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-{\L}ojasiewicz (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.

翻译：基于梯度的凸优化一阶算法在机器学习等众多领域具有广泛适用性。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，我们提出了一种广义框架用于设计加速优化算法，该算法具有最强的收敛保证，并可进一步扩展至非凸函数子类。具体而言，我们提出了GenFlow算法及其动量变体，该算法能够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。对于存在非退化鞍点的函数，我们证明所提出的GenFlow算法规避这些鞍点所需的时间对所有初始条件一致有界。最后，对于最优解为鞍点的强凸-强凹极小极大问题，我们证明类似方案同样能在固定时间内收敛至最优解。通过在多个基准数据集上的实验验证了本算法优越的收敛性能。