SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Yuanyuan Liu $^{1}$ , Fanhua Shang $^{2,*}$ , Weixin An $^{1}$ , Junhao Liu $^{1}$ , Hongying Liu $^{3,6}$ , Zhouchen Lin $^{4,5,6}$
<sup>1</sup>Key Laboratory of Intelligent Perception and Image Understanding of Ministry of Education, School of Artificial Intelligence, Xidian University, China
$^{2}$ School of Computer Science and Technology, College of Intelligence and Computing, Tianjin University Medical College, Tianjin University, China
$^{4}$ National Key Lab of General AI, School of Intelligence Science and Technology, Peking University Institute for Artificial Intelligence, Peking University
Peng Cheng Laboratory
yyliu@xidian.edu.cn, fhshang@tju.edu.cn, hyliu2009@tju.edu.cn  
zlin@pku.edu.cn
In this paper, we propose a novel extra-gradient difference acceleration algorithm for solving constrained nonconvex-nonconcave (NC-NC) minimax problems. In particular, we design a new extra-gradient difference step to obtain an important quasi-cocoercivity property, which plays a key role to significantly improve the convergence rate in the constrained NC-NC setting without additional structural assumption. Then momentum acceleration is also introduced into our dual accelerating update step. Moreover, we prove that, to find an  $\epsilon$ -stationary point of the function  $f$ , the proposed algorithm attains the complexity  $\mathcal{O}(\epsilon^{-2})$  in the constrained NC-NC setting, while the best-known complexity bound is  $\widetilde{\mathcal{O}}(\epsilon^{-4})$ , where  $\widetilde{\mathcal{O}}(\cdot)$  hides logarithmic factors compared to  $\mathcal{O}(\cdot)$ . As the special cases of the constrained NC-NC setting, the proposed algorithm can also obtain the same complexity  $\mathcal{O}(\epsilon^{-2})$  for both the nonconvex-concave (NC-C) and convex-nonconcave (C-NC) cases, while the best-known complexity bounds are  $\widetilde{\mathcal{O}}(\epsilon^{-2.5})$  for the NC-C case and  $\widetilde{\mathcal{O}}(\epsilon^{-4})$  for the C-NC case. For fair comparison with existing algorithms, we also analyze the complexity bound to find  $\epsilon$ -stationary point of the primal function  $\phi$  for the constrained NC-C problem, which shows that our algorithm can improve the complexity bound from  $\widetilde{\mathcal{O}}(\epsilon^{-3})$  to  $\mathcal{O}(\epsilon^{-2})$ . To the best of our knowledge, this is the first time that the proposed algorithm improves the best-known complexity bounds from  $\widetilde{\mathcal{O}}(\epsilon^{-4})$  and  $\widetilde{\mathcal{O}}(\epsilon^{-3})$  to  $\mathcal{O}(\epsilon^{-2})$  in both the NC-NC and NC-C settings.
This paper considers the following smooth minimax optimization problem,
$$
\min  _ {x \in \mathcal {X}} \max  _ {y \in \mathcal {Y}} f (x, y), \tag {1}
$$
*Corresponding authors
where  $\mathcal{X} \subseteq \mathbb{R}^m$  and  $\mathcal{Y} \subseteq \mathbb{R}^n$  are nonempty closed and convex feasible sets, and  $f: \mathbb{R}^m \times \mathbb{R}^n \to \mathbb{R}$  is a smooth function. In recent years, this problem has drawn considerable interest from machine learning and other engineering communities such as generative adversarial networks [14], adversarial machine learning [15, 28], game theory [3], reinforcement learning [12, 41], empirical risk minimization [54, 39], and robust optimization [36, 13, 5]. While there is an extensive body of literature on minimax optimization, most prior works such as [54, 39, 42, 33, 50] focus on the convex-concave setting, where  $f(x,y)$  is convex in  $x$  and concave in  $y$ . However, in many nonconvex minimax machine learning problems as in [35, 6],  $f(x,y)$  is nonconvex in  $x$  and (strongly) concave in  $y$ , or  $f(x,y)$  is (strongly) convex in  $x$  and nonconcave in  $y$ . For nonconvex-strongly-concave (NC-SC) minimax problems, a number of efficient algorithms such as [34, 25, 27] were proposed, and their complexity can be  $\widetilde{\mathcal{O}}(\kappa_y^2\epsilon^{-2})$  for achieving an  $\epsilon$ -stationary point  $\hat{x}$  (e.g.,  $\|\nabla\phi(\hat{x})\| \leq \epsilon$ ) of the primal function  $\phi(\cdot) := \max_{y \in \mathcal{Y}} f(\cdot,y)$ , where  $\kappa_y$  is the condition number for  $f(x,\cdot)$ , and  $\widetilde{\mathcal{O}}$  hides logarithmic factors. More recently, [24] proposed an accelerated algorithm, which improves the gradient complexity to  $\widetilde{\mathcal{O}}(\sqrt{\kappa_y\epsilon^{-2}})$ , which exactly matches the lower complexity bound in [52, 23]. Therefore, we mainly consider the problem (1) in nonconvex-nonconcave (NC-NC), nonconvex-concave (NC-C) and convex-nonconcave (C-NC) settings.
