Set-Valued Sensitivity Analysis of Deep Neural Networks

This paper proposes a sensitivity analysis framework based on set-valued mapping for deep neural networks (DNN) to understand and compute how the solutions (model weights) of DNN respond to perturbations in the training data. As a DNN may not exhibit a unique solution (minima) and the algorithm of solving a DNN may lead to different solutions with minor perturbations to input data, we focus on the sensitivity of the solution set of DNN, instead of studying a single solution. In particular, we are interested in the expansion and contraction of the solution set in response to data perturbations. If the change of solution set can be bounded by the extent of the data perturbation, the model is said to exhibit the Lipschitz-like property. This 'set-to-set' analysis approach provides a deeper understanding of the robustness and reliability of DNNs during training. Our framework incorporates both isolated and non-isolated minima, and critically, does not require the assumption that the Hessian of loss function is non-singular. By developing set-level metrics such as distance between sets, convergence of sets, derivatives of set-valued mapping, and stability across the solution set, we prove that the solution set of the Fully Connected Neural Network holds Lipschitz-like properties. For general neural networks (e.g. Resnet), we introduce a graphical-derivative-based method to estimate the new solution set following data perturbation without retraining.