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Conformal Prediction in The Loop: A Feedback-Based Uncertainty Model for Trajectory Optimization

Han Wang, Chao Ning

TL;DR

The paper introduces Fb-CP, a closed-loop framework that couples conformal prediction with shrinking-horizon trajectory optimization to enforce a joint time-wide safety constraint while leveraging feedback from realized trajectories. It constructs CP regions via a CP-based posterior risk calculation and adjusts prediction regions online through Lipschitz-based constraint reformulations, preserving coverage guarantees. A risk-allocation scheme, notably Iterative Risk Allocation (IRA), iteratively tightens inactive constraints and reallocates residual risk to active times, with theoretical convergence and overall safety guarantees. Empirical results across multiple robotic platforms show that Fb-CP, especially with IRA, achieves significant performance improvements over sequential CP baselines while maintaining safety; the approach also extends to handle distribution shift via weighting schemes. Overall, Fb-CP offers a principled, data-driven pathway to safer and more efficient decision-making under obstacle trajectory uncertainty.

Abstract

Conformal Prediction (CP) is a powerful statistical machine learning tool to construct uncertainty sets with coverage guarantees, which has fueled its extensive adoption in generating prediction regions for decision-making tasks, e.g., Trajectory Optimization (TO) in uncertain environments. However, existing methods predominantly employ a sequential scheme, where decisions rely unidirectionally on the prediction regions, and consequently the information from decision-making fails to be fed back to instruct CP. In this paper, we propose a novel Feedback-Based CP (Fb-CP) framework for shrinking-horizon TO with a joint risk constraint over the entire mission time. Specifically, a CP-based posterior risk calculation method is developed by fully leveraging the realized trajectories to adjust the posterior allowable risk, which is then allocated to future times to update prediction regions. In this way, the information in the realized trajectories is continuously fed back to the CP, enabling attractive feedback-based adjustments of the prediction regions and a provable online improvement in trajectory performance. Furthermore, we theoretically prove that such adjustments consistently maintain the coverage guarantees of the prediction regions, thereby ensuring provable safety. Additionally, we develop a decision-focused iterative risk allocation algorithm with theoretical convergence analysis for allocating the posterior allowable risk which closely aligns with Fb-CP. Furthermore, we extend the proposed method to handle distribution shift. The effectiveness and superiority of the proposed method are demonstrated through benchmark experiments.

Conformal Prediction in The Loop: A Feedback-Based Uncertainty Model for Trajectory Optimization

TL;DR

The paper introduces Fb-CP, a closed-loop framework that couples conformal prediction with shrinking-horizon trajectory optimization to enforce a joint time-wide safety constraint while leveraging feedback from realized trajectories. It constructs CP regions via a CP-based posterior risk calculation and adjusts prediction regions online through Lipschitz-based constraint reformulations, preserving coverage guarantees. A risk-allocation scheme, notably Iterative Risk Allocation (IRA), iteratively tightens inactive constraints and reallocates residual risk to active times, with theoretical convergence and overall safety guarantees. Empirical results across multiple robotic platforms show that Fb-CP, especially with IRA, achieves significant performance improvements over sequential CP baselines while maintaining safety; the approach also extends to handle distribution shift via weighting schemes. Overall, Fb-CP offers a principled, data-driven pathway to safer and more efficient decision-making under obstacle trajectory uncertainty.

Abstract

Conformal Prediction (CP) is a powerful statistical machine learning tool to construct uncertainty sets with coverage guarantees, which has fueled its extensive adoption in generating prediction regions for decision-making tasks, e.g., Trajectory Optimization (TO) in uncertain environments. However, existing methods predominantly employ a sequential scheme, where decisions rely unidirectionally on the prediction regions, and consequently the information from decision-making fails to be fed back to instruct CP. In this paper, we propose a novel Feedback-Based CP (Fb-CP) framework for shrinking-horizon TO with a joint risk constraint over the entire mission time. Specifically, a CP-based posterior risk calculation method is developed by fully leveraging the realized trajectories to adjust the posterior allowable risk, which is then allocated to future times to update prediction regions. In this way, the information in the realized trajectories is continuously fed back to the CP, enabling attractive feedback-based adjustments of the prediction regions and a provable online improvement in trajectory performance. Furthermore, we theoretically prove that such adjustments consistently maintain the coverage guarantees of the prediction regions, thereby ensuring provable safety. Additionally, we develop a decision-focused iterative risk allocation algorithm with theoretical convergence analysis for allocating the posterior allowable risk which closely aligns with Fb-CP. Furthermore, we extend the proposed method to handle distribution shift. The effectiveness and superiority of the proposed method are demonstrated through benchmark experiments.
Paper Structure (38 sections, 9 theorems, 71 equations, 3 figures, 14 tables, 1 algorithm)

This paper contains 38 sections, 9 theorems, 71 equations, 3 figures, 14 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related work
  3. Problem formulation and background
  4. Problem formulation
  5. Conformal prediction
  6. Feedback-based conformal prediction
  7. Constraint reformulation using conformal prediction region
  8. Posterior probability conditional on past decisions
  9. Optimization problem reformulation
  10. Shrinking-horizon trajectory optimization using Fb-CP
  11. Experiments
  12. Conclusion and Limitations
  13. Auxiliary Results
  14. Conformal Prediction Lemma
  15. Upper bound of the expectation of posterior risk
  16. ...and 23 more sections

Key Result

Lemma 4.1

(chance constraint) If Assumption assume2 holds, the constraint function $c$ is $L$-Lipschitz continuous and $c(x_{\tau},\hat{Y}_{\tau|t}) \geq LC_{\tau|t}^{1-\alpha_{\tau}}$ is satisfied where $C_{\tau|t}^{1-\alpha_{\tau}}$ is calculated by (eq_e2ecp3.2), then the individual chance constraint $\mat

Figures (3)

  • Figure 1: Shrinking-horizon trajectory optimization framework using Fb-CP.
  • Figure 2: Trajectories of the vehicle with different TO methods. (Numbers on the circles denote the indices of obstacles. The diamond and pentagon symbols represent the initial and target points of the vehicle, respectively. The translucent circles represent the planned positions of the vehicle and the prediction regions for further time. In particular, the colored and transparent circles with black edges denote the planned positions and the prediction regions for $\tau=9$, respectively.)
  • Figure 3: Left: prediction region radius for $\tau=20$ at each time $t$ ($C_{20|t}$) using the vehicle model with different methods across 1,000 simulations. Right: distributions of $C_{20|9}$.

Theorems & Definitions (11)

  • Lemma 4.1
  • Lemma 4.2
  • Lemma 5.1
  • Lemma 5.2
  • Theorem 5.3
  • Theorem 5.4
  • Remark 5.5
  • Lemma A.1
  • Corollary A.2
  • Remark A.3
  • ...and 1 more