Implicit Bias of Policy Gradient in Linear Quadratic Control: Extrapolation to Unseen Initial States (2402.07875v2)

Abstract: In modern machine learning, models can often fit training data in numerous ways, some of which perform well on unseen (test) data, while others do not. Remarkably, in such cases gradient descent frequently exhibits an implicit bias that leads to excellent performance on unseen data. This implicit bias was extensively studied in supervised learning, but is far less understood in optimal control (reinforcement learning). There, learning a controller applied to a system via gradient descent is known as policy gradient, and a question of prime importance is the extent to which a learned controller extrapolates to unseen initial states. This paper theoretically studies the implicit bias of policy gradient in terms of extrapolation to unseen initial states. Focusing on the fundamental Linear Quadratic Regulator (LQR) problem, we establish that the extent of extrapolation depends on the degree of exploration induced by the system when commencing from initial states included in training. Experiments corroborate our theory, and demonstrate its conclusions on problems beyond LQR, where systems are non-linear and controllers are neural networks. We hypothesize that real-world optimal control may be greatly improved by developing methods for informed selection of initial states to train on.

• The paper reveals that policy gradient methods favor solutions with better extrapolation when training includes ample exploration.
• It combines theoretical analysis with experiments on both linear and non-linear systems to quantify the role of initial state exploration.
• The findings advocate for revising training protocols in optimal control to improve generalization to new, unseen scenarios.

Understanding the Implicit Bias of Policy Gradient in Linear Quadratic Control

Insights into Extrapolation and Training Algorithms

The dynamics of learning in machine learning models, particularly in contexts requiring decision-making under uncertainty, hinges significantly on the model's ability to generalize beyond its training data. This attribute, known as extrapolation, is crucial in fields where the deployed model encounters situations markedly different from those it was trained on, such as autonomous driving or robotic navigation. A paper explores understanding how policy gradient methods, a cornerstone of reinforcement learning, harbor an implicit bias that affects their ability to extrapolate in Linear Quadratic Regulator (LQR) problems.

Theoretical Exploration

At the heart of this paper is the Linear Quadratic Regulator problem, a fundamental model in optimal control theory. The LQR problem involves designing a controller to regulate a system's behavior to minimize a defined quadratic cost. Notably, this problem admits linear solutions, making it a valuable testbed for theoretical analysis. The researchers focused on underdetermined LQR problems, where multiple controllers can achieve the minimum training cost, to explore how different initial states influence the learned controller's ability to extrapolate.

The analysis unveiled that the extent to which a learned controller can extrapolate to unseen initial states is heavily influenced by the system's exploration level from the initial states encountered during training. Specifically, if the system induces adequate exploration, the learned controller exhibits significant potential for extrapolation, an effect that amplifies with a more extensive exploration.

Conversely, in systems with minimal exploration from the initial states seen in training, extrapolation to unseen states does not occur. This phenomenon underscores the implicit bias introduced by the policy gradient method, revealing that not all solutions with minimum training cost are equally likely. The findings suggest that the policy gradient tends toward solutions that are better at extrapolating, provided the system engenders sufficient exploration during training.

Experimental Corroboration

The theoretical insights were substantiated through experiments with both linear systems (where controllers are linear functions) and non-linear systems controlled by neural network controllers. The experiments verified the theory's predictions, demonstrating that the extent of extrapolation is indeed contingent on the degree of exploration induced by the training initial states. For linear systems, different settings altered the level of exploration and, consequently, the extrapolation capability. The framework was extendable to non-linear systems, where neural networks controllers also exhibited extrapolation under suitable conditions.

Implications and Future Directions

This research makes a compelling case for the significance of exploration in training to the extrapolation abilities of learned controllers. The findings prompt a reassessment of training regimes in optimal control and reinforcement learning, advocating for strategies that enhance exploration to improve models' generalization to new, unseen scenarios.

The paper opens several avenues for future research, including developing methods to quantify exploration in non-linear systems and designing training protocols that systematically exploit this implicit bias for better extrapolation. Furthermore, understanding the distinctions in implicit bias across different learning algorithms could provide deeper insights into designing more robust and adaptable machine learning models for control tasks.

This exploration into the implicit bias of policy gradient methods in LQR problems not only enriches our understanding of generalization in machine learning models but also sets the stage for more informed approaches to training models for real-world decision-making applications.