1.1 Algorithms in NC-C and C-NC Settings
For nonconvex-concave (NC-C, but not strongly concave) and convex-nonconcave (C-NC) problems, there are two types of algorithms (i.e., multi-loop (including double-loop and triple-loop) and single-loop algorithms), and most of them are multi-loop algorithms such as [18, 31, 40, 24]. [31] proposed a multi-step framework that finds an  $\epsilon$ -first order Nash equilibrium of  $f(x,y)$  with the complexity  $\widetilde{\mathcal{O}} (\epsilon^{-3.5})$ . [40] designed a proximal dual implicit accelerated algorithm and proved that their algorithm finds an  $\epsilon$ -stationary point of  $\phi$  with the complexity  $\widetilde{\mathcal{O}} (\epsilon^{-3})$ . More recently, [24] proposed an accelerated algorithm, which achieves the complexity  $\widetilde{\mathcal{O}} (\epsilon^{-2.5})$  to find an  $\epsilon$ -stationary point of  $f$ . These multi-loop algorithms require at least  $\mathcal{O}(\epsilon^{-2})$  outer iterations, and thus their complexities are more than  $\mathcal{O}(\epsilon^{-2})$ . Even though single-loop methods are more popular in practice due to their simplicity, few single-loop algorithms have been proposed for NC-C setting. The most natural approach is the gradient descent-ascent (GDA) method, which performs a gradient descent step on  $x$  and a gradient ascent step on  $y$  at each iteration. However, GDA fails to converge even for simple bilinear zero-sum games [22]. Subsequently, several improved GDA algorithms such as [8, 25, 27, 44] were proposed. For instance, [25] proved that the complexity of their two-time-scale GDA to find an  $\epsilon$ -stationary point of  $\phi$  is  $\mathcal{O}(\epsilon^{-6})$  for NC-C problems. Moreover, [27] presented an efficient algorithm, which obtains an  $\epsilon$ -stationary point of  $f$  with the complexity  $\mathcal{O}(\epsilon^{-4})$ . In fact, the complexity only counts the number of times the maximization subproblem is solved, and does not consider the complexity of solving this subproblem. [44] proposed a unified algorithm, and proved that the complexity of their algorithm to find an  $\epsilon$ -stationary point of  $f$  is  $\mathcal{O}(\epsilon^{-4})$  for both NC-C and C-NC problems. More recently, [51] presented a smoothed-GDA algorithm and proved that the complexity can be improved to  $\mathcal{O}(\epsilon^{-2})$  for optimizing a special case of NC-C problems (i.e., minimizing the pointwise maximum of a finite collection of nonconvex functions). However, its complexity is still  $\mathcal{O}(\epsilon^{-4})$  for both NC-C and C-NC problems. One natural question is: can we design a single-loop accelerated algorithm with the optimal complexity bound  $\mathcal{O}(\epsilon^{-2})$  for both NC-C and C-NC problems?
1.2 Algorithms in NC-NC Settings
This paper mainly considers constrained NC-NC minimax problems, i.e.,  $f(x,y)$  is nonconvex in  $x$  and nonconcave in  $y$  in the constrained setting (called constrained NC-NC). In recent years, some works such as [7, 31, 46, 38, 20] focus on structured NC-NC problems. That is, the saddle gradient operator of such minimax problems or their objectives must satisfy one of the structured assumptions: the minty variational inequality (MVI) condition, the weak MVI condition, or negative comonotone condition and the Polyak-Lojasiewicz condition. However, structured NC-NC problems have limited practical applications because they are required to satisfy strong structural assumptions. For more practical constrained NC-NC problems or more general NC-NC problems, the convergence guarantee of the algorithms is still a challenge. In this paper, we also focus on convergence analysis
Table 1: Comparison of complexities of the minimax algorithms to find an  $\epsilon$ -stationary point of  $f(\cdot, \cdot)$  in the NC-C, C-NC and NC-NC settings. Note that Smoothed-GDA [51] can find an  $\epsilon$ -stationary point for a special problem of (1) with  $\mathcal{O}(\epsilon^{-2})$ , and  $\widetilde{\mathcal{O}}$  hides logarithmic factors compared to  $\mathcal{O}(\cdot)$ .
