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Conformalized Data-Driven Reachability Analysis with PAC Guarantees

Yanliang Huang, Zhen Zhang, Peng Xie, Zhuoqi Zeng, Amr Alanwar

Abstract

Data-driven reachability analysis computes over-approximations of reachable sets directly from noisy data. Existing deterministic methods require either known noise bounds or system-specific structural parameters such as Lipschitz constants. We propose Conformalized Data-Driven Reachability (CDDR), a framework that provides Probably Approximately Correct (PAC) coverage guarantees through the Learn Then Test (LTT) calibration procedure, requiring only that calibration trajectories be independently and identically distributed. CDDR is developed for three settings: linear time-invariant (LTI) systems with unknown process noise distributions, LTI systems with bounded measurement noise, and general nonlinear systems including non-Lipschitz dynamics. Experiments on a 5-dimensional LTI system under Gaussian and heavy-tailed Student-t noise and on a 2-dimensional non-Lipschitz system with fractional damping demonstrate that CDDR achieves valid coverage where deterministic methods do not provide formal guarantees. Under anisotropic noise, a normalized score function reduces the reachable set volume while preserving the PAC guarantee.

Conformalized Data-Driven Reachability Analysis with PAC Guarantees

Abstract

Data-driven reachability analysis computes over-approximations of reachable sets directly from noisy data. Existing deterministic methods require either known noise bounds or system-specific structural parameters such as Lipschitz constants. We propose Conformalized Data-Driven Reachability (CDDR), a framework that provides Probably Approximately Correct (PAC) coverage guarantees through the Learn Then Test (LTT) calibration procedure, requiring only that calibration trajectories be independently and identically distributed. CDDR is developed for three settings: linear time-invariant (LTI) systems with unknown process noise distributions, LTI systems with bounded measurement noise, and general nonlinear systems including non-Lipschitz dynamics. Experiments on a 5-dimensional LTI system under Gaussian and heavy-tailed Student-t noise and on a 2-dimensional non-Lipschitz system with fractional damping demonstrate that CDDR achieves valid coverage where deterministic methods do not provide formal guarantees. Under anisotropic noise, a normalized score function reduces the reachable set volume while preserving the PAC guarantee.
Paper Structure (19 sections, 2 theorems, 13 equations, 1 figure, 3 tables, 1 algorithm)

This paper contains 19 sections, 2 theorems, 13 equations, 1 figure, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Contributions
  4. Preliminaries
  5. Notations
  6. Set Representation
  7. Conformal Prediction and Learn Then Test
  8. Problem Formulation
  9. Conformalized Data-Driven Reachability
  10. LTI Systems
  11. LTI Systems with Measurement Noise
  12. General Nonlinear Systems
  13. Theoretical Guarantee
  14. Numerical Experiments
  15. Experimental Setup
  16. ...and 4 more sections

Key Result

Lemma 1

For the three system settings A, B, and C, if $x_{\mathrm{new}}(0) \in \mathcal{X}_0$, $u_{\mathrm{new}}(k) \in \mathcal{U}$ for all $k$, and $\|r_{\mathrm{new},k}\|_\infty \leq \hat{q}_k$ for all $k = 0, \ldots, N\!-\!1$, then $x_{\mathrm{new}}(k) \in \mathcal{R}^{\mathrm{CP}}_k$ for all $k = 0, \l

Figures (1)

  • Figure 1: Reachable sets for all methods. (a) 5D LTI under Gaussian noise ($x_1$-$x_2$). (b) 5D LTI under Student-$t$ noise ($x_1$-$x_2$). (c) 2D Nonlinear system with fractional damping; CDDR (per-dim) is omitted as the system is only 2-dimensional.

Theorems & Definitions (9)

  • Definition 1
  • Definition 2
  • Lemma 1: Deterministic inclusion
  • proof
  • Theorem 1: $({\alpha},{\delta})$-PAC multi-step coverage
  • proof
  • Remark 1
  • Remark 2: Bonferroni correction
  • Remark 3: Per-dimension and normalized score functions