Efficient Gradient-Based Optimization for the Discounted Discrete-Time Linear Quadratic Regulator with Unknown Parameters

Core Concepts

This paper proposes a new algorithm that provably achieves ε-optimality for the discounted discrete-time Linear Quadratic Regulator (LQR) problem with unknown parameters, using only O(1/ε) function evaluations, without relying on two-point gradient estimates.

Abstract

The paper focuses on the discounted discrete-time Linear Quadratic Regulator (LQR) problem, where the system parameters are unknown. The key contributions are:

The authors propose a new gradient estimation scheme inspired by the REINFORCE method, which relies on appropriately sampling deterministic policies. This allows them to achieve high-probability upper bounds on the gradient estimates using moment concentration inequalities.

By adopting time-varying learning rates, the authors' methodology enables them to reach an O(1/ε) convergence rate, circumventing the need for two-point gradient estimations, which are known to be unrealistic in many settings.

The authors provide a detailed analysis of the regularity properties of the LQR cost function, including local Lipschitz continuity, local smoothness, and the Polyak-Lojasiewicz (PL) condition. These properties are crucial for establishing the convergence guarantees.

The authors show that their proposed algorithm achieves ε-optimality with a sample complexity of Õ(1/ε), substantially improving upon the previous best-known results, which either had a sample complexity of O(1/ε^2) or relied on additional stability assumptions.

The paper presents a significant advancement in the understanding and optimization of the discounted discrete-time LQR problem in the model-free setting, with potential applications in various control and reinforcement learning domains.

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Key Insights Distilled From

Sample Complexity of the Linear Quadratic Regulator: A Reinforcement Learning Lens

by Amirreza Nes... at arxiv.org 04-18-2024

https://arxiv.org/pdf/2404.10851.pdf

Deeper Inquiries

How can the proposed gradient estimation scheme be extended or adapted to handle more general classes of control problems beyond the LQR setting

The proposed gradient estimation scheme can be extended or adapted to handle more general classes of control problems beyond the Linear Quadratic Regulator (LQR) setting by incorporating different policy representations and function approximators. In the context of reinforcement learning, the policy gradient method can be applied to a wider range of control problems by using neural networks to represent the policy function. This allows for more complex and high-dimensional state and action spaces to be handled effectively. Additionally, the algorithm can be modified to incorporate different exploration strategies, such as epsilon-greedy or Boltzmann exploration, to balance exploration and exploitation in more challenging environments. Furthermore, the algorithm can be extended to handle continuous action spaces by using techniques like actor-critic methods or deterministic policy gradients.

What are the potential limitations or challenges in applying the proposed algorithm in real-world scenarios with practical constraints, such as limited computational resources or noisy observations

While the proposed algorithm shows promise in theory, there are potential limitations and challenges in applying it to real-world scenarios with practical constraints. One major challenge is the computational complexity of the algorithm, especially when dealing with high-dimensional state and action spaces. The algorithm may require significant computational resources and time to converge, making it less practical for real-time applications or systems with limited computational capabilities. Additionally, the algorithm's performance may be affected by noisy observations or uncertainties in the system dynamics, leading to suboptimal policies or convergence issues. Adapting the algorithm to handle noisy observations and uncertainties effectively is crucial for its practical applicability in real-world scenarios. Moreover, the algorithm's sensitivity to hyperparameters and initialization may pose challenges in tuning the algorithm for optimal performance in different environments.

Can the ideas developed in this work be leveraged to address other non-convex optimization problems in machine learning and control theory

The ideas developed in this work can be leveraged to address other non-convex optimization problems in machine learning and control theory by applying similar reinforcement learning techniques and gradient estimation methods. The policy gradient approach can be extended to tackle a variety of non-convex optimization problems, such as deep reinforcement learning tasks, robotic control, and multi-agent systems. By incorporating neural networks as function approximators and leveraging advanced optimization algorithms, the proposed algorithm can be adapted to handle complex non-convex optimization landscapes. Additionally, the insights gained from this work, such as the use of time-varying learning rates and gradient estimates, can be applied to a wide range of optimization problems to improve convergence rates and sample efficiency.