SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Morteza Boroun*, Erfan Yazdandoost Hamedani*, Afrooz Jalilzadeh*
Department of Systems & Industrial Engineering
The University of Arizona
Tucson, AZ 85721
morteza@arizona.edu, erfany@arizona.edu, afrooz@arizona.edu
In this paper, we investigate a class of constrained saddle point (SP) problems where the objective function is nonconvex-concave and smooth. This class of problems has wide applicability in machine learning, including robust multi-class classification and dictionary learning. Several projection-based primal-dual methods have been developed to tackle this problem; however, the availability of methods with projection-free oracles remains limited. To address this gap, we propose efficient single-loop projection-free methods reliant on first-order information. In particular, using regularization and nested approximation techniques, we propose a primal-dual conditional gradient method that solely employs linear minimization oracles to handle constraints. Assuming that the constraint set in the maximization is strongly convex, our method achieves an  $\epsilon$ -stationary solution within  $\mathcal{O}(\epsilon^{-6})$  iterations. When the projection onto the constraint set of maximization is easy to compute, we propose a one-sided projection-free method that achieves an  $\epsilon$ -stationary solution within  $\mathcal{O}(\epsilon^{-4})$  iterations. Moreover, we present improved iteration complexities of our methods under a strong concavity assumption. To the best of our knowledge, our proposed algorithms are among the first projection-free methods with convergence guarantees for solving nonconvex-concave SP problems.
Let  $(\mathcal{X},\| \cdot \|_{\mathcal{X}})$  and  $(\mathcal{Y},\| \cdot \|_{\mathcal{Y}})$  be finite-dimensional, real normed spaces with the corresponding dual spaces denoted by  $(\mathcal{X}^{*},\| \cdot \|_{\mathcal{X}^{*}})$  and  $(\mathcal{Y}^{*},\| \cdot \|_{\mathcal{Y}^{*}})$ , respectively. In this paper, we study a saddle point (SP) problem of the following form:
$$
\min  _ {x \in X} \max  _ {y \in Y} \mathcal {L} (x, y), \tag {1}
$$
where  $\mathcal{L}:\mathcal{X}\times \mathcal{Y}\to \mathbb{R}$  is possibly nonconvex in  $x$  for any  $y\in Y$  and concave in  $y$  for any  $x\in X$  with certain differentiability properties (see Assumption 2.6); moreover,  $X\subseteq \mathcal{X}$  and  $Y\subseteq \mathcal{V}$  are convex and compact sets. Such a problem has a broad range of applications including robust optimization [4], reinforcement learning [11], and adversarial learning [17], to name just a few.
There has been vast research conducted on developing algorithms for solving SP problems [37, 41, 8, 42, 18]. Despite the extensive studies conducted on the development of algorithms to solve problem (1) a common drawback of these methods is their reliance on costly operations involving the projection onto constraint sets. In the optimization literature, the shortcomings associated with projection-based techniques prompted the emergence and advancement of projection-free approaches. Namely, Frank Wolfe-based algorithms [14] are first-order methods designed for minimizing smooth constrained optimization problems where instead of projection onto a constraint set only a linear minimization
over the set needs to be solved. Such an operation can significantly reduce the computational cost of the algorithm as in the case of nuclear-norm ball constraints.
Nonconvex SP problems have already been extensively investigated in the literature and various methods have been proposed for different settings. Namely, the vanilla projected gradient descent ascent has been shown to achieve iteration complexity  $\mathcal{O}(\epsilon^{-2})$  for finding an  $\epsilon$ -stationary of nonconvex-strongly concave problem [27] matching the lower bound in this setting. However, this method fails to address nonconvex-concave problems, therefore, many researchers studied various smoothing [51] and acceleration [40] techniques to ensure a convergence rate guarantee. Most of these methods are built upon the idea of solving regularized subproblems in exactly resulting in a multi-loop method that can pose implementation challenges. To avoid such a complication, single-loop primal-dual methods have gained popularity. Recently, a unified projected primal-dual gradient using the regularization technique has been studied [49]. The proposed method can achieve the complexities of  $\mathcal{O}(\epsilon^{-2})$  and  $\mathcal{O}(\epsilon^{-4})$  for nonconvex-strongly concave and nonconvex-concave settings. While the studies on primal-dual projected gradient-based methods are rich and fruitful, less is known about projected-free methods for SP problems. With only a handful of studies focusing on (strongly) convex-(strongly) concave problems [16, 46, 9], no projection-free method for solving nonconvex SP problems is currently available, to the best of our knowledge. Moreover, recently there has been a growing interest in developing algorithms for solving bilevel optimization problems [43, 20, 13] which subsumes SP problems as a special case. However, it is important to note that most of the existing methods for solving bilevel optimization problems consider an additional assumption on the lower-level objective function satisfying strong convexity or Polyak Lojasiewicz (PL) condition [20, 2, 31]. These assumptions in the context of SP problems translate into strong concavity or PL condition for  $\mathcal{L}(x,\cdot)$  which cannot handle merely concave setting considered in this paper. Therefore, in this paper, we focus on developing projection-free methods for solving (1). In particular, our main research question is
Can we develop a one-sided or fully projection-free primal-dual method with a convergence rate guarantee for solving nonconvex-(strongly) concave SP problem in (1)?
