| \\ \leq \| z ^ {t} - x ^ {t + 1} \| + \| x ^ {t + 1} - x \left(y ^ {t}, z ^ {t}, v ^ {t}\right) \| + \| x \left(y ^ {t}, z ^ {t}, v ^ {t}\right) - x \left(z ^ {t}, v ^ {t}\right) \| + \| x \left(z ^ {t}, v ^ {t}\right) - x ^ {*} \left(z ^ {t}\right) \| \\ \leq \frac {\left\| z ^ {t} - z ^ {t + 1} \right\|}{\beta} + \sigma_ {6} \left\| x ^ {t} - x ^ {t + 1} \right\| + \sigma_ {1} \sigma_ {8} \left\| y ^ {t} - y _ {+} ^ {t} \left(z ^ {t}, v ^ {t}\right) \right\| + \omega_ {2} \left\| v ^ {t} - y \left(z ^ {t}, v ^ {t}\right) \right\| \\ \leq \frac {\left\| z ^ {t} - z ^ {t + 1} \right\|}{\beta} + \sigma_ {6} \left\| x ^ {t} - x ^ {t + 1} \right\| + \sigma_ {1} \sigma_ {8} \left\| y ^ {t} - y _ {+} ^ {t} \left(z ^ {t}, v ^ {t}\right) \right\| + \omega_ {2} \left(\frac {\left\| v ^ {t} - v _ {+} ^ {t} \left(z ^ {t + 1}\right) \right\|}{\mu} + \right. \\ \sigma_ {3} \left\| z ^ {t} - z ^ {t + 1} \right\| + 2 L _ {y} \alpha \sigma_ {6} \left\| x ^ {t + 1} - x ^ {t} \right\| + 2 \left(\sigma_ {8} + 1\right) \left\| y _ {+} ^ {t} \left(z ^ {t}, v ^ {t}\right) - y ^ {t} \right\|\). \\ \end{array}
$$
Since  $\lim_{t\to \infty}\| z^t -z^{t + 1}\| = 0,\lim_{t\to \infty}\| x^t -x^{t + 1}\| = 0,\lim_{t\to \infty}\| y^t -y_+^t (z^t,v^t)\| = 0,\lim_{t\to \infty}\| v^t -$ $v_{+}^{t}(z^{t + 1})\| = 0$  , we conclude  $\{z^t\}$  convergs to an OS.
J Proof of Theorem 3
Here, without the generality, we assume  $L_{x} = L_{y} = L$  and  $r_1 = r_2 = r$ . Then from the Proposition 1 and Proposition 5, we can easily derive the bounds for all the parameters as follows:
$$
r \geq 2 L, 0 <   c = \alpha = \min  \left\{\frac {4}{3 (L + r)}, \frac {1}{6 L _ {d}}, \frac {1}{5 \sqrt {6} L} \right\} <   1, \alpha \geq \frac {3 L (2 \alpha r + 1) ^ {2}}{2 (r - L) ^ {2}}, \tag {27a}
$$
$$
0 <   \beta = \mu \leq \min  \left\{\frac {2 4 r}{3 6 0 r + 5 r ^ {2} + (2 L + 5 r) ^ {2}}, \frac {c L ^ {2}}{9 2 1 6 r}, \frac {1 2}{1 2 + L ^ {2}} \right\} \leq \mathcal {O} \left(T ^ {- \frac {1}{2}}\right). \tag {27b}
$$
There are numerous feasible choices for  $\beta$  and  $\mu$  that satisfy equation (27b). Next, we will demonstrate the existence of a viable solution for  $c$  and  $\alpha$  by establishing the intersection of the last two constraints in equation (27a). Solving these constraints under the condition  $r\geq 2L$  yields
$$
\frac {- (r - L) \sqrt {L ^ {2} - 1 4 L r + r ^ {2}} + L ^ {2} - 8 L r + r ^ {2}}{1 2 L r ^ {2}} \leq c = \alpha \leq \frac {(r - L) \sqrt {L ^ {2} - 1 4 L r + r ^ {2}} + L ^ {2} - 8 L r + r ^ {2}}{1 2 L r ^ {2}}
$$
where  $r$  should further satisfy  $r \geq 14L$ . Combining these with the penultimate constraint in equation (27a), we can provide a set of feasible parameters as follows:
$$
r \geq 2 0 L, \frac {- (r - L) \sqrt {L ^ {2} - 1 4 L r + r ^ {2}} + L ^ {2} - 8 L r + r ^ {2}}{1 2 L r ^ {2}} \leq c = \alpha \leq \frac {1}{6 L _ {d}}. \tag {28}
$$
Armed with equation (27b) and equation (28), we finish the proof of Theorem 3.
K Relationship Between Different Notions of Stationary Points
In this section, we illustrate the quantitative relationship among several notions of stationary measure.
