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

TL;DR

This paper introduces the GenFlow algorithm, a novel optimization method with fixed-time convergence guarantees for convex and certain non-convex functions, and demonstrates its ability to efficiently evade saddle points.

Contribution

The paper proposes the GenFlow algorithm with fixed-time convergence and saddle point evasion capabilities, extending stability theory to optimization.

Findings

Provably converges to the optimal solution in fixed time.

Uniformly bounded time to evade non-degenerate saddle points.

Validated experimentally on benchmark datasets.

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.