# Safety Verification and Robustness Analysis of Neural Networks via Quadratic Constraints and Semidefinite Programming

## TL;DR

This paper introduces a semidefinite programming framework using quadratic constraints to certify neural network safety and robustness against input uncertainties and adversarial attacks, applicable to general activation functions.

## Contribution

It presents a novel SDP-based method that abstracts activation functions with quadratic constraints for safety verification of neural networks.

## Key findings

- Effective in bounding neural network outputs under input uncertainties.
- Balances conservatism and computational efficiency.
- Applicable to various neural network architectures and activation functions.

## Abstract

Certifying the safety or robustness of neural networks against input uncertainties and adversarial attacks is an emerging challenge in the area of safe machine learning and control. To provide such a guarantee, one must be able to bound the output of neural networks when their input changes within a bounded set. In this paper, we propose a semidefinite programming (SDP) framework to address this problem for feed-forward neural networks with general activation functions and input uncertainty sets. Our main idea is to abstract various properties of activation functions (e.g., monotonicity, bounded slope, bounded values, and repetition across layers) with the formalism of quadratic constraints. We then analyze the safety properties of the abstracted network via the S-procedure and semidefinite programming. Our framework spans the trade-off between conservatism and computational efficiency and applies to problems beyond safety verification. We evaluate the performance of our approach via numerical problem instances of various sizes.