Two Completely Parameter-Free Alternating Gradient Projection Algorithms for Nonconvex-(strongly) Concave Minimax Problems

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, we propose a completely parameter-free alternating gradient projection (PF-AGP) algorithm to solve the smooth nonconvex-(strongly) concave minimax problems using a backtracking strategy, which does not require prior knowledge of parameters such as the Lipschtiz constant $L$ or the strongly concave constant $\mu$. The PF-AGP algorithm utilizes a parameter-free gradient projection step to alternately update the outer and inner variables in each iteration. We show that the total number of gradient calls of the PF-AGP algorithm to obtain an $\varepsilon$-stationary point for nonconvex-strongly concave minimax problems is upper bounded by $\mathcal{O}\left( L\kappa^3\varepsilon^{-2} \right)$ where $\kappa$ is the condition number, while the total number of gradient calls to obtain an $\varepsilon$-stationary point for nonconvex-concave minimax problems is upper bounded by $\mathcal{O}\left( L^4\varepsilon^{-4} \right)$. As far as we know, this is the first completely parameter-free algorithm for solving nonconvex-strongly concave minimax problems, and it is also the completely parameter-free algorithm which achieves the best iteration complexity in single loop method for solving nonconvex-concave minimax problems. Numerical results validate the efficiency of the proposed PF-AGP algorithm.

Zeroth-Order primal-dual Alternating Projection Gradient Algorithms for Nonconvex Minimax Problems with Coupled linear Constraints

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 zero-order primal-dual alternating projected gradient (ZO-PDAPG) algorithm and the zero-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.

An accelerated first-order regularized momentum descent ascent algorithm for stochastic nonconvex-concave minimax problems

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.

Primal Dual Alternating Proximal Gradient Algorithms for Nonsmooth Nonconvex Minimax Problems with Coupled Linear Constraints

Nonconvex minimax problems have attracted wide attention in machine learning, signal processing and many other fields in recent years. In this paper, we propose a primal dual alternating proximal gradient (PDAPG) algorithm and a primal dual proximal gradient (PDPG-L) algorithm for solving nonsmooth nonconvex-strongly concave and nonconvex-linear minimax problems with coupled linear constraints, respectively. The corresponding iteration complexity of the two algorithms are proved to be $\mathcal{O}\left( \varepsilon ^{-2} \right)$ and $\mathcal{O}\left( \varepsilon ^{-3} \right)$ to reach an $\varepsilon$-stationary point, respectively. To our knowledge, they are the first two algorithms with iteration complexity guarantee for solving the two classes of minimax problems.

A Unified Single-loop Alternating Gradient Projection Algorithm for Nonconvex-Concave and Convex-Nonconcave Minimax Problems

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 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.