Our response to this question is affirmative, and next we outline the key contributions of our work.
In this paper, we consider a broad class of constrained SP problems where the objective function is nonconvex-concave and smooth. Motivated by the pressing need for developing projection-free algorithms from an application perspective as well as the lack of efficient methods with theoretical guarantees, we propose primal-dual projection-free methods for solving nonconvex-concave SP problems. Using regularization and nested approximation techniques we first develop a single-loop primal-dual method that solely uses linear minimization oracle (LMO) to find a direction for both primal and dual steps. Next, assuming that the projection oracle (PO) is available for the maximization component of the objective function, we develop a one-sided projection-free method with regularized projection gradient ascent. Our main theoretical contributions are summarized below.
1) Assuming the constraint set in the maximization component is strongly convex, we show that our proposed primal-dual conditional gradient method that employs LMOs for both variables achieves an  $\epsilon$ -stationary solution within  $\mathcal{O}(\epsilon^{-6})$  iterations. To the best of our knowledge, this is the first complexity result for primal-dual methods with only LMOs for nonconvex-concave SP problems. Moreover, when the objective function is nonconvex-strongly concave, our proposed method achieves  $\epsilon$ -primal and  $\epsilon$ -dual gaps within  $\mathcal{O}(\epsilon^{-4})$  and  $\mathcal{O}(\epsilon^{-2})$  iterations, respectively.
2) Considering the case where projection onto the constraint set of the maximization component is easy to compute, we demonstrate that our one-sided projection-free method achieves an  $\epsilon$ -stationary solution within  $\mathcal{O}(\epsilon^{-4})$  iterations matching the best-known results for single-loop projection-based primal-dual methods. For nonconvex-strongly concave setting, the iteration complexity improves to  $\mathcal{O}(\epsilon^{-2})$ .
SP problems have been subject to extensive studies, primarily in the context of the convex-concave setting [37, 41, 8, 18]. However, the practical applications of SP problems in nonconvex settings,
particularly in the context of robust optimization and adversarial learning have sparked a demand for a thorough investigation and further exploration. Next, we briefly discuss the state-of-the-art methods in the literature for nonconvex-(strongly) concave problems. We divide our review into two sections of projected gradient-based and projection-free-based algorithms.
Projected gradient based algorithms: Various algorithms have been proposed for nonconvex-concave setting under different assumptions [30, 7, 40, 52, 6, 3]. Authors in [44] proposed a double loop algorithm to solve nonsmooth weakly convex-concave with a complexity of  $\tilde{\mathcal{O}} (\epsilon^{-6})$ . In the pursuit of more efficient solutions, Triple-loop inexact proximal point methods with the faster convergence rate of  $\tilde{\mathcal{O}} (\epsilon^{-3})$  are developed in [48, 23]. Additionally, single-loop algorithms for addressing nonconvex-concave problems were introduced by [29] and [32] where they achieved the iteration complexities of  $\tilde{\mathcal{O}} (\epsilon^{-6})$  and  $\tilde{\mathcal{O}} (\epsilon^{-4})$ , respectively. More recently, authors in [33] proposed optimistic gradient descent ascent and extra-gradient methods and obtained a convergence rate of  $\mathcal{O}(\epsilon^{-6})$ . Employing Nesterov's smoothing technique [38], the iteration complexity of  $\mathcal{O}(\epsilon^{-4})$  was obtained by [51] for general nonconvex-concave problem. In addition, [50] achieved the same convergence rate with linear coupled equality or inequality constraints. Recently, [49] proposed an alternating gradient projection (AGP) algorithm where it utilizes gradient projection for updating  $x$  and  $y$  and it achieves the suboptimal complexity of  $\mathcal{O}(\epsilon^{-4})$ . All the aforementioned methods rely on projection onto the constraint sets.
