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Safety Beyond the Training Data: Robust Out-of-Distribution MPC via Conformalized System Level Synthesis

Anutam Srinivasan, Antoine Leeman, Glen Chou

TL;DR

The paper addresses safe planning with learned dynamics when operating outside the training distribution. It introduces CP-SLS-MPC, which binds model error via weighted conformal prediction bound ellipsoids and enforces safety through system level synthesis-based reachable tubes, enabling robust constraint satisfaction over a horizon $T$ with high probability. The method provides theoretical guarantees on coverage and robustness under distribution drift, and demonstrates improved safety and prediction accuracy on nonlinear 4D car and 12D quadcopter trajectories, including OOD scenarios and disjoint training domains. By combining online calibration and an active uncertainty reduction cost, the approach remains computationally efficient for real-time planning and can guide exploration toward regions with lower model error, enhancing practical impact in robotics and autonomous systems.

Abstract

We present a novel framework for robust out-of-distribution planning and control using conformal prediction (CP) and system level synthesis (SLS), addressing the challenge of ensuring safety and robustness when using learned dynamics models beyond the training data distribution. We first derive high-confidence model error bounds using weighted CP with a learned, state-control-dependent covariance model. These bounds are integrated into an SLS-based robust nonlinear model predictive control (MPC) formulation, which performs constraint tightening over the prediction horizon via volume-optimized forward reachable sets. We provide theoretical guarantees on coverage and robustness under distributional drift, and analyze the impact of data density and trajectory tube size on prediction coverage. Empirically, we demonstrate our method on nonlinear systems of increasing complexity, including a 4D car and a {12D} quadcopter, improving safety and robustness compared to fixed-bound and non-robust baselines, especially outside of the data distribution.

Safety Beyond the Training Data: Robust Out-of-Distribution MPC via Conformalized System Level Synthesis

TL;DR

The paper addresses safe planning with learned dynamics when operating outside the training distribution. It introduces CP-SLS-MPC, which binds model error via weighted conformal prediction bound ellipsoids and enforces safety through system level synthesis-based reachable tubes, enabling robust constraint satisfaction over a horizon with high probability. The method provides theoretical guarantees on coverage and robustness under distribution drift, and demonstrates improved safety and prediction accuracy on nonlinear 4D car and 12D quadcopter trajectories, including OOD scenarios and disjoint training domains. By combining online calibration and an active uncertainty reduction cost, the approach remains computationally efficient for real-time planning and can guide exploration toward regions with lower model error, enhancing practical impact in robotics and autonomous systems.

Abstract

We present a novel framework for robust out-of-distribution planning and control using conformal prediction (CP) and system level synthesis (SLS), addressing the challenge of ensuring safety and robustness when using learned dynamics models beyond the training data distribution. We first derive high-confidence model error bounds using weighted CP with a learned, state-control-dependent covariance model. These bounds are integrated into an SLS-based robust nonlinear model predictive control (MPC) formulation, which performs constraint tightening over the prediction horizon via volume-optimized forward reachable sets. We provide theoretical guarantees on coverage and robustness under distributional drift, and analyze the impact of data density and trajectory tube size on prediction coverage. Empirically, we demonstrate our method on nonlinear systems of increasing complexity, including a 4D car and a {12D} quadcopter, improving safety and robustness compared to fixed-bound and non-robust baselines, especially outside of the data distribution.
Paper Structure (41 sections, 5 theorems, 51 equations, 11 figures, 2 tables, 1 algorithm)

This paper contains 41 sections, 5 theorems, 51 equations, 11 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Predictive Control
  4. Planning with Learned Dynamics
  5. Conformal Prediction (CP)
  6. Preliminaries and Problem Statement
  7. Conformal Prediction (CP)
  8. SLS
  9. Problem Statement
  10. Method
  11. Model Training and Model Error Bounding via Conformal Prediction
  12. Informing SLS-based Planning with CP Model Error Bounds
  13. Practical Implementation via SCP
  14. Theoretical Analysis: Bounding the Coverage Gap
  15. Results
  16. ...and 26 more sections

Key Result

theorem 1

Given a length-$T$ trajectory of the true dynamics $f$eq:dynamics$\tau_f:= \{(x_k, u_k)\}_{k=1}^{T}$ such that $(x_k, u_k) \in \mathcal{X} \times \mathcal{U}$ for all $k \in \{1, \ldots, T\}$, desired miscoverage $\{\alpha_k\}_{k=1}^T$, total miscoverage $\sigma_k = \alpha_k + 2\sum_{i=1}^{N_\textrm

Figures (11)

  • Figure 1: OOD Car. (Top) Minimum distance between prediction error and the ellipsoid edge is plotted for varying $\mathcal{D}_{\mathrm{calib}}$ with a fixed start and goal; negative values (blue) indicate points inside the ellipsoid, i.e., valid coverage. (Bottom) Corresponding one-step $\theta$ tubes are shown. Smaller $\mathcal{D}_{\mathrm{calib}}$ keeps prediction errors inside the ellipsoid more often and results in larger tubes.
  • Figure 2: Friction Car. We plot all approaches and forecasted MPC steps & tubes for CP-Ellipsoid.
  • Figure 3: Active Uncertainty. We plot the histogram of summed log-volumes for each forecasted MPC plan. With active uncertainty reduction, the distribution shifts left, indicating less uncertainty at runtime.
  • Figure 4: Quadcopter. A rollout of the quadcopter with V-MPC and CP-Ellipsoid. CP-Ellipsoid maintains sufficient proximity to avoid crashing into the obstacle, while V-MPC crashes.
  • Figure 5: Toy Example. We plot the effect of the calibration set size, $N_\textrm{calib}$, and compare the weightage provided to points using the exponential weight decay function at different $\rho$ values. The results are averaged across 100 trials of randomly sampling $N_\textrm{calib}$ points in the unit circle, with an example of the calibration data location in the top row. The second row (a histogram) shows that, under uniform sampling on the unit circle, the number of points at each radius (Distance r) increases linearly. The remaining rows provide the weight contribution of each histogram bin by normalizing the weights (as discussed in Section \ref{['sec:cp_err_bound']}) and summing the normalized weights for each bin. We observe that smaller $\rho$ values place more emphasis on points closer to the test point, and the weight of the test point, represented by a point mass at $\infty$ in the empirical score distribution, increases. Lastly, in the plot, red denotes the test point, while orange highlights the bin of points closest to it.
  • ...and 6 more figures

Theorems & Definitions (8)

  • theorem 1
  • theorem 2
  • theorem 3
  • theorem 4
  • proof
  • corollary 1
  • proof
  • proof