Recently, convex-concave bilinear Saddle Point Problems (SPP) is widely used in lasso problems, Support Vector Machines, game theory, and so on. Previous researches have proposed many methods to solve SPP, and present their convergence rate theoretically. To achieve linear convergence, analysis in those previouse studies requires strong convexity of φ( z ). But, we find the linear convergence can also be achieved even for a general convex but not strongly convex φ( z ). In the article, by exploiting the strong duality of SPP, we propose a new method to solve SPP, and achieve the linear convergence. We present a new general sufficient condition to achieve linear convergence, but do not require the strong convexity of φ( z ). Furthermore, a more efficient method is also proposed, and its convergence rate is analyzed in theoretical. Our analysis shows that the well conditioned φ( z ) is necessary to improve the efficiency of our method. Finally, we conduct extensive empirical studies to evaluate the convergence performance of our methods.

中文翻译：

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鞍点问题线性收敛的理论回顾

最近，凸凹双线性鞍点问题（SPP）被广泛应用于套索问题、支持向量机、博弈论等。以往的研究已经提出了多种求解SPP的方法，并从理论上提出了它们的收敛速度。为了实现线性收敛，之前的研究中的分析需要 φ(z）。但是，我们发现即使对于一般凸但不是强凸的 φ(z）。在本文中，我们利用 SPP 的强对偶性，提出了一种求解 SPP 的新方法，并实现了线性收敛。我们提出了一个新的一般充分条件来实现线性收敛，但不需要 φ(z）。此外，还提出了一种更有效的方法，并对其收敛速度进行了理论分析。我们的分析表明，条件良好的 φ(z) 是提高我们方法效率所必需的。最后，我们进行了广泛的实证研究来评估我们方法的收敛性能。