Source-linked AI summary
Data-Efficient Reinforcement Learning with Probabilistic Model Predictive Control
Sanket Kamthe, Marc Peter Deisenroth
TL;DR
The paper addresses RL’s high interaction requirements and the need to operate under state and control constraints. It proposes probabilistic MPC with GP transition models and uncertainty propagation, using a deterministic reformulation for PMP-based planning. The authors report state-of-the-art data efficiency alongside constrained control and theoretical optimality guarantees.
Problem
RL algorithms often require impractically many environment interactions, while real systems may impose state and control constraints.
Method
The method learns probabilistic GP dynamics, propagates model uncertainty through long-term predictions, and uses probabilistic MPC with PMP for constrained control planning.
Results
The approach achieves state-of-the-art data efficiency while handling state and control constraints and retaining a probabilistic formulation.
Takeaways & Limitations
Probabilistic MPC provides a theoretically justified framework for data-efficient RL in constrained environments.
Takeaways & Limitations
The PMP formulation relies on moment matching to produce deterministic system equations, requiring a unique solution for each control sequence.
Abstract
from arXiv · showhide
Trial-and-error based reinforcement learning (RL) has seen rapid advancements in recent times, especially with the advent of deep neural networks. However, the majority of autonomous RL algorithms require a large number of interactions with the environment. A large number of interactions may be impractical in many real-world applications, such as robotics, and many practical systems have to obey limitations in the form of state space or control constraints. To reduce the number of system interactions while simultaneously handling constraints, we propose a model-based RL framework based on probabilistic Model Predictive Control (MPC). In particular, we propose to learn a probabilistic transition model using Gaussian Processes (GPs) to incorporate model uncertainty into long-term predictions, thereby, reducing the impact of model errors. We then use MPC to find a control sequence that minimises the expected long-term cost. We provide theoretical guarantees for first-order optimality in the GP-based transition models with deterministic approximate inference for long-term planning. We demonstrate that our approach does not only achieve state-of-the-art data efficiency, but also is a principled way for RL in constrained environments.
1 Introduction
The paper targets reinforcement learning’s severe interaction burden and constraint-handling challenges with a probabilistic MPC framework that learns uncertain dynamics models. It combines Gaussian-process uncertainty propagation with optimal-control principles to improve data efficiency while supporting constrained control.
- Motivation: Current RL algorithms often require impractically many environment interactions, limiting their use in real control and robotic systems.Even low-dimensional, benign tasks may require thousands of trials without task-specific priors.
- Motivation: Learning system dynamics can improve data efficiency, but model errors make probabilistic models useful for explicitly representing uncertainty.Using learned models as proxies for real environments can reduce additional interactions when uncertainty is accounted for.
- Limitations of Prior Work: PILCO is data efficient but requires full-horizon planning for feedback stabilization, a parameterized policy, and lacks state-constraint handling.Its control constraints rely on a differentiable squashing function applied to the policy.
- Limitations of Prior Work: Constraint handling through differentiable control squashing can produce unreliable predictions near constraint boundaries.This limitation is reported for prior GP-based policy control methods.
- Proposed Approach: The proposed probabilistic MPC algorithm propagates uncertainty through learned GP dynamics, uses PMP for constrained planning, and plans over relatively short horizons.Short-horizon MPC limits computational burden while supporting infinite-horizon control applications.
- Proposed Approach: The framework is designed to improve data efficiency, robustness to model errors, and principled handling of state and control constraints without requiring a parameterized policy.Its deterministic reformulation enables PMP-based open-loop planning with GP models.
2 Controller Learning via Probabilistic MPC
The controller learns probabilistic GP dynamics, propagates uncertainty through deterministic moment matching, and optimizes expected long-term cost with open-loop planning. MPC repeatedly replans from the current state, updating the model with new experience to obtain implicit feedback control.
- 2.1 Probabilistic Transition Model: The method learns a probabilistic GP model of unknown transition dynamics from state-control inputs and successor-state targets.Each GP predicts a distribution with a mean and covariance; target dimensions are assumed independent.
- 2.2 Optimal Open-Loop Control Sequence: The planner computes expected long-term cost from the predicted state distributions and selects the open-loop controls that minimize it.The stage-cost expectation and its partial derivatives are chosen to be analytically computable.
- 2.2 Optimal Open-Loop Control Sequence: For a candidate control sequence, moment matching deterministically approximates successive state distributions while propagating model uncertainty.The approximation is available in closed form with the Gaussian kernel.
- 2.3 Feedback Control with MPC: MPC applies only the first control of an H-step plan, updates the GP with the new transition, and replans from the current state.Repeated replanning creates an implicit feedback controller, typically with H much smaller than the total horizon T.
- 2.3 Feedback Control with MPC: The framework explicitly uses GP uncertainty for long-term predictions and handles constrained control signals without warping or squashing them.The model is updated immediately during interaction, which the paper connects to robustness against model inaccuracies.
3 Theoretical Justification
The paper reformulates uncertainty-propagating GP dynamics as a deterministic moment-matching system so Pontryagin’s Maximum Principle can analyze and optimize constrained trajectories. This yields necessary first-order conditions and efficient gradient computation for the expected long-term cost.
