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Gradient-based first-order convex optimization algorithms find widespreadapplicability in a variety of domains, including machine learning tasks.Motivated by the recent advances in fixed-time stability theory ofcontinuous-time dynamical systems, we introduce a generalized framework fordesigning accelerated optimization algorithms with strongest convergenceguarantees that further extend to a subclass of non-convex functions. Inparticular, we introduce the GenFlow algorithm and its momentum variant thatprovably converge to the optimal solution of objective functions satisfying thePolyak-{\L}ojasiewicz (PL) inequality in a fixed time. Moreover, for functionsthat admit non-degenerate saddle-points, we show that for the proposed GenFlowalgorithm, the time required to evade these saddle-points is uniformly boundedfor all initial conditions. Finally, for strongly convex-strongly concaveminimax problems whose optimal solution is a saddle point, a similar scheme isshown to arrive at the optimal solution again in a fixed time. The superiorconvergence properties of our algorithm are validated experimentally on avariety of benchmark datasets.

Generalized Gradient Flows with Provable Fixed-Time Convergence and Fast Evasion of Non-Degenerate Saddle Points