Huiling Zhang

Huiling Zhang, Zi Xu, Yu-Hong Dai

The study of nonconvex minimax games has gained significant momentum in machine learning and decision science communities due to their fundamental connections to adversarial training scenarios. This work develops a primal-dual alternating proximal gradient (PDAPG) algorithm framework for resolving iterative minimax games featuring nonsmooth nonconvex objectives subject to coupled linear constraints. We establish rigorous convergence guarantees for both nonconvex-strongly concave and nonconvex-concave game configurations, demonstrating that PDAPG achieves an $\varepsilon$-stationary solution within $\mathcal{O}\left( \varepsilon ^{-2} \right)$ iterations for strongly concave settings and $\mathcal{O}\left( \varepsilon ^{-4} \right)$ iterations for concave scenarios. Our analysis provides the first known iteration complexity bounds for this class of constrained minimax games, particularly addressing the critical challenge of coupled linear constraints that induce inherent interdependencies among strategy variables. The proposed game-theoretic framework advances existing solution methodologies by simultaneously handling nonsmooth components and coordinated constraint structures through alternating primal-dual updates.

Junlin Wang, Zi Xu, Huiling Zhang

In this paper, we study second-order algorithms for the convex-concave minimax problem, which has attracted much attention in many fields such as machine learning in recent years. We propose a Lipschitz-free cubic regularization (LF-CR) algorithm for solving the convex-concave minimax optimization problem without knowing the Lipschitz constant. It can be shown that the iteration complexity of the LF-CR algorithm to obtain an $ε$-optimal solution with respect to the restricted primal-dual gap is upper bounded by $\mathcal{O}(ρ^{2/3}\|z_0-z^*\|^2ε^{-2/3})$ , where $z_0=(x_0,y_0)$ is a pair of initial points, $z^*=(x^*,y^*)$ is a pair of optimal solutions, and $ρ$ is the Lipschitz constant. We further propose a fully parameter-free cubic regularization (FF-CR) algorithm that does not require any parameters of the problem, including the Lipschitz constant and the upper bound of the distance from the initial point to the optimal solution. We also prove that the iteration complexity of the FF-CR algorithm to obtain an $ε$-optimal solution with respect to the gradient norm is upper bounded by $\mathcal{O}(ρ^{2/3}\|z_0-z^*\|^{4/3}ε^{-2/3}) $. Numerical experiments show the efficiency of both algorithms. To the best of our knowledge, the proposed FF-CR algorithm is a completely parameter-free second-order algorithm, and its iteration complexity is currently the best in terms of $ε$ under the termination criterion of the gradient norm.

Junnan Yang, Huiling Zhang, Zi Xu

Due to their importance in various emerging applications, efficient algorithms for solving minimax problems have recently received increasing attention. However, many existing algorithms require prior knowledge of the problem parameters in order to achieve optimal iteration complexity. In this paper, three completely parameter-free single-loop algorithms, namely PF-AGP-NSC algorithm, PF-AGP-NC algorithm and PF-AGP-NL algorithm, are proposed to solve the smooth nonconvex-strongly concave, nonconvex-concave minimax problems and nonconvex-linear minimax problems respectively using line search without requiring any prior knowledge about parameters such as the Lipschtiz constant $L$ or the strongly concave modulus $μ$. Furthermore, we prove that the total number of gradient calls required to obtain an $\varepsilon$-stationary point for the PF-AGP-NSC algorithm, the PF-AGP-NC algorithm, and the PF-AGP-NL algorithm are upper bounded by $\mathcal{O}\left( L^2κ^3\varepsilon^{-2} \right)$, $\mathcal{O}\left( \log^2(L)L^4\varepsilon^{-4} \right)$, and $\mathcal{O}\left( L^3\varepsilon^{-3} \right)$, respectively, where $κ$ is the condition number. To the best of our knowledge, PF-AGP-NC and PF-AGP-NL are the first completely parameter-free algorithms for solving nonconvex-concave and nonconvex-linear minimax problems, respectively. PF-AGP-NSC is a completely parameter-free algorithm for solving nonconvex-strongly concave minimax problems, achieving the best known complexity with respect to $\varepsilon$. Numerical results demonstrate the efficiency of the three proposed algorithms.

