  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.
