Fast Estimation of Shapley Value by Stratified Sampling and Its Application in Explaining Fault Diagnosis Neural Network

Abstract

There are two problems when the Shapley value is employed to interpret deep neural networks. The first issue is that the computational complexity increases exponentially with the number of players. The other issue is that the contribution evaluation index cannot effectively reflect the nonlinearity of the classification function (i.e., SoftMax), which is often neglected in previous studies. To address these challenges, a method for fast estimating the Shapley value based on the stratified sampling and the Mann–Whitney test (SSMW-Shap) is proposed in this work. In SSMW-Shap, a new contribution index is designed to accurately measure the contribution of each player by leveraging the distance between the outputs of two specific neurons, accounting for the nonlinearity of SoftMax and the efficiency of the Shapley value. Based on the proposed index, a simplified two-player coalition evaluation method is built to select important affiliates for each player, significantly reducing the computational complexity of the Shapley value. Then, the Shapley value is fast estimated by combining the stratified sampling and the Mann–Whitney test. In this process, the Mann–Whitney test is employed to estimate the difference between the samples and the population, and sample expansion is executed for the failed test, improving the estimation accuracy. Finally, a simple but reasonable method based on the proposed index is designed to quantitatively evaluate the explanation accuracy of each method. The proposed method is verified using two classic classification networks trained on two bearing datasets.