  Data valuation has found various applications in machine learning, such as
data filtering, efficient learning and incentives for data sharing. The most
popular current approach to data valuation is the Shapley value. While popular
for its various applications, Shapley value is computationally expensive even
to approximate, as it requires repeated iterations of training models on
different subsets of data. In this paper we show that the Shapley value of data
points can be approximated more efficiently by leveraging the structural
properties of machine learning problems. We derive convergence guarantees on
the accuracy of the approximate Shapley value for different learning settings
including Stochastic Gradient Descent with convex and non-convex loss
functions. Our analysis suggests that in fact models trained on small subsets
are more important in the context of data valuation. Based on this idea, we
describe $\delta$-Shapley -- a strategy of only using small subsets for the
approximation. Experiments show that this approach preserves approximate value
and rank of data, while achieving speedup of up to 9.9x. In pre-trained
networks the approach is found to bring more efficiency in terms of accurate
evaluation using small subsets.
