Breaking the Convergence Barrier: Optimization via Fixed-Time Convergent Flows

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 framework for achieving acceleration, based on recently introduced notion fixed-time stability dynamical systems. The method presents itself as generalization simple suitably scaled to achieve convergence optimizer fixed-time, independent initialization. this by first leveraging continuous-time designing stable systems, later providing consistent discretization strategy, such equivalent discrete-time algorithm tracks practically fixed number iterations. also provide theoretical analysis behavior proposed flows, their robustness additive disturbances range functions obeying strong convexity, strict possibly nonconvexity but satisfying Polyak-Łojasiewicz inequality. show regret bound rate is constant virtue convergence. hyperparameters have intuitive interpretations can be tuned fit requirements desired rates. validate accelerated properties schemes numerical examples against state-of-the-art algorithms. Our work provides insights developing novel algorithms via flows.