具有可证明固定时间收敛与非退化鞍点快速逃离的广义梯度流
Generalized Gradient Flows with Provable Fixed-Time Convergence and Fast Evasion of Non-Degenerate Saddle Points
AI总结：
提出GenFlow算法及其动量变体，基于固定时间稳定性理论设计加速优化框架，在固定时间内收敛至满足PL不等式的最优解，并一致有界地快速逃离非退化鞍点，解决极小极大问题。
摘要：
基于梯度的凸优化一阶算法在包括机器学习任务在内的众多领域有着广泛应用。受连续时间动力系统固定时间稳定性理论最新进展的启发，我们引入了一个广义框架，用于设计具有最强收敛保证的加速优化算法，并进一步扩展至非凸函数的子类。具体而言，我们提出了GenFlow算法及其动量变体，可证明它们能在固定时间内收敛至满足Polyak-Łojasiewicz (PL) 不等式的目标函数的最优解。此外，对于存在非退化鞍点的函数，我们证明了对于所提出的GenFlow算法，逃离这些鞍点所需的时间对所有初始条件都是一致有界的。最后，对于最优解为鞍点的强凸-强凹极小极大问题，证明了类似方案同样能在固定时间内达到最优解。我们在多个基准数据集上通过实验验证了算法的优越收敛特性。
英文摘要：
Gradient-based first-order convex optimization algorithms find widespread applicability in a variety of domains, including machine learning tasks. Motivated by the recent advances in fixed-time stability theory of continuous-time dynamical systems, we introduce a generalized framework for designing accelerated optimization algorithms with strongest convergence guarantees that further extend to a subclass of non-convex functions. In particular, we introduce the GenFlow algorithm and its momentum variant that provably converge to the optimal solution of objective functions satisfying the Polyak-Łojasiewicz (PL) inequality in a fixed time. Moreover, for functions that admit non-degenerate saddle-points, we show that for the proposed GenFlow algorithm, the time required to evade these saddle-points is uniformly bounded for all initial conditions. Finally, for strongly convex-strongly concave minimax problems whose optimal solution is a saddle point, a similar scheme is shown to arrive at the optimal solution again in a fixed time. The superior convergence properties of our algorithm are validated experimentally on a variety of benchmark datasets.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
