Two trust region type algorithms for solving nonconvex-strongly concave minimax problems

In this paper, we propose a Minimax Trust Region (MINIMAX-TR) algorithm and a Minimax Trust Region Algorithm with Contractions and Expansions(MINIMAX-TRACE) algorithm for solving nonconvex-strongly concave minimax problems. Both algorithms can find an $(\epsilon, \sqrt{\epsilon})$-second order stationary point(SSP) within $\mathcal{O}(\epsilon^{-1.5})$ iterations, which matches the best well known iteration complexity.

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.

Zeroth-Order Alternating Gradient Descent Ascent Algorithms for a Class of Nonconvex-Nonconcave Minimax Problems

In this paper, we consider a class of nonconvex-nonconcave minimax problems, i.e., NC-PL minimax problems, whose objective functions satisfy the Polyak-$\L$ojasiewicz (PL) condition with respect to the inner variable. We propose a zeroth-order alternating gradient descent ascent (ZO-AGDA) algorithm and a zeroth-order variance reduced alternating gradient descent ascent (ZO-VRAGDA) algorithm for solving NC-PL minimax problem under the deterministic and the stochastic setting, respectively. The number of iterations to obtain an $\epsilon$-stationary point of ZO-AGDA and ZO-VRAGDA algorithm for solving NC-PL minimax problem is upper bounded by $\mathcal{O}(\varepsilon^{-2})$ and $\mathcal{O}(\varepsilon^{-3})$, respectively. To the best of our knowledge, they are the first two zeroth-order algorithms with the iteration complexity gurantee for solving NC-PL minimax problems.

Zeroth-Order Alternating Randomized Gradient Projection Algorithms for General Nonconvex-Concave Minimax Problems

In this paper, we study zeroth-order algorithms for nonconvex-concave minimax problems, which have attracted widely attention in machine learning, signal processing and many other fields in recent years. We propose a zeroth-order alternating randomized gradient projection (ZO-AGP) algorithm for smooth nonconvex-concave minimax problems, and its iteration complexity to obtain an $\varepsilon$-stationary point is bounded by $\mathcal{O}(\varepsilon^{-4})$, and the number of function value estimation is bounded by $\mathcal{O}(d_{x}\varepsilon^{-4}+d_{y}\varepsilon^{-6})$ per iteration. Moreover, we propose a zeroth-order block alternating randomized proximal gradient algorithm (ZO-BAPG) for solving block-wise nonsmooth nonconvex-concave minimax optimization problems, and the iteration complexity to obtain an $\varepsilon$-stationary point is bounded by $\mathcal{O}(\varepsilon^{-4})$ and the number of function value estimation per iteration is bounded by $\mathcal{O}(K d_{x}\varepsilon^{-4}+d_{y}\varepsilon^{-6})$. To the best of our knowledge, this is the first time that zeroth-order algorithms with iteration complexity gurantee are developed for solving both general smooth and block-wise nonsmooth nonconvex-concave minimax problems. Numerical results on data poisoning attack problem validate the efficiency of the proposed algorithms.

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.