Optimality Criteria	References	NC-C	C-NC	NC-NC	Simplicity
Stationarity of fwith smoothnessand compact setsassumptions	Lu et al. [27]	\(\widetilde{\mathcal{O}}(\epsilon^{-4})\)	-	-	Single-LoopMulti-Loop
Nouiehed et al. [31]	\(\widetilde{\mathcal{O}}(\epsilon^{-3.5})\)	-	-
Lin et al. [24]	\(\widetilde{\mathcal{O}}(\epsilon^{-2.5})\)	-	-	Multi-Loop
Zhang et al. [51]	\(\mathcal{O}(\epsilon^{-4})\)	-	-	Single-Loop
Xu et al. [44]	\(\mathcal{O}(\epsilon^{-4})\)	\(\mathcal{O}(\epsilon^{-4})\)	-	Single-Loop
This work (Theorem 1)	\(\mathcal{O}(\epsilon^{-2})\)	\(\mathcal{O}(\epsilon^{-2})\)	\(\mathcal{O}(\epsilon^{-2})\)	Single-Loop
Table 2: Comparison of complexities of existing minimax algorithms and the proposed algorithm to find an  $\phi(\cdot) := \max_{y \in \mathcal{Y}} f(\cdot, y)$  in the nonconvex-concave (NC-C) setting. This table only highlights the dependence of  $\epsilon$ , and compared with  $\mathcal{O}(\cdot)$ ,  $\widetilde{\mathcal{O}}(\cdot)$  hides logarithmic factors.
NC-C Settings	References	Compact set	Complexity	Simplicity
NC-C (Stationarity of φ)	Rafique et al. [34], Jin et al. [17]	x, y	O(ε-6)	Multi-Loop
Lin et al. [25]	x, y	O(ε-6)	Single-Loop
Thekumprampil et al. [40]	x, y	O(ε-3)	Multi-Loop
Zhao [55], Lin et al. [24]	x, y	O(ε-3)	Multi-Loop
This work (Theorem 2)	y	O(ε-2)	Single-Loop
for solving more practical constrained NC-NC problems. Another natural question is: can we design a single-loop accelerated algorithm to further improve the bound in the constrained NC-NC setting?
1.3 Motivations and Our Contributions
Motivations: For NC-C minimax problems, can we design a single-loop directly accelerated algorithm with the gradient complexity lower than the best-known result  $\widetilde{\mathcal{O}} (\epsilon^{-2.5})$ ? Though Smoothed-GDA [51] can obtain the complexity  $\mathcal{O}(\epsilon^{-2})$  for a special case of Problem (1), it only attains the complexity  $\mathcal{O}(\epsilon^{-4})$  for NC-C minimax problems. For C-NC minimax problems, for any given  $x$ , to solve the nonconcave maximization subproblem with respect to  $y$  is NP-hard. As a result, all existing multi-loop algorithms will lose their theoretical guarantees as discussed in [44]. Can we propose a single-loop directly accelerated algorithm with the complexity lower than the best-known result  $\mathcal{O}(\epsilon^{-4})$  for C-NC and NC-NC minimax problems?
Our Contributions: This paper proposes a novel single-loop extra-gradient difference acceleration algorithm to push towards optimal gradient complexities for constrained NC-NC minimax problems (1), and answer the above-mentioned problems.
We summarize the major contributions of this paper.
- We design a new single-loop accelerating algorithm for solving constrained NC-NC problems. In the proposed algorithm, we design a new extra-gradient difference scheme, and combine the gradient ascent and momentum acceleration steps for the dual variable update. In our algorithm, we present an important quasi-cocoercivity property. By leveraging the quasi-cocoercivity property, we can improve the complexity bound in our theoretical analysis.
