Set-Valued Sensitivity Analysis of Deep Neural Networks

TL;DR

This paper introduces a set-valued sensitivity analysis framework for deep neural networks to evaluate how solution sets respond to data perturbations, enhancing understanding of model robustness without relying on non-singular Hessian assumptions.

Contribution

It develops a novel set-valued analysis approach for DNNs that accounts for multiple solutions and provides Lipschitz-like bounds on solution set changes under data perturbations.

Findings

Solution sets of fully connected neural networks exhibit Lipschitz-like properties.

Graphical-derivative-based method estimates solution set changes in ResNets without retraining.

The framework applies to isolated and non-isolated minima, broadening robustness analysis.

Abstract

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 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.