Huiling Zhang, Zi Xu

Stochastic nonconvex minimax problems have attracted wide attention in machine learning, signal processing and many other fields in recent years. In this paper, we propose an accelerated first-order regularized momentum descent ascent algorithm (FORMDA) for solving stochastic nonconvex-concave minimax problems. The iteration complexity of the algorithm is proved to be $\tilde{\mathcal{O}}(\varepsilon ^{-6.5})$ to obtain an $\varepsilon$-stationary point, which achieves the best-known complexity bound for single-loop algorithms to solve the stochastic nonconvex-concave minimax problems under the stationarity of the objective function.

Huiling Zhang, Zi Xu, Yuhong Dai

In this paper, we study zeroth-order algorithms for nonconvex minimax problems with coupled linear constraints under the deterministic and stochastic settings, which have attracted wide attention in machine learning, signal processing and many other fields in recent years, e.g., adversarial attacks in resource allocation problems and network flow problems etc. We propose two single-loop algorithms, namely the zeroth-order primal-dual alternating projected gradient (ZO-PDAPG) algorithm and the zeroth-order regularized momentum primal-dual projected gradient algorithm (ZO-RMPDPG), for solving deterministic and stochastic nonconvex-(strongly) concave minimax problems with coupled linear constraints. The iteration complexity of the two proposed algorithms to obtain an $\varepsilon$-stationary point are proved to be $\mathcal{O}(\varepsilon ^{-2})$ (resp. $\mathcal{O}(\varepsilon ^{-4})$) for solving nonconvex-strongly concave (resp. nonconvex-concave) minimax problems with coupled linear constraints under deterministic settings and $\tilde{\mathcal{O}}(\varepsilon ^{-3})$ (resp. $\tilde{\mathcal{O}}(\varepsilon ^{-6.5})$) under stochastic settings respectively. To the best of our knowledge, they are the first two zeroth-order algorithms with iterative complexity guarantees for solving nonconvex-(strongly) concave minimax problems with coupled linear constraints under the deterministic and stochastic settings.

Zi Xu, Huiling Zhang, Yang Xu et al.

Much recent research effort has been directed to the development of efficient algorithms for solving minimax problems with theoretical convergence guarantees due to the relevance of these problems to a few emergent applications. In this paper, we propose a unified single-loop alternating gradient projection (AGP) algorithm for solving smooth nonconvex-(strongly) concave and (strongly) convex-nonconcave minimax problems. AGP employs simple gradient projection steps for updating the primal and dual variables alternatively at each iteration. We show that it can find an $\varepsilon$-stationary point of the objective function in $\mathcal{O}\left( \varepsilon ^{-2} \right)$ (resp. $\mathcal{O}\left( \varepsilon ^{-4} \right)$) iterations under nonconvex-strongly concave (resp. nonconvex-concave) setting. Moreover, its gradient complexity to obtain an $\varepsilon$-stationary point of the objective function is bounded by $\mathcal{O}\left( \varepsilon ^{-2} \right)$ (resp., $\mathcal{O}\left( \varepsilon ^{-4} \right)$) under the strongly convex-nonconcave (resp., convex-nonconcave) setting. To the best of our knowledge, this is the first time that a simple and unified single-loop algorithm is developed for solving both nonconvex-(strongly) concave and (strongly) convex-nonconcave minimax problems. Moreover, the complexity results for solving the latter (strongly) convex-nonconcave minimax problems have never been obtained before in the literature. Numerical results show the efficiency of the proposed AGP algorithm. Furthermore, we extend the AGP algorithm by presenting a block alternating proximal gradient (BAPG) algorithm for solving more general multi-block nonsmooth nonconvex-(strongly) concave and (strongly) convex-nonconcave minimax problems. We can similarly establish the gradient complexity of the proposed algorithm under these four different settings.