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

• URL: http://arxiv.org/abs/2212.03765v2
• Date: Sun, 22 Oct 2023 08:17:03 GMT
• Title: Generalized Gradient Flows with Provable Fixed-Time Convergence and Fast Evasion of Non-Degenerate Saddle Points
• Authors: Mayank Baranwal, Param Budhraja, Vishal Raj, Ashish R. Hota
• Abstract summary: 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 theory of optimal time, we introduce a framework for designing accelerated optimization algorithms. For functions that admit non-de saddle-points, we show that the time required to evade these saddle-points is uniformly bounded for all initial conditions.
• Score: 8.452349885923507
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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