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Distributionally Robust Statistical Verification with Imprecise Neural Networks

Souradeep Dutta, Michele Caprio, Vivian Lin, Matthew Cleaveland, Kuk Jin Jang, Ivan Ruchkin, Oleg Sokolsky, Insup Lee

TL;DR

The paper tackles the challenge of providing guarantees for high-dimensional autonomous systems under distributional uncertainty. It introduces distributionally robust statistical verification using Imprecise Neural Networks (INN) to quantify uncertainty and a Sherlock-based active-learning loop to explore input regions efficiently. By constructing an α-contaminated distribution family 𝒫 and deriving a worst-case lower bound Φ^l, the authors prove guarantees that hold across all distributions in 𝒫 with probability at least 1 − 1/λ. Empirically, the approach scales to complex Mujoco control tasks and exhibits robustness to distribution shift, outperforming standard conformal prediction in shifted settings. The framework offers a practical, provable pathway for pre-deployment validation and online monitoring of high-dimensional autonomous systems.

Abstract

A particularly challenging problem in AI safety is providing guarantees on the behavior of high-dimensional autonomous systems. Verification approaches centered around reachability analysis fail to scale, and purely statistical approaches are constrained by the distributional assumptions about the sampling process. Instead, we pose a distributionally robust version of the statistical verification problem for black-box systems, where our performance guarantees hold over a large family of distributions. This paper proposes a novel approach based on uncertainty quantification using concepts from imprecise probabilities. A central piece of our approach is an ensemble technique called Imprecise Neural Networks, which provides the uncertainty quantification. Additionally, we solve the allied problem of exploring the input set using active learning. The active learning uses an exhaustive neural-network verification tool Sherlock to collect samples. An evaluation on multiple physical simulators in the openAI gym Mujoco environments with reinforcement-learned controllers demonstrates that our approach can provide useful and scalable guarantees for high-dimensional systems.

Distributionally Robust Statistical Verification with Imprecise Neural Networks

TL;DR

The paper tackles the challenge of providing guarantees for high-dimensional autonomous systems under distributional uncertainty. It introduces distributionally robust statistical verification using Imprecise Neural Networks (INN) to quantify uncertainty and a Sherlock-based active-learning loop to explore input regions efficiently. By constructing an α-contaminated distribution family 𝒫 and deriving a worst-case lower bound Φ^l, the authors prove guarantees that hold across all distributions in 𝒫 with probability at least 1 − 1/λ. Empirically, the approach scales to complex Mujoco control tasks and exhibits robustness to distribution shift, outperforming standard conformal prediction in shifted settings. The framework offers a practical, provable pathway for pre-deployment validation and online monitoring of high-dimensional autonomous systems.

Abstract

A particularly challenging problem in AI safety is providing guarantees on the behavior of high-dimensional autonomous systems. Verification approaches centered around reachability analysis fail to scale, and purely statistical approaches are constrained by the distributional assumptions about the sampling process. Instead, we pose a distributionally robust version of the statistical verification problem for black-box systems, where our performance guarantees hold over a large family of distributions. This paper proposes a novel approach based on uncertainty quantification using concepts from imprecise probabilities. A central piece of our approach is an ensemble technique called Imprecise Neural Networks, which provides the uncertainty quantification. Additionally, we solve the allied problem of exploring the input set using active learning. The active learning uses an exhaustive neural-network verification tool Sherlock to collect samples. An evaluation on multiple physical simulators in the openAI gym Mujoco environments with reinforcement-learned controllers demonstrates that our approach can provide useful and scalable guarantees for high-dimensional systems.
Paper Structure (18 sections, 3 theorems, 5 equations, 2 figures, 8 tables, 2 algorithms)

This paper contains 18 sections, 3 theorems, 5 equations, 2 figures, 8 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem and Approach
  3. Problem Formulation
  4. Overall Approach
  5. Related Work
  6. Statistical guarantees for autonomy
  7. Uncertainty quantification and active learning
  8. Distributionally Robust Optimization
  9. Imprecision in Neural Networks
  10. Methodology
  11. Preliminaries
  12. Algorithms
  13. Theory
  14. INN Uncertainty
  15. Experimental Evaluation
  16. ...and 3 more sections

Key Result

theorem 1

Pick any $\lambda>0$ and any pair $(x,y)$ sampled from a distribution in $\mathcal{P}$. Let $A=\{\underline{\Phi}(x)-\lambda\beta \leq y \leq \overline{\Phi}(x)+\lambda\beta\}$. If $\mathcal{P}_\mathcal{X}$ and $\mathcal{P}_{\mathcal{Y}\mid \mathcal{X}}$ are compact, thenHere and in the rest of the

Figures (2)

  • Figure 1: Here $\mathcal{X}_0$ is explored across several iterations by employing an active learning strategy based on Imprecise Neural Networks (INN). At each iteration, given the current set of explored points (white dots) $\mathcal{X}_e \in \mathcal{X}_0$, an INN model is trained, which outputs the network uncertainty $\mathcal{U}(x)$ over $\mathcal{X}_o$. Using Sherlock dutta_output_2018, a sample that maximizes the uncertainty, $x^*$, is obtained. Next, a local $\delta$-ball region, $X^*$, is sampled uniformly to be explored. The INN model is updated by including the newly explored samples. Finally, at the end of all iterations INN model (after M iterations) and $\mathcal{X}_e$ are used to compute the lower bound of performance $\Phi^l$.
  • Figure 2: Examples from the Mujoco OpenAI control suite.

Theorems & Definitions (5)

  • definition 1: Imprecise Neural Network
  • definition 2: Network Uncertainty
  • theorem 1
  • corollary 1
  • corollary 2