Many fundamental machine learning tasks can be formulated as min-max optimization. This motivates us to design effective and efficient first-order methods that provably converge to the global min-max points. For this purpose, this thesis focuses on designing practical algorithms for several specific machine learning tasks. We considered some different settings: unconstrained or constrained strongly-convex (strongly-)concave, constrained convex-concave, and nonconvex-concave problems. We tackle

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... he following concrete questions by studying the above problems: 1. Can we reformulate a single minimization problem to two-player games to help reduce the computational complexity of finding global optimal points? 2. Can projection-free algorithms achieve last-iterate convergence for constrained min-max optimization problems with the convex-concave landscape? 3. Can we show that stochastic gradient descent-ascent, a method commonly used in practice for GAN training, actually finds global optima and can learn a target distribution? We make progress on these questions by proposing practical algorithms with theoretical guarantees. We also present extensive empirical studies to verify the effectiveness of our proposed methods.