SlowGuess/NeurIPS_2023_OCR huggingface dataset:
(c) Population Strong PD Risk: The population strong PD risk for  $(\boldsymbol{\omega}^u, \boldsymbol{\nu}^u)$  by algorithm  $\bar{A}_{c - c}$  is
$$
\triangle^ {s} \left(\boldsymbol {\omega} ^ {u}, \boldsymbol {\nu} ^ {u}\right) \leq \mathcal {O} \left(\left(L ^ {3} \ell^ {3} \rho / \lambda^ {6} + L ^ {2} \ell^ {2} / \lambda^ {3}\right) \cdot \frac {m ^ {2} \sqrt {d \log (1 / \delta)}}{n ^ {2} \epsilon} + \frac {m L ^ {2}}{\lambda n} + \frac {L ^ {2} \ell}{\lambda^ {2} n} + \lambda \left(B _ {\boldsymbol {\omega}} ^ {2} + B _ {\boldsymbol {\nu}} ^ {2}\right)\right). \tag {28}
$$
$$
\begin{array}{l} \lambda = \max  \left\{\frac {L}{\sqrt {B _ {\omega} ^ {2} + B _ {\nu} ^ {2}}} \sqrt {\frac {m}{n}}, \left(\frac {L ^ {2} \ell}{\left(B _ {\omega} ^ {2} + B _ {\nu} ^ {2}\right) n}\right), \left(\frac {L ^ {2} \ell^ {2} m ^ {2} \sqrt {d \log (1 / \delta)}}{\left(B _ {\omega} ^ {2} + B _ {\nu} ^ {2}\right) n ^ {2} \epsilon}\right) ^ {1 / 4}, \right. \tag {29} \\ \left. \left(\frac {L ^ {3} \ell^ {3} \rho m ^ {2} \sqrt {d \log (1 / \delta)}}{\left(B _ {\boldsymbol {\omega}} ^ {2} + B _ {\boldsymbol {\nu}} ^ {2}\right) n ^ {2} \epsilon}\right) ^ {1 / 7} \right\}, \\ \end{array}
$$
we have the following population strong PD risk,
$$
\triangle^ {s} \left(\boldsymbol {\omega} ^ {u}, \boldsymbol {\nu} ^ {u}\right) \leq \mathcal {O} \left(c _ {1} \sqrt {\frac {m}{n}} + c _ {2} \frac {1}{\sqrt [ 3 ]{n}} + c _ {3} \left(\frac {d \log (1 / \delta)}{\epsilon^ {2}}\right) ^ {1 / 8} \sqrt {\frac {m}{n}} + c _ {4} \left(\frac {\sqrt {d \log (1 / \delta)}}{\epsilon}\right) ^ {1 / 7} \left(\frac {m}{n}\right) ^ {2 / 7}\right), \tag {30}
$$
where  $c_{1}, c_{2}, c_{3}, c_{4}$  are constants that depend only on  $L, l, \rho, B_{\omega}$  and  $B_{\nu}$ .
(d) Deletion Capacity: The deletion capacity of Algorithm  $\bar{A}_{c - c}$  is
$$
m _ {\epsilon , \delta , \gamma} ^ {A, \bar {A}} \left(d _ {1}, d _ {2}, n\right) \geq c \cdot \frac {n \sqrt {\epsilon}}{\left(d \log (1 / \delta)\right) ^ {1 / 4}}, \tag {31}
$$
where the constant  $c$  depends on the constants  $L, l, \rho, B_{\omega}$  and  $B_{\nu}$ .
In this paper, we have studied the certified machine unlearning for minimax models with a focus on the generalization rates and deletion capacity, while existing works in this area largely focus on standard statistical learning models. We have provided a new minimax unlearning algorithm composed of the total Hessian-based complete Newton update and the Gaussian mechanism-based perturbation, which comes with rigorous  $(\epsilon, \delta)$ -unlearning certification. We have established generalization results in terms of the population weak and strong primal-dual risk and the correspondingly defined deletion capacity results for the strongly-convex-strongly-concave loss functions, both of which match the state-of-the-art results obtained for standard statistical learning models. We have also provided extensions to other loss types like the convex-concave loss function. In addition, we have provided a more computationally efficient extension by getting rid of the total Hessian re-computation during the minimax unlearning phase, which can be more appealing for the successive data deletion setting. Although our bound for deletion capacity is better than that of DP by an order of  $d^{1/4}$  and matches the state-of-the-art result established for unlearning under the STL setting, it remains unclear whether this bound is tight or not. In future work, we plan to extend to more general settings like the nonconvex-nonconcave loss function setting.
