  Influence functions (IF) have been seen as a technique for explaining model
predictions through the lens of the training data. Their utility is assumed to
be in identifying training examples "responsible" for a prediction so that, for
example, correcting a prediction is possible by intervening on those examples
(removing or editing them) and retraining the model. However, recent empirical
studies have shown that the existing methods of estimating IF predict the
leave-one-out-and-retrain effect poorly.
  In order to understand the mismatch between the theoretical promise and the
practical results, we analyse five assumptions made by IF methods which are
problematic for modern-scale deep neural networks and which concern convexity,
numeric stability, training trajectory and parameter divergence. This allows us
to clarify what can be expected theoretically from IF. We show that while most
assumptions can be addressed successfully, the parameter divergence poses a
clear limitation on the predictive power of IF: influence fades over training
time even with deterministic training. We illustrate this theoretical result
with BERT and ResNet models.
  Another conclusion from the theoretical analysis is that IF are still useful
for model debugging and correcting even though some of the assumptions made in
prior work do not hold: using natural language processing and computer vision
tasks, we verify that mis-predictions can be successfully corrected by taking
only a few fine-tuning steps on influential examples.
