Learning causal effects from many randomized experiments using regularized instrumental variables

Scientific and business practices are increasingly resulting in large collections of randomized experiments. Analyzed together, these collections can tell us things that individual experiments in the collection cannot. We study how to learn causal relationships between variables from such collections when the number experiments is large, many experiments have very small effects, and the analyst lacks metadata (e.g., descriptions of the interventions). Here we use experimental groups as instrumental variables (IV) and show that a standard method (two-stage least squares) is biased even when the number of experiments is infinite. We show how a sparsity-inducing $l_0$ regularization can --- in a reversal of the standard bias--variance tradeoff in regularization --- reduce bias and MSE of interventional predictions. We propose a cross-validation procedure (IVCV) to feasibly select the regularization parameter. We show, using a trick from Monte Carlo sampling, that the cross-validation can be done using summary statistics instead of raw data, thus making it simple to use in many real-world applications.