We would like to thank Gautam Kamath for his valuable comments on the presentation of the results in the previous version of the paper. Additionally, we extend our thanks to the reviewers and area chair of NeurIPS 2023 for their constructive comments and feedback. This work was supported in part by the National Natural Science Foundation of China (62072395, 62206207, U20A20178), and the National Key Research and Development Program of China (2020AAA0107705, 2021YFB3100300).
A Additional Definitions and Supporting Lemmas
In this section, we provide additional definitions and supporting lemmas. In the next two sections, Sec.B contains missing proofs in Sec.4 and the online extension to support successive unlearning setting. Sec.C contains missing proofs in Sec.5, as well as detailed algorithm descriptions for the general convex-concave loss function setting.
A.1 Additional Definitions
We first recall the following standard definitions for the loss function  $f(\omega, \nu; z)$  from optimization literature.
Definition 4 (Function Lipschitz Continuity). The function  $f(\omega, \nu; z)$  is  $L$ -Lipschitz, i.e., there exists a constant  $L > 0$  such that for all  $\omega, \omega' \in \mathcal{W}$ ,  $\nu, \nu' \in \mathcal{V}$  and  $z \in \mathcal{Z}$ , it holds that
$$
\left\| f \left(\omega , \nu ; z\right) - f \left(\omega^ {\prime}, \nu^ {\prime}; z\right) \right\| ^ {2} \leq L ^ {2} \left(\left\| \omega - \omega^ {\prime} \right\| ^ {2} + \left\| \nu - \nu^ {\prime} \right\| ^ {2}\right). \tag {32}
$$
Definition 5 (Gradient Lipschitz Continuity). The function  $f(\omega, \nu; z)$  has  $\ell$ -Lipschitz gradients, i.e., there exists a constant  $\ell > 0$  such that for all  $\omega, \omega' \in \mathcal{W}$ ,  $\nu, \nu' \in \mathcal{V}$  and  $z \in \mathcal{Z}$ , it holds that
$$
\left\| \nabla f (\boldsymbol {\omega}, \boldsymbol {\nu}; z) - \nabla f \left(\boldsymbol {\omega} ^ {\prime}, \boldsymbol {\nu} ^ {\prime}; z\right) \right\| ^ {2} \leq \ell^ {2} \left(\left\| \boldsymbol {\omega} - \boldsymbol {\omega} ^ {\prime} \right\| ^ {2} + \left\| \boldsymbol {\nu} - \boldsymbol {\nu} ^ {\prime} \right\| ^ {2}\right), \tag {33}
$$
where recall that  $\nabla f(\omega, \nu; z) = \left[ \begin{array}{c} \nabla_{\omega} f(\omega, \nu; z) \\ \nabla_{\nu} f(\omega, \nu; z) \end{array} \right]$ .
Definition 6 (Hessian Lipschitz Continuity). The function  $f(\omega, \nu; z)$  has  $\rho$ -Lipschitz Hessian, i.e., there exists a constant  $\rho > 0$  such that for all  $\omega, \omega' \in \mathcal{W}$ ,  $\nu, \nu' \in \mathcal{V}$  and  $z \in \mathcal{Z}$ , it holds that
$$
\left\| \nabla^ {2} f (\boldsymbol {\omega}, \boldsymbol {\nu}; z) - \nabla^ {2} f \left(\boldsymbol {\omega} ^ {\prime}, \boldsymbol {\nu} ^ {\prime}; z\right) \right\| ^ {2} \leq \rho^ {2} \left(\left\| \boldsymbol {\omega} - \boldsymbol {\omega} ^ {\prime} \right\| ^ {2} + \left\| \boldsymbol {\nu} - \boldsymbol {\nu} ^ {\prime} \right\| ^ {2}\right), \tag {34}
$$
where recall that  $\nabla^2 f(\pmb {\omega},\pmb {\nu};z) = \left[ \begin{array}{ll}\partial_{\pmb{\omega}\pmb{\omega}}f(\pmb {\omega},\pmb {\nu};z) & \partial_{\pmb{\omega}\pmb{\nu}}f(\pmb {\omega},\pmb {\nu};z)\\ \partial_{\pmb{\nu}\pmb{\omega}}f(\pmb {\omega},\pmb {\nu};z) & \partial_{\pmb{\nu}\pmb{\nu}}f(\pmb {\omega},\pmb {\nu};z) \end{array} \right].$
