With the rapid adoption of machine learning systems in sensitive
applications, there is an increasing need to make black-box models explainable.
Often we want to identify an influential group of training samples in a
particular test prediction for a given machine learning model. Existing
influence functions tackle this problem by using first-order approximations of
the effect of removing a sample from the training set on model parameters. To
compute the influence of a group of training samples (rather than an individual
point) in model predictions, the change in optimal model parameters after
removing that group from the training set can be large. Thus, in such cases,
the first-order approximation can be loose. In this paper, we address this
issue and propose second-order influence functions for identifying influential
groups in test-time predictions. For linear models, across different sizes and
types of groups, we show that using the proposed second-order influence
function improves the correlation between the computed influence values and the
ground truth ones. We also show that second-order influence functions could be
used with optimization techniques to improve the selection of the most
influential group for a test-sample.

本文探讨了二阶影响函数的概念，提出使用二阶影响函数来识别测试预测中有影响的训练样本组，特别是在涉及较大的训练样本集合时，相较于现有的一阶影响函数有更为显著的提高，增强了影响值与实际值之间的相关性。同时展示了如何利用优化技术来选择最具影响力的训练样本组。