We propose a new stochastic method SAPD+ for solving nonconvex-concave minimax problems of the form $\min\max\mathcal{L}(x,y)=f(x)+\Phi(x,y)-g(y)$, where $f,g$ are closed convex and $\Phi(x,y)$ is a smooth function that is weakly convex in $x$, (strongly) concave in $y$. For both strongly concave and merely concave settings, SAPD+ achieves the best known oracle complexities of $\mathcal{O}(L\kappa_y\epsilon^{-4})$ and $\mathcal{O}(L^3\epsilon^{-6})$, respectively, without assuming compactness of the problem domain, where $\kappa_y$ is the condition number, and $L$ is the Lipschitz constant. We also propose SAPD+ with variance reduction, which enjoys the best known oracle complexity of $\mathcal{O}(L\kappa_y^2\epsilon^{-3})$ for weakly convex-strongly concave setting. We demonstrate the efficiency of SAPD+ on a distributionally robust learning problem with a nonconvex regularizer and also on a multi-class classification problem in deep learning.

We propose a new stochastic method SAPD for solving nonconvex-concave minimax problems of the form \min\max\mathcal{L}(x,y) f(x) \Phi(x,y)-g(y), where f,g are closed convex and \Phi(x,y) is a smooth function that is weakly convex in x, (strongly) concave in y . For both strongly concave and merely concave settings, SAPD achieves the best known oracle complexities of \mathcal{O}(L\kappa_y\epsilon {-4}) and \mathcal{O}(L 3\epsilon {-6}), respectively, without assuming compactness of the problem domain, where \kappa_y is the condition number, and L is the Lipschitz constant. We also propose SAPD with variance reduction, which enjoys the best known oracle complexity of \mathcal{O}(L\kappa_y 2\epsilon {-3}) for weakly convex-strongly concave setting. We demonstrate the efficiency of SAPD on a distributionally robust learning problem with a nonconvex regularizer and also on a multi-class classification problem in deep learning.