Definition 2 The point  $(x,y)\in \mathcal{X}\times \mathcal{Y}$  is said to be an
-  $\epsilon$ -optimization game stationary point (OS) if
$$
\left\| \operatorname {p r o x} _ {\frac {\operatorname* {m a x} _ {y \in \mathcal {Y}} f (\cdot , y)}{r _ {1}} + \mathbf {1} _ {\mathcal {X}}} (x) - x \right\| \leq \epsilon ;
$$
-  $\epsilon$ -game stationary point (GS) if
$$
\operatorname {d i s t} (\mathbf {0}, \nabla_ {x} f (x, y) + \partial \mathbf {1} _ {\mathcal {X}} (x)) \leq \epsilon , \operatorname {d i s t} (\mathbf {0}, - \nabla_ {y} f (x, y) + \partial \mathbf {1} _ {\mathcal {Y}} (y)) \leq \epsilon .
$$
Recalling that  $x^{*}(z) = \operatorname{argmin}_{x \in \mathcal{X}} \max_{y \in \mathcal{Y}} f(x, y) + \frac{r_1}{2} \| x - z \|^2$ , we have  $\| \operatorname{prox}_{\frac{\max_{y \in \mathcal{Y}} f(\cdot, y)}{r_1}}(x) - x \| = \| x^{*}(x) - x \|$ . With Proposition 3, we are ready to prove the relationship between OS and GS.
Proposition 8 Suppose that Assumption 2 holds. If  $(x,y) \in \mathcal{X} \times Y$  is a  $\epsilon$ -GS, then it is a  $\mathcal{O}(\epsilon^{\min\{1,\frac{1}{2\theta}\}})$ -OS.
Building on the proposition 3, we have
$$
\begin{array}{l} \left\| x ^ {*} (x) - x \right\| \leq \left\| x ^ {*} (x) - x (x, y) \right\| + \left\| x (x, y) - x \right\| \\ \leq \omega_ {2} \| y - y (x, y) \| ^ {\frac {1}{2 \theta}} + \| x (x, y) - x \| \tag {29} \\ \leq \omega_ {2} \| y - y (x, y) \| ^ {\frac {1}{2 \theta}} + \| x (y, x, v) - x \| + \sigma_ {1} \| y - y (x, y) \|. \\ \end{array}
$$
By the nonexpansiveness of projection operator and error bounds in Lemmas 3 and 4, we can further bound  $\| y - y(z,v) \|$  as follows:
$$
\begin{array}{l} \| y - y (z, v) \| \leq \sigma_ {8} \| y - y _ {+} (z, v) \| \\ = \sigma_ {8} \| y - \operatorname {p r o j} _ {\mathcal {Y}} (y + \alpha \nabla_ {y} F (x (y, z, v), y, z, v)) \| \\ \leq \sigma_ {8} \| y - \operatorname {p r o j} _ {\mathcal {Y}} (y + \alpha \nabla_ {y} F (x, y, z, v)) \| + \\ \sigma_ {8} \| \operatorname {p r o j} _ {\mathcal {Y}} (y + \alpha \nabla_ {y} F (x, y, z, v)) - \operatorname {p r o j} _ {\mathcal {Y}} (y + \alpha \nabla_ {y} F (x (y, z, v), y, z, v)) \| \\ \leq \sigma_ {8} \| y - \operatorname {p r o j} _ {\mathcal {Y}} (y + \alpha \nabla_ {y} F (x, y, z, v)) \| + L _ {y} \alpha \sigma_ {8} \| x - x (y, z, v) \| \\ \leq \sigma_ {8} \| y - y (x, z, v) \| + \sigma_ {8} \| y (x, z, v) - \operatorname {p r o j} _ {\mathcal {Y}} (y + \alpha \nabla_ {y} F (x, y, z, v)) \| + L _ {y} \alpha \sigma_ {8} \| x - x (y, z, v) \| \\ \leq \sigma_ {8} \| y - y (x, z, v) \| + \sigma_ {8} \left(1 + \alpha L _ {y} + \alpha r _ {2}\right) \| y (x, z, v) - y \| + L _ {y} \alpha \sigma_ {8} \| x - x (y, z, v) \| \\ \leq L _ {y} \alpha \sigma_ {8} \| x - x (y, z, v) \| + \left(2 + \alpha L _ {y} + \alpha r _ {2}\right) \sigma_ {8} \| y - y (x, z, v) \|. \\ \end{array}
$$
Plugging this bound into (29), we get
$$
\begin{array}{l} \left\| x ^ {*} (x) - x \right\| \leq \sigma_ {8} \sigma_ {1} \left(2 + \alpha L _ {y} + \alpha r _ {2}\right) \| y - y (x, x, y) \| + \left(L _ {y} \alpha \sigma_ {8} \sigma_ {1} + 1\right) \| x - x (y, x, y) \| + \\ \omega_ {2} \left(L _ {y} \alpha \sigma_ {8} \| x - x (y, x, y) \| + \left(2 + \alpha L _ {y} + \alpha r _ {2}\right) \sigma_ {8} \| y - y (x, x, y) \|\right) ^ {\frac {1}{2 \theta}}. \\ \end{array}
$$
Next, we will explore the relationship between  $\| x - x(y,x,v)\|$  and  $\mathrm{dist}(\mathbf{0},\nabla_xf(x,y) + \partial \mathbf{1}_{\mathcal{X}}(x))$ . Let  $x_{+}(y,x,v)\coloneqq \mathrm{proj}_{\mathcal{X}}(x - c\nabla_{x}F(x,y,x,v))$ . Then, from the primal error bound (see [49]), we know that
$$
\| x - x (y, x, v) \| \leq \frac {c L _ {x} + c r _ {1} + 1}{c r _ {1} - c L _ {x}} \| x - x _ {+} (y, x, v) \|.