Definition 7 (Strongly-Convex-Strongly-Concave Objective Function). The function  $f(\omega, \nu; z)$  is  $\mu_{\omega}$ -strongly convex on  $\mathcal{W}$  and  $\mu_{\nu}$ -strongly concave on  $\mathcal{V}$ , i.e., there exist constants  $\mu_{\omega} > 0$  and  $\mu_{\nu} > 0$  such that for all  $\omega, \omega' \in \mathcal{W}$ ,  $\nu, \nu' \in \mathcal{V}$  and  $z \in \mathcal{Z}$ , it holds that
$$
\left\{ \begin{array}{l} f (\omega , \nu ; z) \geq f \left(\omega^ {\prime}, \nu ; z\right) + \left\langle \nabla_ {\omega} f \left(\omega^ {\prime}, \nu ; z\right), \omega - \omega^ {\prime} \right\rangle + \frac {\mu_ {\omega}}{2} \| \omega - \omega^ {\prime} \| ^ {2}, \\ f (\omega , \nu ; z) \leq f \left(\omega , \nu^ {\prime}; z\right) + \left\langle \nabla_ {\nu} f \left(\omega , \nu^ {\prime}; z\right), \nu - \nu^ {\prime} \right\rangle - \frac {\mu_ {\nu}}{2} \| \nu - \nu^ {\prime} \| ^ {2}. \end{array} \right. \tag {35}
$$
Definition 8 (Best Response Auxiliary Functions). We introduce auxiliary functions  $\mathrm{V}_S(\omega)$  and  $\mathrm{V}_{S^{\backslash}}(\omega)$ , given by
$$
V _ {S} (\boldsymbol {\omega}) := \underset {\boldsymbol {\nu} \in \mathcal {V}} {\operatorname {a r g m a x}} F _ {S} (\boldsymbol {\omega}, \boldsymbol {\nu}), \quad V _ {S \backslash} (\boldsymbol {\omega}) := \underset {\boldsymbol {\nu} \in \mathcal {V}} {\operatorname {a r g m a x}} F _ {S \backslash} (\boldsymbol {\omega}, \boldsymbol {\nu}), \tag {36}
$$
and we have  $\pmb{\nu}_{S}^{*} = \nabla_{S}\big(\pmb{\omega}_{S}^{*}\big)$  and  $\pmb{\nu}_{S\backslash}^{*} = \nabla_{S\backslash}\big(\pmb{\omega}_{S\backslash}^{*}\big)$ . We can similarly introduce  $\mathbb{W}_S(\pmb {\nu})$  and  $\mathbb{W}_{S\setminus}(\pmb {\nu})$  as
$$
\mathrm {W} _ {S} (\boldsymbol {\nu}) := \underset {\boldsymbol {\omega} \in \mathcal {W}} {\operatorname {a r g m i n}} F _ {S} (\boldsymbol {\omega}, \boldsymbol {\nu}), \quad \mathrm {W} _ {S ^ {\backslash}} (\boldsymbol {\nu}) := \underset {\boldsymbol {\omega} \in \mathcal {W}} {\operatorname {a r g m i n}} F _ {S ^ {\backslash}} (\boldsymbol {\omega}, \boldsymbol {\nu}), \tag {37}
$$
and we have  $\omega_{S}^{*} = \mathbb{W}_{S}(\pmb{\nu}_{S}^{*})$  and  $\omega_{S\backslash}^{*} = \mathbb{W}_{S\backslash}(\pmb{\nu}_{S\backslash}^{*})$  by this definition.
In addition, we define the primal function  $P(\omega) \coloneqq \max_{\nu \in \mathcal{V}} F_S(\omega, \nu) = F_S(\omega, \mathsf{V}_S(\omega))$ , which has gradient  $\nabla P(\omega) = \nabla_{\omega} F_S(\omega, \mathsf{V}_S(\omega))$  and Hessian  $\nabla_{\omega \omega}^2 P(\omega) = \mathsf{D}_{\omega \omega} F_S(\omega, \mathsf{V}_S(\omega))$  (i.e., the total Hessian of  $F_S$ ). The dual function, its gradient, and Hessian can be similarly defined, e.g.,  $D(\nu) \coloneqq \min_{\omega \in \mathcal{W}} F_S(\omega, \nu) = F_S(\mathbb{W}_S(\nu), \nu)$ .
A.2 Supporting Lemmas
The following lemma provides the distance between  $\mathsf{V}_S(\pmb{\omega}_S^*)$  and  $\mathsf{V}_{S\setminus}(\pmb{\omega}_S^*)$ . Similar result can be derived for the distance between  $\mathsf{W}_S(\pmb{\nu}_S^*)$  and  $\mathsf{W}_{S\setminus}(\pmb{\nu}_S^*)$ .