- We analyze the convergence properties of the proposed algorithm for constrained NC-NC problems. Theorem 1 shows that to find an  $\epsilon$ -stationary point of  $f$ , the proposed algorithm can obtain the gradient complexity  $\mathcal{O}(\epsilon^{-2})$ , which is the first time to attain the complexity bound in constrained NC-NC setting. The constrained NC-C and C-NC problems can be viewed as two special cases of the constrained NC-NC problem, and the proposed algorithm is also applicable to these two special problems. And its complexity is still  $\mathcal{O}(\epsilon^{-2})$  for both NC-C and C-NC problems, which significantly improves the gradient complexity from  $\mathcal{O}(\epsilon^{-4})$  of existing single-loop algorithms or  $\widetilde{\mathcal{O}}(\epsilon^{-2.5})$  of existing multi-loop algorithms to  $\mathcal{O}(\epsilon^{-2})$ . The complexities of some recently proposed algorithms are listed in Table 1.
- In order to make a comprehensive comparison with existing algorithms, we also provide the theoretical analysis of our algorithm in terms of another convergence criteria (i.e., an  $\epsilon$ -stationary point of  $\phi$ ) for constrained NC-C problems. The result shows that our algorithm improves the best-known result as in [40, 24, 55] from  $\widetilde{\mathcal{O}}(\epsilon^{-3})$  to  $\mathcal{O}(\epsilon^{-2})$ , as shown in Table 2.
2 Preliminaries and Related Works
Notation: Throughout this paper, we use lower-case letters to denote vectors such as  $x, y$ , and calligraphic upper-case letters to denote sets such as  $\mathcal{X}, \mathcal{Y}$ . For a differentiable function  $f$ ,  $\nabla f(x)$  is the gradient of  $f$  at  $x$ . For a function  $f(\cdot, \cdot)$  of two variables,  $\nabla_x f(x, y)$  (or  $\nabla_y f(x, y)$ ) is the partial gradient of  $f$  with respect to the first variable (or the second variable) at  $(x, y)$ . For a vector  $x$ ,  $\|x\|$  denotes its  $\ell_2$ -norm. We use  $\mathcal{P}_{\mathcal{X}}$  and  $\mathcal{P}_{\mathcal{Y}}$  to denote projections onto the sets  $\mathcal{X}$  and  $\mathcal{Y}$ .
Assumption 1 (Smoothness).  $f(\cdot, \cdot)$  is continuously differentiable, and there exists a positive constant  $L$  such that
$$
\| \nabla_ {x} f (x _ {1}, y _ {1}) - \nabla_ {x} f (x _ {2}, y _ {2}) \| \leq L \| x _ {1} - x _ {2} \|, \| \nabla_ {y} f (x _ {1}, y _ {1}) - \nabla_ {y} f (x _ {2}, y _ {2}) \| \leq L \| y _ {1} - y _ {2} \|
$$
holds for all  $x_{1},x_{2}\in \mathbb{R}^{m}$ $y_{1},y_{2}\in \mathbb{R}^{n}$
Definitions of the monotone operators: A operator  $F(\cdot): \mathbb{R}^n \to \mathbb{R}^n$  is monotone, if  $[F(s) - F(t)]^T (s - t) \geq 0$ ,  $\forall s, t \in \mathbb{R}^n$ . If  $[F(s) - F(t)]^T (s - t) \leq 0$ ,  $F$  is negative monotone.
A mapping  $F(\cdot)$  is co-coercive if there is a positive constant  $\alpha$ , such that
$$
\left[ F (s) - F (t) \right] ^ {T} (s - t) \geq \alpha \| F (s) - F (t) \| ^ {2}, \forall s, t \in \mathbb {R} ^ {n}.
$$
Nonconvex-Concave Minimax Optimization: Due to the nonconvex nature of these minimax problems, finding the global solution is NP-hard in general. The recently proposed algorithms aim to find stationary solutions to such problems. For instance, the first-order Nash equilibrium condition (called game stationary) is used as an optimality condition in [31]. Besides game stationary, there are two main optimality criteria (i.e., an  $\epsilon$ -stationary point of  $f(\cdot, \cdot)$  or  $\phi(\cdot) := \max_{y \in \mathcal{Y}} f(\cdot, y)$ ) for the convergence analysis of the algorithms such as [4, 44, 24].
For solving NC-C minimax problems, there exist a number of efficient multi-loop and single-loop algorithms such as [31, 25, 4, 44, 24]. Most of them are multi-loop algorithms, which either employ an accelerated update rule of  $x$  by adding regularization terms to its subproblem, or use multiple gradient ascent steps for the update of  $y$  to solve the subproblem exactly or in exactly. Compared with multi-loop algorithms, single-loop algorithms are easier to implement. One of the most popular single-loop algorithms is GDA. However, GDA with a constant stepsize can fail to converge even for a simple bilinear minimax problem [29].
