Verifying Safety of High-Dimensional Neural Network Controllers through Statistical Approximation and Formal Reachability Analysis

Core Concepts

This paper proposes a method to verify the safety of high-dimensional neural network controllers by approximating them with multiple low-dimensional controllers and combining formal reachability analysis with statistical inference to provide safety guarantees.

Abstract

The paper addresses the challenge of verifying the safety of high-dimensional neural network controllers (HDCs), which are increasingly used in safety-critical autonomous systems. Due to the high-dimensional nature of the input space, modern verification tools cannot be directly applied to HDCs. The key idea is to approximate the HDC with several low-dimensional controllers (LDCs) and then perform reachability analysis on the LDCs. To balance the approximation accuracy and verifiability of the LDCs, the authors leverage verification-aware knowledge distillation. The paper develops two approaches to quantify the discrepancy between the HDC and LDCs: Trajectory-based discrepancy: An upper bound on the maximum L1 distance between trajectories of the HDC and LDCs. Action-based discrepancy: An upper bound on the difference between the control actions produced by the HDC and LDCs. These discrepancies are computed using conformal prediction, a distribution-free statistical technique, to provide probabilistic guarantees on the safety of the HDC-controlled system. The authors implement and evaluate their methods on two OpenAI Gym benchmarks: inverted pendulum and mountain car. The results show that the multi-LDC approaches outperform the single-LDC approaches and the pure conformal prediction baseline in terms of safety verification performance.

Stats

The initial state space for the inverted pendulum is [0, 2] × [-2, 0]. The initial state space for the mountain car is a grid with position step 0.01 and velocity step 0.001. In the inverted pendulum case, the truly safe-to-unsafe ratio is 0.56, and in the mountain car case, it is 0.78.

Quotes

"Our key insight is that the behavior of a high-dimensional controller can be approximated with several low-dimensional controllers in different regions of the state space." "We leverage the latest verification-aware knowledge distillation to balance the approximation accuracy and verifiability of our low-dimensional controllers." "If low-dimensional reachability results are inflated with statistical approximation errors, they yield a high-confidence reachability guarantee for the high-dimensional controller."

Key Insights Distilled From

Bridging Dimensions: Confident Reachability for High-Dimensional Controllers

by Yuang Geng,J... at arxiv.org 05-02-2024

https://arxiv.org/pdf/2311.04843.pdf

Deeper Inquiries

How can the state-space representation be further improved to capture the uncertainty in the HDC-LDC discrepancy more effectively

To improve the state-space representation and capture the uncertainty in the HDC-LDC discrepancy more effectively, several strategies can be implemented: Dynamic State Partitioning: Instead of using a fixed grid for the initial state space, dynamic partitioning based on the discrepancy values can be employed. Regions with high discrepancies can be further subdivided to better capture the variations in the discrepancy across the state space. Adaptive Sampling: Instead of uniform sampling, adaptive sampling techniques can be utilized to focus more on regions where the HDC-LDC discrepancy is high. This targeted sampling approach can provide more accurate estimates of the discrepancies in critical areas. Incorporating Model Uncertainty: Including a measure of uncertainty in the discrepancy calculations can provide a more nuanced understanding of the HDC-LDC differences. Bayesian methods or ensemble techniques can be used to quantify and incorporate model uncertainty into the discrepancy analysis. Hierarchical State Representations: Introducing hierarchical representations of the state space can help capture the multi-scale nature of the HDC-LDC discrepancies. By organizing the state space into hierarchical levels, the discrepancies can be analyzed at different levels of granularity. By implementing these strategies, the state-space representation can be enhanced to better capture the uncertainty in the HDC-LDC discrepancy and provide more robust verification results.

What are the potential limitations of the conformal prediction approach in handling time-series dependencies in the action-based discrepancy

The conformal prediction approach, while effective in providing uncertainty estimates without strong assumptions, may face limitations in handling time-series dependencies in the action-based discrepancy. Some potential limitations include: Assumption of Independence: Conformal prediction typically assumes independence among the data points, which may not hold true for time-series data where observations are correlated. This assumption can lead to inaccurate uncertainty estimates in the presence of temporal dependencies. Sequential Structure: Time-series data has a sequential structure where the order of observations matters. Conformal prediction may not fully capture the sequential dependencies and may overlook important patterns or trends in the data that impact the action-based discrepancy. Complex Dynamics: Time-series data often exhibits complex dynamics and non-linear relationships that may not be fully captured by conformal prediction models. The discrepancy in action-based predictions may involve intricate temporal patterns that require more sophisticated modeling techniques. Modeling Long-Term Dependencies: Conformal prediction may struggle to model long-term dependencies in time-series data, especially in the context of action-based discrepancies where the impact of past actions on future outcomes is crucial. To address these limitations, advanced time-series modeling techniques, such as recurrent neural networks (RNNs) or attention mechanisms, can be explored to better capture the temporal dependencies in the action-based discrepancy analysis.

Can the proposed framework be extended to handle other types of high-dimensional inputs beyond images, such as point clouds or text

The proposed framework can be extended to handle other types of high-dimensional inputs beyond images, such as point clouds or text, by adapting the verification process to suit the characteristics of these input modalities. Here are some considerations for extending the framework: Feature Extraction: For point clouds, techniques like voxelization or point cloud segmentation can be used to extract relevant features for the controllers. Similarly, for text inputs, natural language processing (NLP) methods can be employed to convert text data into a format suitable for neural network controllers. Model Architecture: The neural network architecture of the controllers may need to be modified to accommodate the specific characteristics of point cloud or text data. For point clouds, methods like PointNet can be utilized, while for text data, recurrent neural networks (RNNs) or transformer models may be more suitable. Discrepancy Calculation: The discrepancy bounds between HDCs and LDCs for point cloud or text inputs may require different calculation methods compared to image data. Techniques tailored to the unique properties of these data types should be developed for accurate discrepancy estimation. Verification Strategy: The verification process may need to be adjusted to account for the intricacies of point cloud or text data. Considerations such as spatial relationships in point clouds or semantic meanings in text inputs should be incorporated into the verification framework. By customizing the framework to handle diverse high-dimensional inputs, such as point clouds or text, the approach can be applied to a wider range of autonomous systems with varying input modalities.