Lemma 2. Under Assumption 1 and Assumption 2, the variables  $\mathbf{V}_{S^{\backslash}}(\boldsymbol{\omega}_S^*)$  and  $\pmb{\nu}_{S}^{*}$  (i.e.,  $\mathrm{V}_S(\pmb{\omega}_S^*)$ ) defined in Algorithm 1 satisfy the following distance bound
$$
\left\| \boldsymbol {\nu} _ {S} ^ {*} - \mathrm {V} _ {S ^ {\backslash}} \left(\boldsymbol {\omega} _ {S} ^ {*}\right) \right\| \leq \frac {2 L m}{\mu_ {\boldsymbol {\nu}} (n - m)}. \tag {38}
$$
Proof. We observe that
$$
\begin{array}{l} (n - m) \left(F _ {S ^ {\backslash}} \left(\boldsymbol {\omega} _ {S} ^ {*}, \mathbf {v} _ {S ^ {\backslash}} \left(\boldsymbol {\omega} _ {S} ^ {*}\right)\right) - F _ {S ^ {\backslash}} \left(\boldsymbol {\omega} _ {S} ^ {*}, \boldsymbol {\nu} _ {S} ^ {*}\right)\right) \\ = \sum_ {z _ {i} \in S \backslash U} f \left(\omega_ {S} ^ {*}, \mathrm {V} _ {S \backslash} \left(\omega_ {S} ^ {*}\right); z _ {i}\right) - \sum_ {z _ {i} \in S \backslash U} f \left(\omega_ {S} ^ {*}, \boldsymbol {\nu} _ {S} ^ {*}; z _ {i}\right) \\ = \sum_ {z _ {i} \in S} f \left(\boldsymbol {\omega} _ {S} ^ {*}, \mathrm {V} _ {S} \backslash \left(\boldsymbol {\omega} _ {S} ^ {*}\right); z _ {i}\right) - \sum_ {z _ {i} \in U} f \left(\boldsymbol {\omega} _ {S} ^ {*}, \mathrm {V} _ {S} \backslash \left(\boldsymbol {\omega} _ {S} ^ {*}\right); z _ {i}\right) - \left(\sum_ {z _ {i} \in S} f \left(\boldsymbol {\omega} _ {S} ^ {*}, \boldsymbol {\nu} _ {S} ^ {*}; z _ {i}\right) - \sum_ {z _ {i} \in U} \left(\boldsymbol {\omega} _ {S} ^ {*}, \boldsymbol {\nu} _ {S} ^ {*}; z _ {i}\right)\right) \\ = n (F _ {S} (\boldsymbol {\omega} _ {S} ^ {*}, \mathsf {V} _ {S \setminus} (\boldsymbol {\omega} _ {S} ^ {*})) - F _ {S} (\boldsymbol {\omega} _ {S} ^ {*}, \boldsymbol {\nu} _ {S} ^ {*})) + \sum_ {z _ {i} \in U} f (\boldsymbol {\omega} _ {S} ^ {*}, \boldsymbol {\nu} _ {S} ^ {*}; z _ {i}) - \sum_ {z _ {i} \in U} f (\boldsymbol {\omega} _ {S} ^ {*}, \mathsf {V} _ {S \setminus} (\boldsymbol {\omega} _ {S} ^ {*}); z _ {i}) \\ \leq \sum_ {z _ {i} \in U} ^ {(i)} f \left(\boldsymbol {\omega} _ {S} ^ {*}, \boldsymbol {\nu} _ {S} ^ {*}; z _ {i}\right) - \sum_ {z _ {i} \in U} f \left(\boldsymbol {\omega} _ {S} ^ {*}, \mathrm {V} _ {S \backslash} \left(\boldsymbol {\omega} _ {S} ^ {*}\right); z _ {i}\right) \stackrel {(i i)} {\leq} m L \left\| \boldsymbol {\nu} _ {S} ^ {*} - \mathrm {V} _ {S \backslash} \left(\boldsymbol {\omega} _ {S} ^ {*}\right) \right\|, \tag {39} \\ \end{array}
$$
where the inequality (i) follows from that  $\pmb{\nu}_{S}^{*}$  is the maximizer of the function  $F_{S}(\omega, \nu)$ , thus  $F_{S}(\omega_{S}^{*}, \mathrm{V}_{S \setminus} (\omega_{S}^{*})) - F_{S}(\omega_{S}^{*}, \pmb{\nu}_{S}^{*}) \leq 0$ . The inequality (ii) is due to the fact that the function  $f$  is  $L$ -Lipschitz. Also note that the function  $F_{S \setminus} (\omega, \nu)$  is  $\mu_{\nu}$ -strongly concave, thus we have
