We consider the problem of identifying a \textbackslash{}emph\{minimal subset\} of training data \$\textbackslash{}mathcal\{S\}\_t\$ such that if the instances comprising \$\textbackslash{}mathcal\{S\}\_t\$ had been removed prior to training, the categorization of a given test point \$x\_t\$ would have been different.
Identifying such a set may be of interest for a few reasons.
First, the cardinality of \$\textbackslash{}mathcal\{S\}\_t\$ provides a measure of robustness (if \$|\textbackslash{}mathcal\{S\}\_t|\$ is small for \$x\_t\$, we might be less confident in the corresponding prediction), which we show is correlated with but complementary to predicted probabilities.
Second, interrogation of \$\textbackslash{}mathcal\{S\}\_t\$ may provide a novel mechanism for \textbackslash{}emph\{contesting\} a particular model prediction: If one can make the case that the points in \$\textbackslash{}mathcal\{S\}\_t\$ are wrongly labeled or irrelevant, this may argue for overturning the associated prediction. 
Identifying \$\textbackslash{}mathcal\{S\}\_t\$ via brute-force is intractable.
We propose comparatively fast approximation methods to find \$\textbackslash{}mathcal\{S\}\_t\$ based on \textbackslash{}emph\{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.

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

Jinghan Yang

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