- 3.3 Efficient Gradient Computation: PMP reduces each control-gradient calculation to variables at neighboring time indices, preserving sparsity in the optimization problem.For GP dynamics, the multipliers can be computed in closed form, reducing the burden of minimizing expected long-term cost.
- 3 Theoretical Justification: Moment matching converts probabilistic GP uncertainty propagation into deterministic system equations with unique trajectories for a given control sequence.The moment-matching mapping is Lipschitz continuous for controls on a compact set, and the resulting difference equation has a unique solution.
- 3 Theoretical Justification: The Hamiltonian minimum condition minimizes the Hamiltonian point-wise over the admissible control set to identify optimal controls.The adjoint vector acts as a Lagrange multiplier for dynamics constraints.
- 3 Theoretical Justification: PMP supplies necessary first-order optimality conditions for admissible controlled trajectories while accommodating state and control constraints.The admissible controls are piecewise continuous over a compact control set.
- 3.3 Efficient Gradient Computation: Gradient descent on the Hamiltonian is equivalent to gradient descent on the total expected cost in this setting.The equivalence follows because the substituted multipliers satisfy the terminal transversality condition and the adjoint equation.
4 Experimental Results
Experiments on cart-pole and double-pendulum benchmarks show that GP-MPC learns efficiently, while probabilistic planning enables constrained task completion with fewer violations.
- Data Efficiency: GP-MPC and the zero-variance MPC approach solve cart-pole in fewer trials than PILCO, with GP-MPC learning faster and more reliably.Both methods reach a 90% success probability after three trials, whereas PILCO needs two additional trials.
- Data Efficiency: 90% success requires only six trials (18 s) with GP-MPC on the double pendulum, while PILCO needs four additional trials and zero-variance MPC fails.The deterministic approach lacks sufficient exploration when the state space is complicated and the model is poor.
- Experimental Interpretation: On-the-fly GP updates are crucial during early learning because they allow newly observed transitions to augment the model immediately.Ignoring these updates still permits successful learning but slightly decreases efficiency.
- State Constraints: The constrained benchmarks restrict cart motion with a wall and inner-pendulum motion to 340°, leaving only one feasible swing-up direction.The cart-pole must swing from the right side, while the double pendulum must first swing clockwise for a counter-clockwise swing-up.
- State Constraints: GP-MPC with chance constraints completes the constrained task with a small acceptable number of violations, whereas mean-only planning fails or struggles.Incorporating predictive variance into planning produces significantly fewer constraint violations and makes learning more reliably safe.
5 Conclusion and Discussion
The paper presents data-efficient reinforcement learning based on probabilistic MPC with learned Gaussian-process transition models. The approach uses uncertainty propagation and Pontryagin’s maximum principle to handle state and control constraints while supporting robustness to model inaccuracies.
- The algorithm combines probabilistic MPC with learned Gaussian-process transition models for data-efficient reinforcement learning.
- Reformulating uncertainty propagation as a deterministic optimal-control problem enables Pontryagin’s maximum principle to handle state and control constraints naturally.
- MPC updates the learned model immediately, increasing robustness to model inaccuracies during learning.
- The framework is reported to be theoretically sound, extremely data efficient, and applicable in settings with hard state constraints.
- Model uncertainty drives targeted exploration and supports risk-averse constraint handling, particularly early in learning.
Lipschitz Continuity
The moment-matching mapping used for GP-based uncertainty propagation is shown to be Lipschitz continuous for controls in a compact control set. The proof establishes bounded gradients with respect to the control inputs.
- Lemma 3 states that the moment-matching mapping fMM is Lipschitz continuous for controls defined over a compact set U.
- The proof reduces Lipschitz continuity to showing that the gradient ∂fMM/∂ut is bounded.
- The gradient with respect to the control input can be computed analytically.
- The GP training-set size is denoted by N, and exi denotes the ith training input in the predictive expressions.
Sequential Quadratic Programming
The paper formulates constrained nonlinear optimization for sequential quadratic programming and derives GP moment-matching statistics used in predictive mean and covariance calculations. The derivations rely on standard Gaussian integration, compact-control assumptions, and conditional independence across GP outputs.
- Sequential Quadratic Programming: Sequential Quadratic Programming solves nonlinear programs with inequality constraints b(x) ≥ 0 and equality constraints c(x) = 0.
- Sequential Quadratic Programming: The nonlinear-program Lagrangian introduces multipliers for the equality and inequality constraints.
- Sequential Quadratic Programming: At each iteration, SQP uses a quadratic objective approximation and linearized constraints around the current iterate xk.
- Sequential Quadratic Programming: The equality-constraint multipliers correspond to those in the control Hamiltonian, while the Hessian exploits the Hamiltonian’s block-diagonal structure.
- Moment matching: Predictive means and covariance matrices are derived using the law of iterated expectations, the law of total covariance, and Gaussian multiplication and integration identities.
- Moment matching: Bounded GP terms, positive-definite matrices, and compact controls establish that the moment-matching mapping is locally Lipschitz continuous.
- Moment matching: Conditional independence of GP outputs removes the additional conditional-covariance term from off-diagonal predictive covariance entries.