$$
F _ {S \backslash} \left(\boldsymbol {\omega} _ {S} ^ {*}, \mathrm {V} _ {S \backslash} \left(\boldsymbol {\omega} _ {S} ^ {*}\right)\right) - F _ {S \backslash} \left(\boldsymbol {\omega} _ {S} ^ {*}, \boldsymbol {\nu} _ {S} ^ {*}\right) \geq \frac {\mu_ {\boldsymbol {\nu}}}{2} \left\| \boldsymbol {\nu} _ {S} ^ {*} - \mathrm {V} _ {S \backslash} \left(\boldsymbol {\omega} _ {S} ^ {*}\right) \right\| ^ {2}. \tag {40}
$$
Eq.(39) and eq.(40) together give that
$$
\frac {\mu_ {\boldsymbol {\nu}} (n - m)}{2} \left\| \boldsymbol {\nu} _ {S} ^ {*} - \mathrm {V} _ {S \backslash} \left(\boldsymbol {\omega} _ {S} ^ {*}\right) \right\| ^ {2} \leq m L \left\| \boldsymbol {\nu} _ {S} ^ {*} - \mathrm {V} _ {S \backslash} \left(\boldsymbol {\omega} _ {S} ^ {*}\right) \right\|, \tag {41}
$$
which implies that  $\| \pmb{\nu}_S^* - \mathsf{V}_{S\setminus}(\pmb{\omega}_S^*)\| \leq \frac{2Lm}{\mu_\nu(n - m)}$ .
The following lemma provides the distance between  $(\omega_{S^{\backslash}}^{*},\nu_{S^{\backslash}}^{*})$  and  $(\omega_{S}^{*},\nu_{S}^{*})$
Lemma 3. Under Assumption 1 and Assumption 2, the variables  $(\omega_{S\setminus}^{*},\nu_{S\setminus}^{*})$  defined in eq.(9) and  $(\omega_{S}^{*},\nu_{S}^{*})$  defined in Algorithm 1 satisfy the following guarantees
$$
\left\| \boldsymbol {\omega} _ {S ^ {\backslash}} ^ {*} - \boldsymbol {\omega} _ {S} ^ {*} \right\| \leq \frac {2 L m}{\mu_ {\boldsymbol {\omega}} n}, \quad a n d \quad \left\| \boldsymbol {\nu} _ {S ^ {\backslash}} ^ {*} - \boldsymbol {\nu} _ {S} ^ {*} \right\| \leq \frac {2 L m}{\mu_ {\boldsymbol {\nu}} n}. \tag {42}
$$
Proof. We begin with the  $\omega$ -part,
$$
\begin{array}{l} n \left[ F _ {S} \left(\boldsymbol {\omega} _ {S ^ {\backslash}} ^ {*}, \boldsymbol {\nu} _ {S ^ {\backslash}} ^ {*}\right) - F _ {S} \left(\boldsymbol {\omega} _ {S} ^ {*}, \boldsymbol {\nu} _ {S ^ {\backslash}} ^ {*}\right) \right] \\ = \sum_ {z _ {i} \in S} f \left(\omega_ {S ^ {\backslash}} ^ {*}, \nu_ {S ^ {\backslash}} ^ {*}; z _ {i}\right) - \sum_ {z _ {i} \in S} f \left(\omega_ {S} ^ {*}, \nu_ {S ^ {\backslash}} ^ {*}; z _ {i}\right) \\ = \sum_ {z _ {i} \in S ^ {\backslash} U} f \left(\omega_ {S ^ {\backslash}} ^ {*}, \nu_ {S ^ {\backslash}} ^ {*}; z _ {i}\right) + \sum_ {z _ {i} \in U} f \left(\omega_ {S ^ {\backslash}} ^ {*}, \nu_ {S ^ {\backslash}} ^ {*}; z _ {i}\right) - \sum_ {z _ {i} \in S ^ {\backslash} U} f \left(\omega_ {S} ^ {*}, \nu_ {S ^ {\backslash}} ^ {*}; z _ {i}\right) - \sum_ {z _ {i} \in U} f \left(\omega_ {S} ^ {*}, \nu_ {S ^ {\backslash}} ^ {*}; z _ {i}\right) \\ = (n - m) \left[ F _ {S ^ {\backslash}} \left(\boldsymbol {\omega} _ {S ^ {\backslash}} ^ {*}, \boldsymbol {\nu} _ {S ^ {\backslash}} ^ {*}\right) - F _ {S ^ {\backslash}} \left(\boldsymbol {\omega} _ {S} ^ {*}, \boldsymbol {\nu} _ {S ^ {\backslash}} ^ {*}\right) \right] + \sum_ {z _ {i} \in U} f \left(\boldsymbol {\omega} _ {S ^ {\backslash}} ^ {*}, \boldsymbol {\nu} _ {S ^ {\backslash}} ^ {*}; z _ {i}\right) - \sum_ {z _ {i} \in U} f \left(\boldsymbol {\omega} _ {S} ^ {*}, \boldsymbol {\nu} _ {S ^ {\backslash}} ^ {*}; z _ {i}\right) \\ \leq \sum_ {z _ {i} \in U} f \left(\boldsymbol {\omega} _ {S ^ {\backslash}} ^ {*}, \boldsymbol {\nu} _ {S ^ {\backslash}} ^ {*}; z _ {i}\right) - \sum_ {z _ {i} \in U} f \left(\boldsymbol {\omega} _ {S} ^ {*}, \boldsymbol {\nu} _ {S ^ {\backslash}} ^ {*}; z _ {i}\right) \stackrel {(i i)} {\leq} m L \left\| \boldsymbol {\omega} _ {S ^ {\backslash}} ^ {*} - \boldsymbol {\omega} _ {S} ^ {*} \right\|, \tag {43} \\ \end{array}
$$
where the inequality (i) holds because  $\omega_{S\backslash}^{*}$  is the minimizer of the function  $F_{S\backslash}(\omega ,\nu)$ , thus  $F_{S\backslash}(\omega_{S\backslash}^{*},\nu_{S\backslash}^{*}) - F_{S\backslash}(\omega_{S}^{*},\nu_{S\backslash}^{*})\leq 0$ , and the inequality (ii) follows from the fact that the function  $f$  is  $L$ -Lipschitz. Since the function  $F_{S}(\omega ,\nu)$  is  $\mu_{\omega}$ -strongly convex, we further get
$$
F _ {S} \left(\boldsymbol {\omega} _ {S ^ {\backslash}} ^ {*}, \boldsymbol {\nu} _ {S ^ {\backslash}} ^ {*}\right) - F _ {S} \left(\boldsymbol {\omega} _ {S} ^ {*}, \boldsymbol {\nu} _ {S ^ {\backslash}} ^ {*}\right) \geq \frac {\mu_ {\boldsymbol {\omega}}}{2} \left\| \boldsymbol {\omega} _ {S ^ {\backslash}} ^ {*} - \boldsymbol {\omega} _ {S} ^ {*} \right\| ^ {2}. \tag {44}
$$
Eq.(43) and eq.(44) together gives that
$$
\frac {\mu_ {\boldsymbol {\omega}} n}{2} \left\| \boldsymbol {\omega} _ {S ^ {\backslash}} ^ {*} - \boldsymbol {\omega} _ {S} ^ {*} \right\| ^ {2} \leq m L \left\| \boldsymbol {\omega} _ {S ^ {\backslash}} ^ {*} - \boldsymbol {\omega} _ {S} ^ {*} \right\|. \tag {45}
$$
Thus, we get  $\| \omega_{S^{\prime}}^{*} - \omega_{S}^{*}\| \leq \frac{2Lm}{\mu_{\omega}n}$
For the  $\pmb{\nu}$  -part, we similarly have
$$
\begin{array}{l} n \left[ F _ {S} \left(\omega_ {S \backslash} ^ {*}, \nu_ {S} ^ {*}\right) - F _ {S} \left(\omega_ {S \backslash} ^ {*}, \nu_ {S \backslash} ^ {*}\right) \right] \\ = \sum_ {z _ {i} \in S} f \left(\omega_ {S ^ {\backslash}} ^ {*}, \nu_ {S ^ {*}} ^ {*}; z _ {i}\right) - \sum_ {z _ {i} \in S} f \left(\omega_ {S ^ {\backslash}} ^ {*}, \nu_ {S ^ {\backslash}} ^ {*}; z _ {i}\right) \\ = \sum_ {z _ {i} \in S \backslash U} f \left(\omega_ {S ^ {\backslash}} ^ {*}, \nu_ {S ^ {\backslash}} ^ {*}; z _ {i}\right) + \sum_ {z _ {i} \in U} f \left(\omega_ {S ^ {\backslash}} ^ {*}, \nu_ {S ^ {\backslash}} ^ {*}; z _ {i}\right) - \sum_ {z _ {i} \in S \backslash U} f \left(\omega_ {S ^ {\backslash}} ^ {*}, \nu_ {S ^ {\backslash}} ^ {*}; z _ {i}\right) - \sum_ {z _ {i} \in U} f \left(\omega_ {S ^ {\backslash}} ^ {*}, \nu_ {S ^ {\backslash}} ^ {*}; z _ {i}\right) \\ = (n - m) \left[ F _ {S \backslash} \left(\boldsymbol {\omega} _ {S \backslash} ^ {*}, \boldsymbol {\nu} _ {S} ^ {*}\right) - F _ {S \backslash} \left(\boldsymbol {\omega} _ {S \backslash} ^ {*}, \boldsymbol {\nu} _ {S \backslash} ^ {*}\right) \right] + \sum_ {z _ {i} \in U} f \left(\boldsymbol {\omega} _ {S \backslash} ^ {*}, \boldsymbol {\nu} _ {S} ^ {*}; z _ {i}\right) - \sum_ {z _ {i} \in U} f \left(\boldsymbol {\omega} _ {S \backslash} ^ {*}, \boldsymbol {\nu} _ {S \backslash} ^ {*}; z _ {i}\right) \\ \leq \sum_ {z _ {i} \in U} f \left(\boldsymbol {\omega} _ {S ^ {\backslash}} ^ {*}, \boldsymbol {\nu} _ {S ^ {\backslash}} ^ {*}; z _ {i}\right) - \sum_ {z _ {i} \in U} f \left(\boldsymbol {\omega} _ {S ^ {\backslash}} ^ {*}, \boldsymbol {\nu} _ {S ^ {\backslash}} ^ {*}; z _ {i}\right) \leq m L \| \boldsymbol {\nu} _ {S} ^ {*} - \boldsymbol {\nu} _ {S ^ {\backslash}} ^ {*} \|, \tag {46} \\ \end{array}
$$
where the inequality  $(i)$  follows from that  $\pmb{\nu}_{S\backslash}^{*}$  is the maximizer of the function  $F_{S\backslash}(\pmb {\omega},\pmb {\nu})$  , thus  $F_{S\backslash}(\pmb{\omega}_{S\backslash}^{*},\pmb{\nu}_{S}^{*}) - F_{S\backslash}(\pmb{\omega}_{S\backslash}^{*},\pmb{\nu}_{S\backslash}^{*})\leq 0$  . The inequality  $(ii)$  is due to the fact that the function  $f$  is  $L$  -Lipschitz. In addition, by the strongly-concave assumption of  $F_{S}(\pmb {\omega},\pmb {\nu})$  is  $\mu_{\nu}$  , we have
$$
F _ {S} \left(\boldsymbol {\omega} _ {S ^ {\backslash}} ^ {*}, \boldsymbol {\nu} _ {S} ^ {*}\right) - F _ {S} \left(\boldsymbol {\omega} _ {S ^ {\backslash}} ^ {*}, \boldsymbol {\nu} _ {S ^ {\backslash}} ^ {*}\right) \geq \frac {\mu_ {\boldsymbol {\nu}}}{2} \left\| \boldsymbol {\nu} _ {S ^ {\backslash}} ^ {*} - \boldsymbol {\nu} _ {S} ^ {*} \right\| ^ {2}. \tag {47}
$$
By eq.(46) and eq.(47), we get that
$$
\frac {\mu_ {\nu} n}{2} \left\| \boldsymbol {\nu} _ {S ^ {\backslash}} ^ {*} - \boldsymbol {\nu} _ {S} ^ {*} \right\| ^ {2} \leq m L \left\| \boldsymbol {\nu} _ {S} ^ {*} - \boldsymbol {\nu} _ {S ^ {\backslash}} ^ {*} \right\|. \tag {48}
$$
Thus, we have  $\| \pmb{\nu}_{S^{\backslash}}^{*} - \pmb{\nu}_{S}^{*}\| \leq \frac{2Lm}{\mu\nu n}$ .
In the following, we recall several lemmas (i.e., Lemma 4 to Lemma 8) from existing minimax optimization literature for completeness.
Lemma 4 ([Lin et al., 2020, Lemma 4.3]). Under Assumption 1 and Assumption 2, for any  $\omega \in \mathcal{W}$ , the function  $\mathrm{V}_S(\omega)$  is  $(\ell/\mu_{\nu})$ -Lipschitz.