Projection-free based algorithms: There has been a surge of interest in the application of projection-free methods within the machine learning and optimization problems [24, 15, 25, 26]. Despite the widespread application of projection-free methods such as the Frank Wolfe (FW) algorithm, there have been limited studies that examine their use in the context of SP problems [1, 19, 47, 16, 46, 9, 22]. In [19] a projection-free algorithm for convex-concave setting is proposed and achieved the complexity of  $\mathcal{O}(\epsilon^{-2})$  LMO calls. This algorithm was subsequently refined by [47] with the development of a parallelizable algorithm. Gidel et al. [16] presented the first convergence result for an FW-type algorithm for nonbilinear SP problems. Assuming that the SP solution lies within the interior of constraint sets, the proposed method achieves a linear rate for strongly convex-strongly concave SP problems. Moreover, the method achieves the same complexity for convex-concave setting, assuming that the constraint sets are strongly convex and gradients of the objective function are uniformly lower-bounded. Roy et al. [46] advanced a projection-free algorithm for the same setting and introduced a variant of the FW algorithm that is particularly suited for the nonstationary stochastic SP problems. Additionally, the authors in [9] proposed a four-loop projection-free algorithm which can be viewed as a variant of Mirror-Prox in which proximal subproblems are solved in exactly using conditional gradient sliding [25] for convex-strongly concave problems. Considering a nonconvex-concave setting, Nouiehed et al. [39] proposed a multi-loop one-sided projection-free primal-dual method that achieves the complexity of  $\tilde{\mathcal{O}} (\epsilon^{-3.5})$  to find an  $\epsilon$ -game stationary solution. Notably, in their method, at each iteration  $k$  the minimization subproblem needs to be computed over the set  $\{x\in \mathcal{X}\mid x + x_k\in X,\| x\| \leq 1\}$ . Kolmogorov et al. [22] developed a two-loop one-sided projection-free primal-dual method for solving bilinear convex-concave SP problems where LMO is solely used for computing the minimization variable. The proposed method achieves an iteration complexity of  $\mathcal{O}(\epsilon^{-1})$ . The summary of related work and their comparison are presented in Table 1.
1.3 Motivating Examples
Example 1. Robust Multiclass Classification: Consider a multiclass classification task for a given training dataset  $\mathcal{D}^{tr} = \{(\mathbf{a}_i, b_i)\}_{i=1}^n$  where  $\mathbf{a}_i \in \mathbb{R}^d$  denotes the feature vector of the  $i$ -th sample and  $b_i \in \{1, \ldots, k\}$  is the corresponding label. The goal is to find a linear predictor with parameter  $\pmb{\theta} = [\theta_1^\top, \theta_2^\top, \ldots, \theta_k^\top] \in \mathbb{R}^{k \times d}$  that is able to predict the labels for an unlabeled dataset  $\mathcal{D}^{test} = \{\mathbf{a}_j\}_{i=1}^{n'}$ . In particular, given a new data point  $\hat{\mathbf{a}} \in \mathcal{D}^{test}$ , the corresponding label can be predicted via  $\hat{b} = \arg \max_{j=1}^{k} \theta_j^\top \hat{\mathbf{a}}$ . It has been shown that [12] the linear predictor  $\pmb{\theta}$  can be found by minimizing a multinomial logistic loss function, i.e.,  $\ell_i(\pmb{\theta}) = \log \left(1 + \exp \left(\sum_{j \neq b_i} (\theta_j^\top \mathbf{a}_i - \theta_{b_i}^\top \mathbf{a}_i)\right)\right)$ , over a nuclear norm constraint, i.e.,  $X = \{\pmb{\theta} | \| \pmb{\theta} \|_* \leq r\}$  for some  $r > 0$ . On the other hand, in many applications where safety and reliability are crucial distributionally robust optimization offers a promising approach for a more robust and reliable prediction that can be used in classification tasks to capture uncertainty in data distribution [35, 34]. This can be achieved by considering a worst-case performance of non-parametric uncertainty set on the underlying data distribution. This leads to a min-max formulation where the maximization is taking over an uncertainty set  $Y$ . For
Table 1: Comparative analysis of optimization schemes for convex-strongly concave (C-SC), convex-concave (C-C), nonconvex-strongly concave (NC-SC) and nonconvex-concave (NC-C) problems, featuring different oracles: Linear Minimization Oracle (LMO) and Projection/Proximal Oracle (PO).  $\tilde{O} (\cdot)$  denotes  $\mathcal{O}(\cdot)$  up to a logarithmic factor. In the column of "additional assumption", the abbreviation SC stands for strongly convex. * The complexities are presented for  $\epsilon$ -primal and  $\epsilon$ -dual, respectively.
