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 individually have useful properties 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 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 (Table II). 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 MLP to facilitate equivalence testing from noisy stimulus-response pairs. Second, a variety of FFNN architectures having between 3 and 7 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 analysis/approximation of nonlinearities in neural networks, opening new directions and opportunities for the theoretical analysis and systematic testing of neural networks.
翻译:在分析神经网络非线性特性方面, 聚合扩张很重要。 已经应用它们来应对在核查、 解释和安全方面众所周知的困难。 现有的方法包括古典泰勒和Chebyshev方法、 杂现和许多数字方法。 我们发现, 虽然这些个别方法具有有用的属性, 如精确错误公式、 可调整域和对未定义衍生物的稳健性, 但是没有办法提供一致的方法, 使所有这些特性都能够产生扩张。 为了解决这个问题, 我们开发了经过分析修改的综合变换扩展( AMITE), 通过利用衍生的趋同标准修改整体变换。 我们展示了总体扩展, 然后展示了两种流行激活功能的应用, 即超曲相色和修正线性单位。 与目前用于此目的的扩展( 即切比谢夫、 泰勒和数字) 相比, AMITE是第一个提供六种相互排斥的扩张属性, 如精确的系数公式和精确的扩展错误( 表二)。 我们在两个案例研究中展示了AMITE的有效性。 首先, 多变的混合混合混合网络的理论分析, 和精确的亚调调调调调调调调调调调调调调调调调调调调调调的货币的货币结构, 。