How Many and Which Training Points Would Need to be Removed to Flip this Prediction?

We consider the problem of identifying a minimal subset of training data St such that if the instances comprising St had been removed prior to training, the categorization of a given test point xt would have been different. Identifying such a set may be of interest for a few reasons. First, the cardinality of St provides a measure of robustness (if |St| is small for xt, we might be less confident in the corresponding prediction), which we show is correlated with but complementary to predicted probabilities. Second, interrogation of St may provide a novel mechanism for contesting a particular model prediction: If one can make the case that the points in St are wrongly labeled or irrelevant, this may argue for overturning the associated prediction. Identifying St via brute-force is intractable. We propose comparatively fast approximation methods to find St based on influence functions, and find that -- for simple convex text classification models -- these approaches can often successfully identify relatively small sets of training examples which, if removed, would flip the prediction.