AMITE: A Novel Polynomial Expansion for Analyzing Neural Network Nonlinearities.

Abstract

Polynomial expansions are important in the analysis of neural network nonlinearities. They have been applied thereto addressing well-known difficulties in verification, explainability, and security. Existing approaches span classical Taylor and Chebyshev methods, asymptotics, and many numerical approaches. We find that, while these have useful properties individually, such as exact error formulas, adjustable domain, and robustness to undefined derivatives, there are no approaches that provide a consistent method, yielding an expansion with all these properties. To address this, we develop an analytically modified integral transform expansion (AMITE), a novel expansion via integral transforms modified using derived criteria for convergence. We show the general expansion and then demonstrate an application for two popular activation functions: hyperbolic tangent and rectified linear units. Compared with existing expansions (i.e., Chebyshev, Taylor, and numerical) employed to this end, AMITE is the first to provide six previously mutually exclusive desired expansion properties, such as exact formulas for the coefficients and exact expansion errors. We demonstrate the effectiveness of AMITE in two case studies. First, a multivariate polynomial form is efficiently extracted from a single hidden layer black-box multilayer perceptron (MLP) to facilitate equivalence testing from noisy stimulus-response pairs. Second, a variety of feedforward neural network (FFNN) architectures having between three and seven layers are range bounded using Taylor models improved by the AMITE polynomials and error formulas. AMITE presents a new dimension of expansion methods suitable for the analysis/approximation of nonlinearities in neural networks, opening new directions and opportunities for the theoretical analysis and systematic testing of neural networks.