Achieving the Tightest Relaxation of Sigmoids for Formal Verification

TL;DR

This paper introduces a novel method called α-sig that provides the tightest convex relaxation of sigmoid functions for neural network verification, improving the efficiency and accuracy of formal verification processes.

Contribution

The paper proposes a new tuneable hyperplane approach in the dual space to achieve the tightest convex relaxation of sigmoid activations for formal verification.

Findings

α-sig relaxations outperform existing methods like LiRPA and α-CROWN

Tighter bounds lead to faster verification times

Enhanced accuracy in neural network verification results

Abstract

In the field of formal verification, Neural Networks (NNs) are typically reformulated into equivalent mathematical programs which are optimized over. To overcome the inherent non-convexity of these reformulations, convex relaxations of nonlinear activation functions are typically utilized. Common relaxations (i.e., static linear cuts) of "S-shaped" activation functions, however, can be overly loose, slowing down the overall verification process. In this paper, we derive tuneable hyperplanes which upper and lower bound the sigmoid activation function. When tuned in the dual space, these affine bounds smoothly rotate around the nonlinear manifold of the sigmoid activation function. This approach, termed $\alpha$-sig, allows us to tractably incorporate the tightest possible, element-wise convex relaxation of the sigmoid activation function into a formal verification framework. We embed these relaxations inside of large verification tasks and compare their performance to LiRPA and $\alpha$-CROWN, a state-of-the-art verification duo.