a measure of how strongly the model parameters or predictions depend on a training instance.
An influence function tells you the effect of a change in one observation on an estimator. It's' useful in studying model robustness and calculating variance-covariance matrices for certain types of estimators, especially when more straightforward methods become hard to implement.
Influence functions are basically an analytical tool that can be used to assess the effect (or "influence") of removing an observation on the value of a statistic without having to re-calculate that statistic. They can also be used to create asymptotic variance estimates.
The influence function could be very useful to understand and debug deep learning models.