To address this issue, only a few single-loop algorithms such as [25, 27, 44] were proposed, and most of them employed a smoothing or proximal point technique. For instance, Smoothed-GDA [51] introduces a smooth function  $\varphi (x,y,z) = f(x,y) + \frac{a}{2}\| x - z\| ^2$  for the update of the primal variable  $x$ , where  $z$  is an auxiliary variable and  $a$  is a constant, and its main update steps are
$$
x _ {t + 1} = \mathcal {P} _ {\mathcal {X}} \left(x _ {t} - \eta_ {x} \nabla_ {x} \varphi \left(x _ {t}, z _ {t}, y _ {t}\right)\right), y _ {t + 1} = \mathcal {P} _ {\mathcal {Y}} \left(y _ {t} + \eta_ {y} \nabla_ {y} f \left(x _ {t + 1}, y _ {t}\right)\right), z _ {t + 1} = z _ {t} + \beta \left(x _ {t + 1} - z _ {t}\right),
$$
where  $\eta_x, \eta_y > 0$  are two stepsizes, and  $0 < \beta \leq 1$ . Smoothed-GDA can obtain the gradient complexity,  $\mathcal{O}(\epsilon^{-4})$ , for nonconvex-concave minimax problems.
Extra-Gradient Methods: There are some extra-gradient methods such as [48] for solving minimax problems. Let  $\eta_0 > 0$  and  $u_{t}$  be the  $t$ -th iterate, the extra-gradient method [53] has the following projection-type prediction-correction step:
Prediction:  $u_{t+1/2} = \mathcal{P}_{\Omega}(u_t - \eta_t F(u_t))$ , Correction:  $u_{t+1} = \mathcal{P}_{\Omega}(u_t - \eta_t F(u_{t+1/2}))$ .
In this paper, we call  $u_{t + 1 / 2}$  as the prediction point at the  $t$ -iteration.
3 Single-Loop Extra-Gradient Difference Acceleration Algorithm
In this section, we propose a single-loop Extra-Gradient Difference Acceleration algorithm (EGDA) for solving constrained NC-NC minimax problems.
Algorithm 1 EGDA for NC-NC minimax problems
1: Initialize:  $x_0, y_0, u_0, u_{-1/2}, y_{-1}, u_{-1/2} \neq y_{-1}, \tau \geq 0.75, \eta_x, \delta, \beta$ .
2: for  $t = 0,1,\ldots ,T - 1$  do
3:  $x_{t + 1} = \mathcal{P}_{\mathcal{X}}[x_t - \eta_x\nabla_xf(x_t,y_t)];\qquad u_{t + 1 / 2} = y_t + \beta [\nabla_yf(x_t,u_{t - 1 / 2}) - \nabla_yf(x_t,y_{t - 1})];$
4:  $u_{t + 1} = \mathcal{P}_{\mathcal{Y}}$ $[y_t + \eta_y^t\nabla_yf(x_t,u_{t - 1 / 2})]$  .  $y_{t + 1} = \tau y_t + (1 - \tau)u_{t + 1};$
5: end for
6: Output:  $(x_{T},y_{T})$
3.1 Extra-Gradient Difference Acceleration
In recent years, many algorithms such as [27, 31, 32, 24, 51, 44] have been proposed for solving NC-C minimax problems. In essence, most of them are "indirect" acceleration algorithms, which are used to optimize the surrogate functions with a smoothing or proximal point term instead of the original function. However, this may hurt the performance of these algorithms both in theory and in practice [2, 1]. To address this issue, we propose a single-loop directly accelerating algorithm to find an  $\epsilon$ -stationary points of  $f$  and  $\phi$  with a significantly improved complexity  $\mathcal{O}(\epsilon^{-2})$ . The main update steps of our EGDA algorithm are designed as follows.
