Incremental Stability of Gradient Flows under Generalized Polyak-Łojasiewicz Inequalities and Its Implications for General Systems
Abstract
Gradient flows of twice continuously differentiable strongly convex and convex objectives are, respectively, incrementally exponentially stable and non-expansive in the Euclidean metric. In this work, we investigate to what extent these properties extend to nonconvex objectives satisfying generalized Polyak-Łojasiewicz inequalities. For objectives with a unique minimizer, we establish conditions under which the gradient flow is contracting in a state-dependent Riemannian metric and incrementally exponentially stable. For objectives with nonunique minimizers, we generalize the non-expansivity of convex objectives to incremental stability for more general nonconvex objectives. We focus on incremental stability due to its implication to convergence to equilibrium points; when paired with a Lyapunov function it implies pointwise convergence to a minimizer and continuity of the endpoint map. Along the way, we derive new structural results for incrementally stable nonlinear systems and extend the analysis to Heavy-Ball and Dynamical Inertial Newton flows.
Related
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BibTeX
@misc{oliveira2026incremental,
title = {{Incremental Stability of Gradient Flows under Generalized Polyak-Łojasiewicz Inequalities and Its Implications for General Systems}},
author = {Andreas Oliveira and Arthur C. B. de Oliveira and Mario Sznaier and Eduardo Sontag},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.40897},
eprint = {2610.40897},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.40897},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Incremental Stability of Gradient Flows under Generalized Polyak-Łojasiewicz Inequalities and Its Implications for General Systems
AU - Andreas Oliveira
AU - Arthur C. B. de Oliveira
AU - Mario Sznaier
AU - Eduardo Sontag
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.40897
UR - https://arxiv.org/abs/2610.40897
AB - Gradient flows of twice continuously differentiable strongly convex and convex objectives are, respectively, incrementally exponentially stable and non-expansive in the Euclidean metric. In this work, we investigate to what extent these properties extend to nonconvex objectives satisfying generalized Polyak-Łojasiewicz inequalities. For objectives with a unique minimizer, we establish conditions under which the gradient flow is contracting in a state-dependent Riemannian metric and incrementally exponentially stable. For objectives with nonunique minimizers, we generalize the non-expansivity of convex objectives to incremental stability for more general nonconvex objectives. We focus on incremental stability due to its implication to convergence to equilibrium points; when paired with a Lyapunov function it implies pointwise convergence to a minimizer and continuity of the endpoint map. Along the way, we derive new structural results for incrementally stable nonlinear systems and extend the analysis to Heavy-Ball and Dynamical Inertial Newton flows.
ER -