This study develops a fixed-time convergent saddle point dynamical system for solving min-max problems under a relaxation of standard convexity-concavity assumption. In particular, it is shown that by leveraging the dynamical systems viewpoint of an optimization algorithm, accelerated convergence to a saddle point can be obtained. Instead of requiring the objective function to be strongly-convex--strongly-concave (as necessitated for accelerated convergence of several saddle-point algorithms), uniform fixed-time convergence is guaranteed for functions satisfying only the two-sided Polyak-{\L}ojasiewicz (PL) inequality. A large number of practical problems, including the robust least squares estimation, are known to satisfy the two-sided PL inequality. The proposed method achieves arbitrarily fast convergence compared to any other state-of-the-art method with linear or even super-linear convergence, as also corroborated in numerical case studies.
翻译:这项研究开发了一种固定时间的集中式马鞍点动态系统,以便在放宽标准凝固度(Confexity-concavility)假设的情况下解决微轴问题,特别是,通过利用动态系统对优化算法的观点,可以加速趋同到一个马鞍点,而不是要求目标功能具有很强的凝固性(因为需要加快使数个马鞍点算法趋同),而保证统一固定时间趋同功能只满足两面的Polyak-L}ojasiewicz(PL)不平等。已知许多实际问题,包括稳健的最小平方估计,可以满足双面的PL不平等。拟议方法与任何其他具有线性甚至超线性趋同状态的方法相比,实现了任意快速趋同,这一点在数字案例研究中也得到了证实。