  We develop a new, principled algorithm for estimating the contribution of
training data points to the behavior of a deep learning model, such as a
specific prediction it makes. Our algorithm estimates the AME, a quantity that
measures the expected (average) marginal effect of adding a data point to a
subset of the training data, sampled from a given distribution. When subsets
are sampled from the uniform distribution, the AME reduces to the well-known
Shapley value. Our approach is inspired by causal inference and randomized
experiments: we sample different subsets of the training data to train multiple
submodels, and evaluate each submodel's behavior. We then use a LASSO
regression to jointly estimate the AME of each data point, based on the subset
compositions. Under sparsity assumptions ($k \ll N$ datapoints have large AME),
our estimator requires only $O(k\log N)$ randomized submodel trainings,
improving upon the best prior Shapley value estimators.
