Imitation Learning of MPC with Neural Networks: Error Guarantees and Sparsification

This paper presents a framework for bounding the approximation error in imitation model predictive controllers utilizing neural networks. Leveraging the Lipschitz properties of these neural networks, we derive a bound that guides dataset design to ensure the approximation error remains at chosen limits. We discuss how this method can be used to design a stable neural network controller with performance guarantees employing existing robust model predictive control approaches for data generation. Additionally, we introduce a training adjustment, which is based on the sensitivities of the optimization problem and reduces dataset density requirements based on the derived bounds. We verify that the proposed augmentation results in improvements to the network’s predictive capabilities and a reduction of the Lipschitz constant. Moreover, on a simulated inverted pendulum problem, we show that the approach results in a closer match of the closed-loop behavior between the imitation and the original model predictive controller.

💡 Research Summary

The paper addresses the problem of approximating a Model Predictive Control (MPC) law with a neural network (NN) while providing rigorous guarantees on the approximation error. The authors first observe that both the exact MPC feedback law κ(x) and the NN approximation κ_NN(x) are Lipschitz continuous. By denoting the Lipschitz constants of the MPC law and the NN as L_MPC and L_NN respectively, they derive a worst‑case error bound that holds over the entire state space X. The key result (Theorem 1) states that if the training dataset D = {(x_i, κ(x_i))} satisfies two simple conditions—(i) the maximum training error ε_D = max_i ‖κ(x_i) – κ_NN(x_i)‖ is smaller than a prescribed tolerance ε, and (ii) every point x ∈ X is within a distance δ = (ε – ε_D)/(L_MPC + L_NN) of at least one training point—then the global error ‖κ(x) – κ_NN(x)‖ ≤ ε for all x ∈ X. This gives a constructive guideline for how densely the state space must be sampled, linking the required data density directly to the Lipschitz constants and the desired error level.

Having a bound on the approximation error enables the use of robust MPC formulations that are tolerant to bounded input disturbances. The authors show that if the original MPC is robust to disturbances of magnitude ε, then the NN‑based controller inherits feasibility and stability guarantees when the bound from Theorem 1 is satisfied (Corollary 1). Thus, the NN can be safely deployed without a separate post‑hoc verification step.

The second major contribution is a training augmentation that reduces both ε_D and L_NN, thereby relaxing the data‑density requirement. The authors compute the parametric sensitivity of the optimal control problem, i.e., the Jacobian ∂κ/∂x, by differentiating the Karush‑Kuhn‑Tucker (KKT) conditions of the underlying nonlinear program. This sensitivity is then added to the training set, forming an enriched dataset \hat D = {(x_i, κ(x_i), ∂κ/∂x_i)}. The loss function combines three terms: (1) a standard mean‑squared error (MSE) on the control outputs, (2) a sensitivity loss that penalizes the mismatch between the NN Jacobian ∂κ_NN/∂x and the true MPC Jacobian, and (3) an ℓ₂ regularization on the weight matrices, which is known to reduce the NN’s Lipschitz constant. The overall loss is L = λ₁·L_MSE + λ₂·L_sens + λ₃·∑‖W_j‖², where λ₁‑λ₃ are user‑chosen weighting factors. Because the NN is differentiable (smooth activations such as tanh are assumed), the Jacobian ∂κ_NN/∂x can be obtained by automatic differentiation during back‑propagation, while the true Jacobian is pre‑computed from the OCP sensitivities.

The methodology is validated on an inverted pendulum with two states (angle and angular velocity) and a single torque input. The MPC is formulated with a quadratic stage cost (Q = diag(10,1), R = 0.1) and a terminal cost equal to the stage cost weight, subject to state constraints (−2π ≤ θ ≤ 2π, −1 ≤ ω ≤ 1). A dataset of 350 uniformly spaced grid points in the admissible state space is generated; for each point the optimal control sequence and the sensitivity ∂κ/∂x are computed. Two feed‑forward NNs with identical architecture (2 hidden layers, 10 tanh neurons each) are trained: one with the plain MSE loss, the other with the proposed composite loss. Results show that the sensitivity‑augmented NN achieves a lower validation MSE (≈40 % reduction) and a smaller estimated Lipschitz constant (≈30 % reduction). In closed‑loop simulations, both NNs reproduce the MPC behavior, but the augmented NN exhibits faster convergence, reduced overshoot, and smoother control inputs, indicating that the theoretical error bound is indeed tighter in practice.

Overall, the paper makes three salient points: (1) it provides an explicit, Lipschitz‑based error bound that can be used a priori to design training datasets; (2) it introduces a practical loss‑function augmentation that simultaneously improves approximation accuracy and reduces the NN’s Lipschitz constant; (3) it demonstrates that these theoretical advances translate into tangible performance gains on a benchmark nonlinear control problem. The approach is compatible with any robust MPC scheme that tolerates bounded input disturbances, making it attractive for safety‑critical applications where guarantees on constraint satisfaction and stability are mandatory. Future work is suggested to extend the framework to higher‑dimensional systems, explore alternative robust MPC formulations, and compare the sensitivity‑based augmentation with other regularization techniques such as spectral normalization.