Setting	Ref	Non-bilinear	Oracle	# of loops	Addt'1 Assump.	Rate
C-SC	Chen et al. [9]	✓	LMO-LMO	4	None	\(\tilde{\mathcal{O}}(\epsilon^{-2})\)
C-C	Gidel et al. [16]	✓	LMO-LMO	1	X and Y are SC sets	\(\mathcal{O}(\log \epsilon^{-1})\)
Kolmogorov et al. [22]	✗	LMO-PO	2	X is SC set	\(\mathcal{O}(\epsilon^{-1})\)
NC-SC	Xu et al. [49]	✓	PO-PO	1	None	\(\mathcal{O}(\epsilon^{-2})\)
CG-RPGA (Theorem 4.2)	✓	LMO-PO	1	None	\(\mathcal{O}(\epsilon^{-2})\)
R-PDCG (Theorem 4.4)	✓	LMO-LMO	1	Y is SC set	\(\mathcal{O}(\epsilon^{-4}),\mathcal{O}(\epsilon^{-2})^{*}\)
NC-C	Xu et al. [49]	✓	PO-PO	1	None	\(\mathcal{O}(\epsilon^{-4})\)
CG-RPGA (Theorem 5.1)	✓	LMO-PO	1	None	\(\mathcal{O}(\epsilon^{-4})\)
R-PDCG (Theorem 5.3)	✓	LMO-LMO	1	Y is SC set	\(\mathcal{O}(\epsilon^{-6})\)
instance,  $Y = \{y \in \Delta_n : V(y, \frac{1}{n} \mathbf{1}_n) \leq \rho\}$  is considered in different papers such as [34], where  $\Delta_n$  is an  $n$ -dimensional simplex set, and  $V(Q, P)$  denotes the divergence measure between two sets of probability measures  $Q$  and  $P$ . Therefore, distributionally robust multiclass classification can be formulated as the following SP problem [9]:
$$
\min  _ {\boldsymbol {\theta} \in X} \max  _ {y = \left[ y _ {i} \right] _ {i = 1} ^ {n}} \sum_ {i = 1} ^ {n} y _ {i} \ell_ {i} (\boldsymbol {\theta}). \tag {2}
$$
This problem can be readily cast as (1) by defining  $\mathcal{L}(\pmb{\theta}, y) = \sum_{i=1}^{n} y_i \ell_i(\pmb{\theta})$ . The constraint of the maximization is the intersection of simplex set and divergence measure constraints. Indeed, one can relax the simplex constraint using the splitting technique and Fenchel duality. The resulting equivalent saddle point problem has a maximization constraint of  $Y = \{y : V(y, \frac{1}{n} \mathbf{1}_n) \leq \rho\}$ . which is only described by the divergence measure constraint. In some popular examples such as the Pearson Chi-square divergence, i.e.,  $V(y, \mathbf{1}_n / n) = \|ny - \mathbf{1}_n\|^2$ ,  $Y$  satisfies the assumption of strongly convex constraint set.
