Mathematics

Semidefinite programming

Gene

Mathematical optimization

Quadratic equation

Mathematical analysis

Artificial intelligence

Chemistry

Geometry

Computer science

Monotonic function

Quadratic programming

Bounded function

Artificial neural network

Biochemistry

Robustness (evolution)

Quadratically constrained quadratic program

IEEE Transactions on Automatic Control

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 article, 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 <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S</i> -procedure and SDP. Our framework spans the tradeoff 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.

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