Lyapunov Exponents on Neural Networks as a Measure for Weight Sensitivity

Weight vulnerability is a major concern for safety and stability of modern machine learning structures such as Large Language Model (LLM) and Deep Neural Networks (DNN) . In this study, we explore weight specific weight sensitivity using Multilayer Perceptron (MLP) as a toy model. Despite various measures of weight sensitivity have been defined previously, most of the measures and analytical formula derived are not weight-specific. However, modern attacks such as Jailbreaking bit-flips or weight scaling on trained LLM or DNN are weight-specific. This study aims to define weight-specific weight sensitivity measures that quantify the sensitivity of neural network output to weight perturbations. This goal is achieved by treating MLP as a dynamical system and considering Lyapunov exponents, which is a stability measure of how fast a perturbation in the initial conditions of a dynamical system propagates through time. In this study, Lyapunov Exponent of a MLP is defined by treating MLP as a discrete dynamical system in two ways : (1) treating each layer of MLP as a vector on a trajectory at one time step; (2) treating the entire feedforward MLP as a discrete Map.