Example 2. Dictionary Learning: Dictionary learning aims to acquire a succinct representation of the input data extracted from a large dataset. Given an input dataset  $\mathbf{A} = [a_1, \ldots, a_n] \in \mathbb{R}^{m \times n}$ , our primary goal is to find a dictionary  $\mathbf{D} = [d_1, \ldots, d_p] \in \mathbb{R}^{m \times p}$  that can accurately approximate the data through linear combinations. Dictionary learning problem can be formulated in nonconvex optimization form [45]:
$$
\min  _ {\mathbf {D} \in \mathbb {R} ^ {m \times p}} \min  _ {\mathbf {C} \in \mathbb {R} ^ {p \times n}} \| \mathbf {A} - \mathbf {D C} \| _ {F} ^ {2}, \quad \text {s . t .} \quad \| \mathbf {C} \| _ {*} \leq r; \| d _ {j} \| _ {2} \leq 1, \forall j \in \{1, \dots , p \}, \tag {3}
$$
where  $\mathbf{C} \in \mathbb{R}^{p \times n}$  denotes the coefficient matrix. In various learning scenarios, such as lifelong learning, the learner engages in a sequential series of tasks, aiming to accumulate knowledge from previous tasks in order to enhance performance in subsequent tasks. Suppose that we have learned a dictionary  $\mathbf{D} \in \mathbb{R}^{m \times p}$  and its corresponding coefficient matrix  $\mathbf{C} \in \mathbb{R}^{p \times n}$  for the dataset  $\mathbf{A}$ . Given a new dataset  $\mathbf{A}' \in \mathbb{R}^{m \times n'}$ , we aim to refine a new dictionary  $\mathbf{D}' \in \mathbb{R}^{m \times q}$  such that it still provides an accurate representation of old dataset  $\mathbf{A}$  with the learned coefficient matrix  $\mathbf{C}$ . Consequently, this problem 3 can be written as the following non-convex problem with a convex nonlinear constraint:
$$
\min _ {\mathbf {D} ^ {\prime}} \min _ {\mathbf {C} ^ {\prime}} \| \mathbf {A} ^ {\prime} - \mathbf {D} ^ {\prime} \mathbf {C} ^ {\prime} \| _ {F} ^ {2}, \text {s . t .} \| \mathbf {A} - \mathbf {D} ^ {\prime} \tilde {\mathbf {C}} \| _ {F} ^ {2} \leq \delta ; \| \mathbf {C} ^ {\prime} \| _ {*} \leq r; \left\| d _ {j} ^ {\prime} \right\| _ {2} \leq 1, \forall j \in \{1, \dots , q \}, \tag {4}
$$
where  $\delta > 0$  denotes the user-predefined accuracy for the representation of the old dataset and  $\tilde{\mathbf{C}} \in \mathbb{R}^{q \times n}$  is the extension of  $\mathbf{C}$  by adding  $q - p$  columns of zeros. This problem can be formulated equivalently as an SP problem (1) using a Lagrangian duality. In particular, it can be cast as (1) by setting  $\mathcal{L}((\mathbf{D}', \mathbf{C}')', y) = \| \mathbf{A}' - \mathbf{D}' \mathbf{C}' \|_F^2 + y (\| \mathbf{A} - \mathbf{D}' \tilde{\mathbf{C}} \|_F^2 - \delta)$ ,  $X = \{(\mathbf{D}', \mathbf{C}') \mid \| \mathbf{C}' \|_* \leq r, \| d_j' \|_2 \leq 1, \forall j \in \{1, \dots, q\}\}$ , and  $Y = \mathbb{R}_+$ . Note that  $\mathcal{L}$  is nonconvex-concave and smooth. Moreover, due to the existence of a Slater point for problem (4) a dual bound, i.e.,  $\| y \| \leq B$  for some  $B > 0$ , can be constructed efficiently as suggested in [36, 18] to ensure boundedness of set  $Y$ .
Other than min-max formulation, the nuclear-norm constraint in the above examples makes the problem computationally demanding for projection-based algorithms. Therefore, it is imperative to address this challenge by developing and utilizing projection-free methods, which can effectively overcome the computational limitations associated with these problems.
In this section, we outline the notations and required assumptions that we need for the analysis of our proposed methods as well as some important definitions.
Notations. Given two sets  $X$  and  $Y$ ,  $X \times Y$  denotes the Cartesian product of  $X$  and  $Y$ . The lower-case  $x$  and  $y$ , as well as their accented variants, are reserved for vectors in the space of  $\mathcal{X}$  and  $\mathcal{Y}$ , respectively; moreover, the letter  $z$  will be used to indicate the concatenation of those vectors, i.e.,  $z \triangleq [x^{\top}y^{\top}]^{\top}$ . The upper-case  $X$  and  $Y$  are reserved for constraint sets; moreover,  $Z$  will be used to indicate their Cartesian product. Given a differentiable function  $\mathcal{L}(x,y)$ ,  $\nabla_{x}\mathcal{L}(x,y)$  and  $\nabla_{y}\mathcal{L}(x,y)$  denote the partial derivatives of  $\mathcal{L}$  with respect to  $x$  and  $y$ , respectively. Given a set  $X$ ,  $\mathbb{1}_X(\cdot)$  denotes the indicator function. Given a finite-dimensional normed vector space,  $\| \cdot \|_p$  for some  $p \geq 1$  denotes the  $\ell_p$ -norm.