$$
Moreover, since  $\nabla_x F(x,y,x,v) = \nabla_x f(x,y)$ , it follows from [39, Lemma 4.1] that
$$
\begin{array}{l} \| x - x (y, x, v) \| \leq \frac {c L _ {x} + c r _ {1} + 1}{c r _ {1} - c L _ {x}} \| x - x _ {+} (y, x, v) \| \\ = \frac {c L _ {x} + c r _ {1} + 1}{c r _ {1} - c L _ {x}} \| x - \operatorname {p r o j} _ {\mathcal {X}} (x - c \nabla_ {x} f (x, y)) \| \\ \leq \frac {c L _ {x} + c r _ {1} + 1}{r _ {1} - L _ {x}} \operatorname {d i s t} (\mathbf {0}, \nabla_ {x} f (x, y) + \partial \mathbf {1} _ {\mathcal {X}} (x)). \\ \end{array}
$$
A similar analysis can be applied to derive the bounds for  $\| y - y(x,z,y)\|$ . Thus, if  $(x,y)$  is a  $\epsilon$ -GS, then it is an  $\mathcal{O}(\epsilon^{\min\{1,\frac{1}{2\theta}\}})$ -OS.
L Details about Examples in Section 4.3 and Behaviors of Wrong Selection of Smoothing Sides in S-GDA
This section will check all the regularity conditions for the examples mentioned in Section 4.3. The violation of the global KL condition can be easily verified by plotting the figures for these examples. Therefore, we mainly discuss whether they satisfy "weak MVI" and "α-interaction dominant" conditions, which are two representative classes of conditions in the nonconvex-nonconcave setting. Moreover, we will also give an example showing the slow convergence caused by the wrong selection of S-GDA.
L.1 Proof of Proposition for "Forsaken" Example
This subsection considers the "Forsaken" example in [32, Example 5.2] on the constraint sets  $\mathcal{X} = \mathcal{Y} = \{z: -1.5 \leq z \leq 1.5\}$ . In [50], it is verified that the "Forsaken" example violates "weak MVI" condition with  $\rho < -\frac{1}{2L}$ . Therefore, we only consider the  $\alpha$ -interaction dominant condition here. By a simple calculation, we get  $\nabla_{xx}^2 f(x,y) = \frac{1}{2} - 6x^2 + 5x^4$ ,  $\nabla_{xy}^2 f(x,y) = \nabla_{yx}^2 f(x,y) = 1$ , and  $\nabla_{yy}^2 f(x,y) = -\frac{1}{2} + 6y^2 - 5y^4$ . Armed with these,  $\alpha$  can be found globally by minimizing the following:
$$
\begin{array}{l} \nabla_ {x x} ^ {2} f (x, y) + \nabla_ {x y} ^ {2} f (x, y) (\eta {\bf 1} - \nabla_ {y y} ^ {2} f (x, y)) ^ {- 1} \nabla_ {y x} ^ {2} f (x, y) \\ = \frac {1}{2} - 6 x ^ {2} + 5 x ^ {4} + \left(\eta + \frac {1}{2} - 6 y ^ {2} + 5 y ^ {4}\right) ^ {- 1}. \\ \end{array}
$$
It is less than zero when  $[x; y] = [1; 0]$ . That is,  $\alpha < 0$  in the constraint set  $\mathcal{X}$ , which means the  $\alpha$ -interaction dominant condition is violated for the primal variable  $x$ . Similar proof could be adapted for the dual variable. This rules out the convergence guarantees of damped PPM, which is validated in Figure 4.
