insight - Optimization - # Robustness of Perturbed Gradient Flows in Linear Quadratic Regulator Problem

Core Concepts

The perturbed gradient flow for the linear quadratic regulator (LQR) problem is shown to be small-disturbance input-to-state stable (ISS) under suitable conditions on the objective function.

Abstract

The paper studies the effect of perturbations on the gradient flow of a general nonlinear programming problem, where the perturbation may arise from inaccurate gradient estimation in the setting of data-driven optimization.
Key highlights:
The authors introduce the concept of "small-disturbance input-to-state stability (ISS)" and provide a necessary and sufficient condition for a system to be small-disturbance ISS.
Under assumptions of coercivity and a "comparison just saturated" (CJS) Polyak-Lojasiewicz (PL) condition on the objective function, the perturbed gradient flow is shown to be small-disturbance ISS.
For the linear quadratic regulator (LQR) problem, the authors establish the CJS-PL property for the objective function, and consequently prove that the standard gradient flow, natural gradient flow, and Newton gradient flow are all small-disturbance ISS.
The new contribution is the establishment of the CJS-PL property for the LQR problem, which extends previous results that only showed a semi-global estimate.

Stats

The following sentences contain key metrics or important figures used to support the author's key logics:
Tr (PK - P*) ≥ α4(||K - K*||_F)
Tr (M_K) ≥ a||K - K*||^2_F + a'Tr (P_K - P*)

Quotes

"Since the objective function J_2 of the LQR problem depends on the initial condition, we are motivated to study an equivalent optimization problem (min_K∈G J_2(K)), which is independent of the initial condition."
"For any stabilizing controller u(t) = -Kx(t), where K ∈ G, and any nonzero initial state x_0 ∈ R^n, the corresponding cost is J_1(x_0, K) = x_0^T P_K x_0, where P_K = P_K^T is the unique positive definite solution of the following Lyapunov equation (A - BK)^T P_K + P_K(A - BK) + Q + K^T RK = 0."

Key Insights Distilled From

by Leilei Cui,Z... at **arxiv.org** 04-17-2024

Deeper Inquiries

The CJS-PL condition, which stands for "comparison just saturated Polyak-Lojasiewicz," can be extended to a broader class of optimization problems beyond the Linear Quadratic Regulator (LQR) setting by considering the fundamental principles underlying its application. The key idea behind the CJS-PL condition is to ensure that the gradient of the objective function is not "too small" compared to the value of the function itself. This condition helps establish stability properties of perturbed gradient flows in optimization problems.
To generalize the CJS-PL condition, one can adapt it to various optimization problems by defining suitable comparison functions that capture the relationship between the gradient and the objective function. By formulating appropriate bounds on the gradient with respect to the objective function, one can ensure convergence properties and robustness in a broader range of optimization scenarios. This extension allows for the analysis of stability and convergence in optimization algorithms beyond the specific constraints of the LQR problem, making it applicable to a wider class of optimization problems.

While the small-disturbance ISS property provides valuable insights into the robustness of optimization algorithms in the presence of perturbations, it may have certain limitations compared to other notions of robustness, such as integral ISS.
One potential limitation of small-disturbance ISS is that it focuses on the behavior of trajectories in response to small perturbations, which may not fully capture the overall robustness of the optimization algorithm. Integral ISS, on the other hand, considers the cumulative effect of disturbances over time, providing a more comprehensive assessment of robustness.
Additionally, small-disturbance ISS may be sensitive to the specific conditions and assumptions of the optimization problem, potentially limiting its applicability to a broader range of scenarios. It may not address certain types of disturbances or uncertainties that integral ISS could handle more effectively.
Overall, while small-disturbance ISS offers valuable insights into the local stability of optimization algorithms, it is essential to consider its limitations and complement it with other robustness measures for a more comprehensive analysis of algorithm performance in the presence of disturbances.

The techniques developed in the paper can be extended to analyze the robustness of policy optimization algorithms in reinforcement learning beyond the LQR problem by adapting the concepts of small-disturbance ISS and the CJS-PL condition to the specific characteristics of reinforcement learning settings.
In reinforcement learning, where agents interact with environments and learn optimal policies through trial and error, the robustness of policy optimization algorithms is crucial for stable and efficient learning. By applying the principles of small-disturbance ISS and the CJS-PL condition to policy optimization in reinforcement learning, researchers can assess the convergence properties and stability of learning algorithms in the presence of perturbations.
By formulating appropriate Lyapunov-like functions and establishing suitable comparison conditions for policy optimization objectives, researchers can analyze the robustness of reinforcement learning algorithms to various sources of disturbances, such as noisy data, approximation errors, or environmental uncertainties. This extension would provide valuable insights into the performance and reliability of policy optimization methods in complex and dynamic environments.

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