Gradient descent:
$$
x _ {t + 1} = \arg \min  _ {x \in \mathcal {X}} \left\{\langle \nabla_ {x} f (x _ {t}, y _ {t}), x \rangle + \| x - x _ {t} \| ^ {2} / \eta_ {x} \right\}. \tag {2}
$$
- Extra-gradient difference prediction:
$$
u _ {t + 1 / 2} = y _ {t} + \beta \left[ \nabla_ {y} f \left(x _ {t}, u _ {t - 1 / 2}\right) - \nabla_ {y} f \left(x _ {t}, y _ {t - 1}\right) \right]. \tag {3}
$$
Gradient ascent correction:
$$
u _ {t + 1} = \arg \max  _ {u \in \mathcal {Y}} \left\{\langle \nabla_ {y} f (x _ {t}, u _ {t - 1 / 2}), u \rangle - \| u - y _ {t} \| ^ {2} / \eta_ {y} ^ {t} \right\}. \tag {4}
$$
- Momentum acceleration:
$$
y _ {t + 1} = \tau y _ {t} + (1 - \tau) u _ {t + 1}. \tag {5}
$$
Here,  $0 < \beta < \frac{1}{L}$ ,  $\eta_x, \eta_y^t$  are two step sizes,  $u_{t+1/2}$  and  $u_{t+1}$  are two auxiliary variables,  $u_{t+1/2}$  is a prediction point, and  $0 < \tau \leq 1$  is a momentum parameter. Our EGDA algorithm is formally presented in Algorithm 1. Our EGDA algorithm first performs one proximal gradient descent step on the primal variable  $x$ , and then we design a new dual-accelerating scheme in Eqs. (3), (4) and (5) for the dual variable  $y$ .
3.2 Advantages of Our Algorithm and Comparison to Related Work
In our EGDA algorithm, we first design a new prediction point  $u_{t+1/2}$  in Eq. (3). Compared with existing extra-gradient-type methods such as [7, 16, 20], one of the main differences is that the proposed prediction point  $u_{t+1/2}$  in (3) is updated by the gradient difference (i.e.,  $\nabla_y f(x_t, u_{t-1/2}) - \nabla_y f(x_t, y_{t-1})$ ), while the prediction point in other extra-gradient-type algorithms is updated only by using the gradient information at the correction point  $u_t$ . Then the gradient at the new prediction point  $u_{t+1/2}$  is used in the gradient ascent step (4). And the dual variable  $y$  is update by the momentum acceleration step in (5).
- Prediction Point: The monotonicity and co-coercivity properties of gradient operators play a crucial role for convergence analysis. However, these important properties do not always hold for nonconvex problems. Some researchers have made great progress in some special nonconvex settings, such as structured nonconvex and weakly convex settings, which require a weaker condition such as weakly monotone [26], pseudo-monotone [16], and MVI [7]. However, such conditions seriously limit the application scope of Problem (1).
To address this challenge, we design a new prediction point scheme in (3), which can help us obtain a useful quasi-cocoercivity property. As a result, it does not require any monotone or structural assumption. Specifically, we find that we only require a weaker property in our theoretical analysis,
that is, the weaker co-coercivity is required at some special points  $\{u_{t + 1 / 2},y_t\}$ $(\langle \nabla_yf(x_t,u_{t + 1 / 2})-$ $\nabla_yf(x_t,y_t),u_{t + 1 / 2} - y_t\rangle \geq \beta \| \nabla_yf(x_t,u_{t + 1 / 2}) - \nabla_yf(x_t,y_t)\| ^2$  with  $\beta >0$ ). Thus, we develop a decoupling idea to construct the prediction point  $u_{t + 1 / 2}$ . That is, we use the gradients with respect to  $y$  at  $u_{t - 1 / 2},y_{t - 1}$  instead of those at the points  $u_{t + 1 / 2},y_t$ . We can obtain a property (called the quasi-coercivity) in Section 4.2, which plays a key role in the theoretical analysis for the proposed algorithm.
- Gradient Difference: We also briefly discuss the underlying intuition of the proposed gradient difference in (3). We find that the proposed update in (3) is similar to the forms in [43] (see Eqs. (12) and (14) in [43]). [43] proposed a first-order procedure (i.e., difference of gradients) to achieve the negative curvature of a Hessian matrix. Therefore, our algorithm has a similar procedure, which contains second-order information. Moreover, we use the difference of gradients in the gradient ascent procedure for the dual update. But our EGDA algorithm only requires the Lipschitz gradient assumption for minimax problems to find first-order stationary points, while [43] requires both the Lipschitz gradient and Lipschitz Hessian assumptions for solving second-order stationary points of nonconvex minimization problems.
- Momentum Acceleration: We design a dual-accelerating update rule in (4) for the dual variable  $y$  in our EGDA algorithm, which is different from standard momentum acceleration schemes as in [31, 40, 24]. That is, the accelerated rules of existing algorithms are for the primal variable  $x$ , while our accelerated rule is designed for the dual variable  $y$ .