data Shapley valuation

How much is one training example worth? Data Shapley answers this by borrowing a fairness idea from cooperative game theory: imagine the training points are players who together earn a payoff (model accuracy), and split the credit so that each point is paid its average marginal contribution across every possible coalition it could join. A point that consistently lifts performance when added gets a high value; a redundant or harmful point gets a low or negative one.

The Shapley value of point i is the average, over all subsets S not containing i, of the performance gain from adding i to S. This is the unique credit assignment satisfying efficiency, symmetry, the null-player property, and additivity. The catch is that the exact sum runs over exponentially many subsets, so in practice you estimate it: truncated Monte Carlo over random permutations, gradient-based or influence-based approximations, or the cheaper K-nearest-neighbor Shapley which admits an exact closed form. Values are defined relative to a learning algorithm and a held-out metric, so they are not an intrinsic property of the data alone.

It is used to price data, detect mislabeled or poisoned points (which earn negative value), and guide acquisition. The honest limits: values are noisy, expensive, and unstable across retraining seeds, and a point's worth depends entirely on which other data is present.