  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.