2.1 Oracle Description
In this paper, we will utilize the following oracles to acquire first-order information and solution of structured subproblems for the proposed algorithms in different settings.
- Given  $(x,y)$ , first-order oracle (FO) returns  $\nabla_{x}\mathcal{L}(x,y)$  and  $\nabla_{y}\mathcal{L}(x,y)$ .
- Given a vector  $\bar{x} \in \mathcal{U}$  and a convex and compact set  $U \subseteq \mathcal{U}$ , linear minimization oracle (LMO) returns a solution of  $\min_{x \in U} \langle \bar{x}, x \rangle$ .
- Given a vector  $\bar{x} \in \mathcal{U}$  and a convex and compact set  $U \subseteq \mathcal{U}$ , projection oracle (PO) returns the solution of  $\min_{x \in U} \| x - \bar{x} \|_{\mathcal{U}}$ .
Based on the above oracles, we define some gap functions to measure the quality of the solution. More specifically, when the proposed algorithm uses LMO we use the following definition.
Definition 2.1 (Gap function for LMO). The stationary gap function  $\mathcal{G}_X: Z \to \mathbb{R}$  for the minimization part of problem (1) is defined as  $\mathcal{G}_X(\bar{x}, \bar{y}) \triangleq \sup_{x \in X} \langle \nabla_x \mathcal{L}(\bar{x}, \bar{y}), \bar{x} - x \rangle$ . Similarly, for the maximization part of problem (1) the stationary gap function  $\mathcal{G}_Y: Z \to \mathbb{R}$  is defined as  $\mathcal{G}_Y(\bar{x}, \bar{y}) \triangleq \sup_{y \in Y} \langle \nabla_y \mathcal{L}(\bar{x}, \bar{y}), y - \bar{y} \rangle$ . Moreover, we define  $\mathcal{G}_Z(\bar{x}, \bar{y}) \triangleq \mathcal{G}_X(\bar{x}, \bar{y}) + \mathcal{G}_Y(\bar{x}, \bar{y})$ .
In the case where the proposed method uses PO for the maximization part of the objective function of (1), we use the same gap function  $\mathcal{G}_X$  for the minimization component while employing the following gap function for the maximization component.
Definition 2.2 (Gap function for PO). The stationary dual gap function  $\mathcal{G}_Y: Z \to \mathbb{R}$  corresponding to the maximization part of problem (1) is defined as  $\mathcal{G}_Y(\bar{x}, \bar{y}) \triangleq \frac{1}{\sigma} \| y - \mathcal{P}_Y(y + \sigma \nabla_y \mathcal{L}(\bar{x}, \bar{y})) \|_y$ . Moreover, we define  $\mathcal{G}_Z(\bar{x}, \bar{y}) \triangleq \mathcal{G}_X(\bar{x}, \bar{y}) + \mathcal{G}_Y(\bar{x}, \bar{y})$  where  $\mathcal{G}_X$  is defined in Definition 2.1.
Definition 2.3. For a given gap function  $\mathcal{G}_Z: Z \to \mathbb{R}$ , a point  $(\bar{x}, \bar{y}) \in Z$  is an  $\epsilon$ -stationary of problem (1) if  $\mathcal{G}_Z(\bar{x}, \bar{y}) \leq \epsilon$ .
Remark 2.4. As indicated in the above definitions, since we use different oracles for the maximization component of the objective function we need to employ different gap functions for the dual iterates. With a slight abuse of notation, we used  $\mathcal{G}_Y$  when using both LMO and PO for the constraint set  $Y$ . The reason is that an  $\epsilon$ -solution in terms of both definitions implies an  $\epsilon$ -game stationary solution. More specifically, if  $\mathcal{G}_Z(\bar{x},\bar{y})\leq \epsilon$  for some  $(\bar{x},\bar{y})\in Z$ , then