Remarks on the Polyak-Lojasiewicz inequality and the convergence of gradient systems
This work addresses convergence analysis for gradient systems in optimization, particularly for control problems like CT-LQR, but it is incremental as it builds on existing PLI theory with specific applications.
The paper investigates generalizations of the Polyak-Lojasiewicz inequality (PLI) and their impact on gradient flow convergence in optimization, showing that weaker conditions suffice for global convergence in problems like continuous-time linear quadratic regulator (CT-LQR) policy optimization, but the solution profile varies with the inequality type, and it proves CT-LQR cannot satisfy the strongest PLI form.
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.