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

2025-03-31

AI Summary

This paper investigates the hierarchical strength of the Polyak–Łojasiewicz inequality (PLI) and its decisive impact on gradient flow convergence behavior, using continuous-time linear quadratic regulator (CT-LQR) policy optimization as a canonical case—where only the weak PLI holds, while the strong PLI fails universally. Method: Leveraging non-convex optimization theory and Lyapunov stability analysis, we systematically characterize convergence rates, trajectory geometry, and decay patterns under varying PLI strengths, and rigorously prove that the CT-LQR objective violates the strong PLI for all parameter configurations. Contribution/Results: We establish a novel global convergence framework under weak PLI, revealing fundamental distinctions between CT-LQR and discrete-time LQR convergence mechanisms. Furthermore, we provide an extensible analytical paradigm applicable to L₁-regularized optimization problems, enabling precise quantification of convergence behavior in settings where strong PLI-based guarantees are unavailable.

Technology Category

Search and Optimization: Non-convex OptimizationMachine Learning: OptimizationReasoning under Uncertainty: Stochastic Optimization

Application Category

Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for rankingSecurity and Privacy: Large-scale security measurementsEconomics, Online Markets and Human Computation: Cost models of using LLMs in production systems

Abstract

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.

Problem

• Generalizes Polyak-Lojasiewicz inequality for gradient flow convergence
• Analyzes weaker PLI conditions in CT-LQR policy optimization
• Explores PLI applicability in L1-regularized proximal gradient flows

Innovation

• Generalizes Polyak-Lojasiewicz inequality for gradient flows
• Analyzes CT-LQR policy optimization under weaker PLI
• Extends framework to L1 regularization via proximal gradients