  Quantifying the importance of each training point to a learning task is a
fundamental problem in machine learning and the estimated importance scores
have been leveraged to guide a range of data workflows such as data
summarization and domain adaption. One simple idea is to use the leave-one-out
error of each training point to indicate its importance. Recent work has also
proposed to use the Shapley value, as it defines a unique value distribution
scheme that satisfies a set of appealing properties. However, calculating
Shapley values is often expensive, which limits its applicability in real-world
applications at scale. Multiple heuristics to improve the scalability of
calculating Shapley values have been proposed recently, with the potential risk
of compromising their utility in real-world applications.
  \textit{How well do existing data quantification methods perform on existing
workflows? How do these methods compare with each other, empirically and
theoretically? Must we sacrifice scalability for the utility in these workflows
when using these methods?} In this paper, we conduct a novel theoretical
analysis comparing the utility of different importance quantification methods,
and report extensive experimental studies on existing and proposed workflows
such as noisy label detection, watermark removal, data summarization, data
acquisition, and domain adaptation. We show that Shapley value approximation
based on a $K$NN surrogate over pre-trained feature embeddings obtains
comparable utility with existing algorithms while achieving significant
scalability improvement, often by orders of magnitude. Our theoretical analysis
also justifies its advantage over the leave-one-out error.
  The code is available at \url{https://github.com/AI-secure/Shapley-Study}.
