  Accelerated gradient methods are the cornerstones of large-scale, data-driven
optimization problems that arise naturally in machine learning and other fields
concerning data analysis. We introduce a gradient-based optimization framework
for achieving acceleration, based on the recently introduced notion of
fixed-time stability of dynamical systems. The method presents itself as a
generalization of simple gradient-based methods suitably scaled to achieve
convergence to the optimizer in a fixed-time, independent of the
initialization. We achieve this by first leveraging a continuous-time framework
for designing fixed-time stable dynamical systems, and later providing a
consistent discretization strategy, such that the equivalent discrete-time
algorithm tracks the optimizer in a practically fixed number of iterations. We
also provide a theoretical analysis of the convergence behavior of the proposed
gradient flows, and their robustness to additive disturbances for a range of
functions obeying strong convexity, strict convexity, and possibly nonconvexity
but satisfying the Polyak-{\L}ojasiewicz inequality. We also show that the
regret bound on the convergence rate is constant by virtue of the fixed-time
convergence. The hyperparameters have intuitive interpretations and can be
tuned to fit the requirements on the desired convergence rates. We validate the
accelerated convergence properties of the proposed schemes on a range of
numerical examples against the state-of-the-art optimization algorithms. Our
work provides insights on developing novel optimization algorithms via
discretization of continuous-time flows.
