Pontryagin Differentiable Programming: An End-to-End Learning and Control Framework (1912.12970v5)

Abstract: This paper develops a Pontryagin Differentiable Programming (PDP) methodology, which establishes a unified framework to solve a broad class of learning and control tasks. The PDP distinguishes from existing methods by two novel techniques: first, we differentiate through Pontryagin's Maximum Principle, and this allows to obtain the analytical derivative of a trajectory with respect to tunable parameters within an optimal control system, enabling end-to-end learning of dynamics, policies, or/and control objective functions; and second, we propose an auxiliary control system in the backward pass of the PDP framework, and the output of this auxiliary control system is the analytical derivative of the original system's trajectory with respect to the parameters, which can be iteratively solved using standard control tools. We investigate three learning modes of the PDP: inverse reinforcement learning, system identification, and control/planning. We demonstrate the capability of the PDP in each learning mode on different high-dimensional systems, including multi-link robot arm, 6-DoF maneuvering quadrotor, and 6-DoF rocket powered landing.

• The paper introduces PDP by leveraging differential PMP and an auxiliary control system to enable end-to-end learning and control.
• It achieves superior performance in inverse reinforcement learning and system identification by computing analytical derivatives that optimize dynamics and policies.
• Experimental results on high-dimensional systems show PDP’s competitive accuracy, faster convergence, and significant computational savings.

An Overview of Pontryagin Differentiable Programming: An End-to-End Learning and Control Framework

The paper introduces Pontryagin Differentiable Programming (PDP) as a unified framework aimed at addressing a diverse range of learning and control tasks. Distinguishing itself from existing methodologies, PDP leverages two innovative techniques: differential Pontryagin Maximum Principle (PMP) and the integration of an auxiliary control system. These techniques enable the computation of analytical derivatives of trajectories concerning tunable parameters within optimal control systems, thereby facilitating end-to-end learning of dynamics models, control policies, and objective functions.

Methodological Contributions

The PDP framework capitalizes on the principle of differentiating through PMP to gain an analytical understanding of how trajectory outcomes react to parameter modifications. This is complemented by the design of an auxiliary control system that iteratively computes these derivatives using established control methods. The proposal explores three contexts: inverse reinforcement learning (IRL), system identification (SysID), and control/planning.

1. Inverse Reinforcement Learning (IRL): The framework models expert behavior by learning both the dynamics and objective functions from demonstration data, minimizing discrepancies between modeled and observed trajectories.
2. System Identification (SysID): PDP allows for the precise estimation of system dynamics by correlating model predictions with observed states and inputs, enhancing the predictability of system behavior.
3. Control/Planning: By parameterizing control policies, PDP derives optimal options that minimize specified costs, adapting both closed-loop and open-loop control settings to situational demands.

Experimental Validation

The paper substantiates its claims across several high-dimensional systems, including a multi-link robot arm, a 6-DoF quadrotor, and a 6-DoF rocket landing scenario. The framework demonstrates superior performance over traditional methods, particularly in environments where the dimensionality and complexity of systems are pronounced. For instance, PDP exhibits greater efficacy than neural policy cloning in IRL tasks by achieving reduced imitation losses and faster convergence speeds. Similarly, its SysID approach outperforms neural dynamics models and DMDc, illustrating enhanced data efficiency and model accuracy.

In the control/planning context, despite PDP's first-order gradient-descent basis potentially limiting convergence speed compared to second-order methods like iLQR or DDP, it provides competitive solutions with considerable computational savings. This advantage is primarily due to the modularity of the auxiliary control system, which simplifies the computational expense associated with trajectory and derivative calculations.

Implications and Future Directions

PDP represents a significant step in integrating optimal control theory with machine learning. Its ability to perform end-to-end learning positions it uniquely for solving large-scale, continuous-space problems found in robotics, autonomous vehicles, and other domains reliant on complex dynamical systems. Additionally, the work advocates for the incorporation of control-theoretic insights within learning paradigms to improve both learning performance and interpretability.

From a theoretical perspective, the contribution underscores the potency of PMP and dynamical systems as lenses through which learning models can be interpreted and enhanced. Practically, PDP paves the way for more efficient model-based reinforcement learning and control solutions.

As the field progresses, exploring the scalability of PDP to even higher-dimensional systems and extending its applicability to systems with stochastic elements could represent fruitful directions. Moreover, investigating the integration of safety constraints into the PDP framework could enhance its deployment in real-world applications, where operational safety remains paramount.

In summary, the PDP framework bridges the gap between learning theories and control applications, offering robust methodologies that harness the strengths of both domains to tackle sophisticated problems. With ongoing development and application, this approach holds promise for broadening our capabilities within the array of tasks central to artificial intelligence and robotics.