# Remarks on the Polyak-Lojasiewicz inequality and the convergence of gradient systems

arXiv: https://arxiv.org/abs/2503.23641 · HTML: https://mlsanity.com/paper/2503.23641/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 28 references that are in the map.
- First posted on arXiv 31 Mar 2025.

## Builds on (most cited first, 11 of 11)

- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Linear Convergence of Gradient and Proximal-Gradient Methods Under the Polyak-Lojasiewicz Condition](https://mlsanity.com/paper/1608.04636/) (2016)
- [Global Convergence of Policy Gradient Methods for the Linear Quadratic Regulator](https://mlsanity.com/paper/1801.05039/) (2018)
- [Global Optimality Guarantees For Policy Gradient Methods](https://mlsanity.com/paper/1906.01786/) (2019)
- [Convergence and sample complexity of gradient methods for the model-free linear quadratic regulator problem](https://mlsanity.com/paper/1912.11899/) (2019)
- [Optimizing Static Linear Feedback: Gradient Method](https://mlsanity.com/paper/2004.09875/) (2020)
- [Training Linear Neural Networks: Non-Local Convergence and Complexity Results](https://mlsanity.com/paper/2002.09852/) (2020)
- [Remarks on input to state stability of perturbed gradient flows, motivated by model-free feedback control learning](https://mlsanity.com/paper/2108.02632/) (2021)
- [Small-Disturbance Input-to-State Stability of Perturbed Gradient Flows: Applications to LQR Problem](https://mlsanity.com/paper/2310.02930/) (2023)
- [Proximal Gradient Dynamics: Monotonicity, Exponential Convergence, and Applications](https://mlsanity.com/paper/2409.10664/) (2024)
- [Revisiting Strong Duality, Hidden Convexity, and Gradient Dominance in the Linear Quadratic Regulator](https://mlsanity.com/paper/2503.10964/) (2025)

## Built on by (most cited first, 1 of 1)

- [Gradient Dominance in the Linear Quadratic Regulator: A Unified Analysis for Continuous-Time and Discrete-Time Systems](https://mlsanity.com/paper/2602.22577/) (2026)

## Abstract (arXiv)

This work explores generalizations of the Polyak-Lojasiewicz inequality (PLI) and their implications for the convergence behavior of gradient flows in optimization problems. Motivated by the continuous-time linear quadratic regulator (CT-LQR) policy optimization problem -- where only a weaker version of the PLI is characterized in the literature -- this work shows that while weaker conditions are sufficient for global convergence to, and optimality of the set of critical points of the cost function, the "profile" of the gradient flow solution can change significantly depending on which "flavor" of inequality the cost satisfies. After a general theoretical analysis, we focus on fitting the CT-LQR policy optimization problem to the proposed framework, showing that, in fact, it can never satisfy a PLI in its strongest form. We follow up our analysis with a brief discussion on the difference between continuous- and discrete-time LQR policy optimization, and end the paper with some intuition on the extension of this framework to optimization problems with L1 regularization and solved through proximal gradient flows.