Reachability of gradient dynamics

2025/09/15 by Josz, Cedric, Ouyang, Wenqing
#14P10 #34A60 #49-XX #FOS: Mathematics #Optimization and Control (math.OC)

paper · doi:10.48550/arxiv.2509.12165

Abstract

We show that gradient dynamics can converge to any local minimum of a semi-algebraic function. Our results cover both discrete and continuous dynamics. For discrete gradient dynamics, we show that it can converge to any local minimum once the stepsize is nonsummable and sufficiently small, and the initial value is properly chosen.

Citations

• On the diameter of subgradient sequences in o-minimal structures
• Convergence of optimizers implies eigenvalues filtering at equilibrium
• Subdifferentiation with symmetry
• Convergence of the momentum method for semialgebraic functions with locally Lipschitz gradients
• Gradient Descent on Neural Networks Typically Occurs at the Edge of\n Stability
• On the stable set of an analytic gradient flow
• Step Size Matters in Deep Learning
• Stochastic subgradient method converges on tame functions
• Convergence of descent methods for semi-algebraic and tame problems: proximal algorithms, forward–backward splitting, and regularized Gauss–Seidel methods
• Curves of Descent
• Clarke subgradients of stratifiable functions
• Geometric categories and o-minimal structures

Cited by

• Lyapunov stability of the Euler method

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