Proof. By the optimality condition of the function  $\mathrm{V}_S(\omega)$ , we have
$$
\left\langle \nabla_ {\nu} F _ {S} \left(\omega_ {1}, \mathrm {V} _ {S} \left(\omega_ {1}\right)\right), \mathrm {V} _ {S} \left(\omega_ {2}\right) - \mathrm {V} _ {S} \left(\omega_ {1}\right) \right\rangle \leq 0,
$$
$$
\left\langle \nabla_ {\nu} F _ {S} \left(\omega_ {2}, \mathrm {V} _ {S} \left(\omega_ {2}\right)\right), \mathrm {V} _ {S} \left(\omega_ {1}\right) - \mathrm {V} _ {S} \left(\omega_ {2}\right) \right\rangle \leq 0.
$$
Summing the two inequalities above yields
$$
\langle \nabla_ {\nu} F _ {S} (\omega_ {1}, \mathrm {V} _ {S} (\omega_ {1})) - \nabla_ {\nu} F _ {S} (\omega_ {2}, \mathrm {V} _ {S} (\omega_ {2})), \mathrm {V} _ {S} (\omega_ {2}) - \mathrm {V} _ {S} (\omega_ {1}) \rangle \leq 0. \tag {49}
$$
Since the function  $F_{S}(\omega ,\nu)$  is  $\mu_{\nu}$  -strongly concave in  $\pmb{\nu}$  , we have
$$
\left\langle \nabla_ {\nu} F _ {S} \left(\omega_ {1}, \mathrm {V} _ {S} \left(\omega_ {2}\right)\right) - F _ {S} \left(\omega_ {1}, \mathrm {V} _ {S} \left(\omega_ {1}\right)\right), \mathrm {V} _ {S} \left(\omega_ {2}\right) - \mathrm {V} _ {S} \left(\omega_ {1}\right) \right\rangle + \mu_ {\nu} \| \mathrm {V} _ {S} \left(\omega_ {2}\right) - \mathrm {V} _ {S} \left(\omega_ {1}\right) \| ^ {2} \leq 0. \tag {50}
$$
By eq.(49) and eq.(50) with the  $\ell$ -Lipschitz continuity of  $\nabla F_S(\omega, \nu)$ , we further get
$$
\mu_ {\nu} \| \mathrm {V} _ {S} (\omega_ {2}) - \mathrm {V} _ {S} (\omega_ {1}) \| ^ {2} \leq \left\langle \nabla_ {\nu} F _ {S} \left(\omega_ {2}, \mathrm {V} _ {S} (\omega_ {2})\right) - \nabla_ {\nu} F _ {S} \left(\omega_ {1}, \mathrm {V} _ {S} (\omega_ {2})\right), \mathrm {V} _ {S} (\omega_ {2}) - \mathrm {V} _ {S} (\omega_ {1}) \right\rangle \tag {51}
$$
$$
\leq \ell \| \boldsymbol {\omega} _ {2} - \boldsymbol {\omega} _ {1} \| \cdot \| \mathrm {V} _ {S} (\boldsymbol {\omega} _ {2}) - \mathrm {V} _ {S} (\boldsymbol {\omega} _ {1}) \|.
$$
$$
\left\| \mathbf {V} _ {S} \left(\boldsymbol {\omega} _ {2}\right) - \mathbf {V} _ {S} \left(\boldsymbol {\omega} _ {1}\right) \right\| \leq \frac {\ell}{\mu_ {\nu}} \| \boldsymbol {\omega} _ {2} - \boldsymbol {\omega} _ {1} \|. \tag {52}
$$
Remark 1. The above lemma can be similarly derived for  $\mathbb{W}_S$  to obtain that the best response auxiliary function  $\mathbb{W}_S(\pmb {\nu})$  is  $(\ell /\mu_{\omega})$  -Lipschitz. In the next three lemmas, we focus on the  $\omega$  -part and omit the  $\pmb{\nu}$  -part.
Lemma 5 ([Luo et al., 2022, Lemma 3]). Denote  $\kappa_{\nu} = \ell / \mu_{\nu}$ . Under Assumption 1 and Assumption 2, for any  $\omega, \omega' \in \mathcal{W}$ , we have
$$
\left\| \mathrm {D} _ {\omega \omega} F _ {S} (\omega , \mathrm {V} _ {S} (\omega)) - \mathrm {D} _ {\omega \omega} F _ {S} \left(\omega^ {\prime}, \mathrm {V} _ {S} \left(\omega^ {\prime}\right)\right) \right\| \leq 4 \sqrt {2} \kappa_ {\nu} ^ {3} \rho \| \omega - \omega^ {\prime} \|. \tag {53}
$$
Lemma 6 ([Nesterov and Polyak, 2006, Lemma 1]). Denote  $\kappa_{\nu} = \ell / \mu_{\nu}$ . Under Assumption 1 and Assumption 2, for any  $\omega, \omega' \in \mathcal{W}$ , we have