Model Predictive Control and Reinforcement Learning: A Unified Framework Based on Dynamic Programming (2406.00592v3)

Abstract: In this paper we describe a new conceptual framework that connects approximate Dynamic Programming (DP), Model Predictive Control (MPC), and Reinforcement Learning (RL). This framework centers around two algorithms, which are designed largely independently of each other and operate in synergy through the powerful mechanism of Newton's method. We call them the off-line training and the on-line play algorithms. The names are borrowed from some of the major successes of RL involving games; primary examples are the recent (2017) AlphaZero program (which plays chess, [SHS17], [SSS17]), and the similarly structured and earlier (1990s) TD-Gammon program (which plays backgammon, [Tes94], [Tes95], [TeG96]). In these game contexts, the off-line training algorithm is the method used to teach the program how to evaluate positions and to generate good moves at any given position, while the on-line play algorithm is the method used to play in real time against human or computer opponents. Significantly, the synergy between off-line training and on-line play also underlies MPC (as well as other major classes of sequential decision problems), and indeed the MPC design architecture is very similar to the one of AlphaZero and TD-Gammon. This conceptual insight provides a vehicle for bridging the cultural gap between RL and MPC, and sheds new light on some fundamental issues in MPC. These include the enhancement of stability properties through rollout, the treatment of uncertainty through the use of certainty equivalence, the resilience of MPC in adaptive control settings that involve changing system parameters, and the insights provided by the superlinear performance bounds implied by Newton's method.

Model Predictive Control and Reinforcement Learning: A Unified Framework Based on Dynamic Programming

Dimitri P. Bertsekas articulates a comprehensive framework bridging Model Predictive Control (MPC) and Reinforcement Learning (RL) through the principles of Dynamic Programming (DP). The paper underscores the synergy between off-line training and on-line play, supported by Newton's method, to draw parallels between the MPC design and successful game-based RL algorithms such as AlphaZero and TD-Gammon.

Framework Overview

Bertsekas proposes a dual-algorithm approach referred to as the off-line training algorithm and the on-line play algorithm. The off-line training algorithm is used to generate cost function approximations, such as those employed in neural networks, while the on-line play algorithm executes control actions in real-time. This distinction emulates successful RL methodologies where off-line training involves substantial data processing to develop policy estimators, and on-line deployment leverages these estimators through lookahead strategies to enhance real-time decision-making.

A principal insight provided by this framework is the reinterpretation of one-step and multi-step lookahead methods within MPC as iterations of Newton’s method, thereby yielding superlinear convergence properties. This interpretation extends the accuracy and stability of control policies in MPC, aligning them closely with optimally derived policies from classical DP paradigms.

Theoretical Underpinnings

The unifying framework positions lookahead methods as the core of both MPC and RL methodologies, benefiting from the robustness and computational efficiency inherently provided by Newton-like iterations. In this context, several key points are emphasized:

1. One-Step Lookahead as Newton's Method: Provides an approximation in value space interpreted as a Newton step for solving BeLLMan's equations, leading to rapid convergence towards optimal policies.
2. Multi-Step Lookahead: This is likened to a step of a combined Newton/SOR method and further enhances the region of convergence, effectively expanding the set of initial conditions leading to stability.
3. Region of Stability: The region within which the policy derived through approximation is stable is intrinsically linked to the region of convergence from Newton’s method, with multi-step lookahead extending this region further.

Practical Extensions and Applications

Stochastic Systems and Certainty Equivalence

For stochastic systems, the framework suggests the integration of certainty equivalence (CE) to manage the computational complexity induced by probabilistic uncertainties. By only treating the immediate stages stochastically and applying CE to subsequent steps, the solution retains the convergence characteristics of Newton's method while simplifying computations.

Adaptive Control Through Rollout

In addressing unknown or dynamically changing system parameters, Bertsekas proposes the use of rollout algorithms as an adaptive control mechanism. Rollout methods employ the current policy as a base and improve real-time decision-making through simulation-based policy iterations, ensuring robust performance even with parameter variations. This approach obviates the need for extensive retraining or optimization, facilitating rapid adaptation in changing environments.

Implications and Future Directions

Bertsekas’s framework suggests a convergence of MPC and RL methodologies enabled by principled approximations rooted in DP. The theoretical underpinnings and practical strategies laid out provide a robust pathway for both fields to adopt and evolve more integrated and adaptive control systems. Future developments could include more advanced learning mechanisms for off-line training and deeper exploration into hybrid algorithms combining the strengths of traditional control theory and modern machine learning.

This unified framework establishes a versatile foundation for addressing complex, uncertain, and dynamically evolving control problems, leveraging the synergies between DP, MPC, and RL to achieve enhanced stability, performance, and computational efficiency.