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.
翻译:最近许多研究都致力于开发高效的电流算法,以解决微离子变异性问题,因为这些问题与一些突发应用程序相关。在本文中,我们提议一个统一的单环交替梯度投影(AGP)算法,以解决平滑的非conex-(强势) conculive和(强力) convex-nonconal mox问题。GP使用简单的梯度投影步骤来更新原始变量和双向变量,或者在每次循环中。我们表明,它可以在 $\ varepex- comlix 中找到一个目标函数的固定点 $\ valexal- laxal- staty 。Axleft (\ varepl) nual- comcal- discial- disquil- lax lax- lax lax discial- discial- lax) lax lax lax- disl- dromax lax- lax- dromax- dromax- lax- lax- lax- lax-max- lax-max-maxxxxxl) max max max maxl max max max max max max max max max maxx maxx max max max max max max max max max max max max max max max max max max max max max max max max max max max max max max max max max max max max max max max max max max max max max max max max max masl max max max maxxxx max max max max