Kernel methods for long term dose response curves

Abstract

A core challenge in causal inference is how to extrapolate long term effects, of possibly continuous actions, from short term experimental data. It arises in artificial intelligence: the long term consequences of continuous actions may be of interest, yet only short term rewards may be collected in exploration. For this estimand, called the long term dose response curve, we propose a simple nonparametric estimator based on kernel ridge regression. By embedding the distribution of the short term experimental data with kernels, we derive interpretable weights for extrapolating long term effects. Our method allows actions, short term rewards, and long term rewards to be continuous in general spaces. It also allows for nonlinearity and heterogeneity in the link between short term effects and long term effects. We prove uniform consistency, with nonasymptotic error bounds reflecting the effective dimension of the data. As an application, we estimate the long term dose response curve of Project STAR, a social program which randomly assigned students to various class sizes. We extend our results to long term counterfactual distributions, proving weak convergence.