L.2 Proof of Proposition for "Blinearly-coupled Minimax" Example
This "Bilinearly-coupled Minimax" example is mentioned as a representative example where  $\alpha$  is in the interaction moderate regime [29]. Experiments also validate that the solution path will be globally trapped into a limit cycle (see Figure 4). For this reason, we only check the "weak MVI" condition. In this example,  $\mathcal{X} = \mathcal{Y} = \{z: -4 \leq z \leq 4\}$ ,  $G(u) = [\nabla_x f(x,y); -\nabla_y f(x,y)] = [4x^3 - 20x + 10y; 4y^3 - 20y - 10x]$ , and  $u^\star = [0;0]$ . Then,  $\rho$  can be found by globally minimizing  $\rho(u) := \frac{\langle G(u), u - u^\star \rangle}{\|G(u)\|^2}$  for all  $u \in \mathcal{X} \times \mathcal{Y}$ . Notice that
$$
\frac {\langle G (u) , u - u ^ {\star} \rangle}{\| G (u) \| ^ {2}} = \frac {x ^ {4} + y ^ {4} - 5 x ^ {2} - 5 y ^ {2}}{(2 x ^ {3} - 1 0 x + 5 y) ^ {2} + (2 y ^ {3} - 1 0 y - 5 x) ^ {2}}.
$$
We have  $\rho(u) = \frac{\langle G(u), u - u^{\star} \rangle}{\|G(u)\|^2} = -\frac{4}{89}$  when  $u = [x; y] = [0; 1]$ , which implies that  $\rho < -\frac{4}{89}$ . Moreover, we find  $L = 172$ , so  $\rho < -\frac{4}{89} < -\frac{1}{344} = -\frac{1}{2L}$ . We conclude that this example does not satisfy the "weak MVI" condition and the limit cycle phenomenon is actually observed in Figure 4.
Figure 5: Convergence behavior of DS-GDA and different smoothing sides for S-GDA.
L.3 Proof of Proposition for "Sixth-order polynomial" Example
We demonstrate that this example violates both the 'weak MVI' and '  $\alpha$  -interaction' conditions. To provide evidence of this violation, we follow the same approach used to prove violations in Sections L.1 and L.2 and present counter-examples. On the one hand, we found that  $\rho \leq \rho (\hat{u}) = -0.0795 < - 0.0368 = -\frac{1}{2L}$  with  $\hat{u} = [x;y] = [-1;0.5]$ , which implies the violation of "weak MVI" condition. On the other hand, we have
$$
\begin{array}{l} \nabla_ {x x} ^ {2} f (x, y) + \nabla_ {x y} ^ {2} f (x, y) (\eta \mathbf {1} - \nabla_ {y y} ^ {2} f (x, y)) ^ {- 1} \nabla_ {y x} ^ {2} f (x, y) \Bigg | _ {[ x; y ] = [ 0; 1 ]} \\ = (2 1 6 0 9 \exp (- 1 / 5 0)) / (6 2 5 (\eta + (7 7 0 6 1 * \exp (- 1 / 1 0 0)) / 2 5 0 0 0)) - (4 9 8 9 \exp (- 1 / 1 0 0)) / 5 0 0. \\ \end{array}
$$
It is less than zero when  $\eta \geq L$ , which indicates the violation of the  $\alpha$ -interaction condition.
L.4 Proof of Proposition for "PolarGame" Example
In this subsection, we check the two conditions for the "PolarGame" example. Firstly, we have  $\rho \leq \rho (\hat{u}) = -0.3722 < -0.0039 = -\frac{1}{2L}$  with  $\hat{u} = [0.8;0]$ . Thus, it does not satisfy the "weak MVI" condition. Next, consider the following at  $[x;y] = [0.8;0]$ :
$$
\begin{array}{l} \nabla_ {x x} ^ {2} f (x, y) + \nabla_ {x y} ^ {2} f (x, y) (\eta {\bf 1} - \nabla_ {y y} ^ {2} f (x, y)) ^ {- 1} \nabla_ {y x} ^ {2} f (x, y) \Bigg | _ {[ x; y ] = [ 0. 8; 0 ]} \\ = 1 / (\eta - 2 7 9 / 6 2 5) - 7 7 9 / 1 2 5. \\ \end{array}
$$
It is less than zero when  $\eta \geq L$ . Thus, the "PolarGame" violates the  $\alpha$ -interaction condition.
L.5 Wrong Smoothing Side of S-GDA
For S-GDA, we show that if we choose the wrong side, it will result in a slow convergence. To validate this, we conduct a new experiment for the KL-NC problem
$$
\min _ {x \in \mathcal {X}} \max _ {y \in \mathcal {Y}} f (x, y) = 2 x ^ {2} - y ^ {2} + 4 x y ^ {6} + \frac {4 y ^ {3}}{3} - \frac {y ^ {4}}{4},
$$
where  $\mathcal{X} = \mathcal{Y} = \{z: -1 \leq z \leq 1\}$ . With a wrong smoothing side, S-GDA (with primal smoothing) leads to a slower convergence compared with dual smoothing (see Figure 5).