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

arXiv: https://arxiv.org/abs/2112.01363 · HTML: https://mlsanity.com/paper/2112.01363/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 23 references that are in the map.
- First posted on arXiv 2 Dec 2021.

## Builds on (most cited first, 8 of 8)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Linear Convergence of Gradient and Proximal-Gradient Methods Under the Polyak-Lojasiewicz Condition](https://mlsanity.com/paper/1608.04636/) (2016)
- [A Differential Equation for Modeling Nesterov's Accelerated Gradient Method: Theory and Insights](https://mlsanity.com/paper/1503.01243/) (2015)
- [A Variational Perspective on Accelerated Methods in Optimization](https://mlsanity.com/paper/1603.04245/) (2016)
- [Accelerated Method for Stochastic Composition Optimization with Nonsmooth Regularization](https://mlsanity.com/paper/1711.03937/) (2017)
- [Fixed-Time Stable Gradient Flows: Applications to Continuous-Time Optimization](https://mlsanity.com/paper/1808.10474/) (2018)
- [Fixed-Time Stable Proximal Dynamical System for Solving MVIPs](https://mlsanity.com/paper/1908.03517/) (2019)
- [A Continuous-Time Nesterov Accelerated Gradient Method for Centralized and Distributed Online Convex Optimization](https://mlsanity.com/paper/2009.12545/) (2020)

## Built on by (most cited first, 3 of 3)

- [Generalized Gradient Flows with Provable Fixed-Time Convergence and Fast Evasion of Non-Degenerate Saddle Points](https://mlsanity.com/paper/2212.03765/) (2022)
- [Distributed Optimization via Energy Conservation Laws in Dilated Coordinates](https://mlsanity.com/paper/2409.19279/) (2024)
- [Using Non-Lipschitz Signum-based Functions for Distributed Optimization and Machine Learning: Trade-off Between Con-vergence Rate and Optimality Gap](https://mlsanity.com/paper/2608.01220/) (2026)

## Abstract (arXiv)

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-Lojasiewicz 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.