Given a data set $mathcal{D}$ containing millions of data points and a data consumer who is willing to pay for $$X$ to train a machine learning (ML) model over $mathcal{D}$, how should we distribute this $$X$ to each data point to reflect its value? In this paper, we define the relative value of data via the Shapley value, as it uniquely possesses properties with appealing real-world interpretations, such as fairness, rationality and decentralizability. For general, bounded utility functions, the Shapley value is known to be challenging to compute: to get Shapley values for all $N$ data points, it requires $O(2^N)$ model evaluations for exact computation and $O(Nlog N)$ for $(epsilon, delta)$-approximation. In this paper, we focus on one popular family of ML models relying on $K$-nearest neighbors ($K$NN). The most surprising result is that for unweighted $K$NN classifiers and regressors, the Shapley value of all $N$ data points can be computed, exactly, in $O(Nlog N)$ time -- an exponential improvement on computational complexity! Moreover, for $(epsilon, delta)$-approximation, we are able to develop an algorithm based on Locality Sensitive Hashing (LSH) with only sublinear complexity $O(N^{h(epsilon,K)}log N)$ when $epsilon$ is not too small and $K$ is not too large. We empirically evaluate our algorithms on up to $10$ million data points and even our exact algorithm is up to three orders of magnitude faster than the baseline approximation algorithm. The LSH-based approximation algorithm can accelerate the value calculation process even further. We then extend our algorithms to other scenarios such as (1) weighed $K$NN classifiers, (2) different data points are clustered by different data curators, and (3) there are data analysts providing computation who also requires proper valuation.
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