  Influence functions estimate the effect of removing a training point on a
model without the need to retrain. They are based on a first-order Taylor
approximation that is guaranteed to be accurate for sufficiently small changes
to the model, and so are commonly used to study the effect of individual points
in large datasets. However, we often want to study the effects of large groups
of training points, e.g., to diagnose batch effects or apportion credit between
different data sources. Removing such large groups can result in significant
changes to the model. Are influence functions still accurate in this setting?
In this paper, we find that across many different types of groups and for a
range of real-world datasets, the predicted effect (using influence functions)
of a group correlates surprisingly well with its actual effect, even if the
absolute and relative errors are large. Our theoretical analysis shows that
such strong correlation arises only under certain settings and need not hold in
general, indicating that real-world datasets have particular properties that
allow the influence approximation to be accurate.
