Synthetic Principle Component Design: Fast Covariate Balancing with Synthetic Controls

In this paper, we target at developing a globally convergent and yet practically tractable optimization algorithm for the optimal experimental design problem with synthetic controls. Specifically, we consider a setting when the pre-treatment outcome data is available. the average treatment effect is estimated via the difference between the weighted average outcomes of the treated and control units, where the weights are learned from the data observed during the pre-treatment periods. We find that if the experimenter has the ability to select an optimal set of non-negative weights, the optimal experimental design problem is identical to to a so-called \textit{phase synchronization} problem. We solve this problem via a normalized variate of the generalized power method with spectral initialization. On the theoretical side, we establish the first global optimality guarantee for experiment design under a realizable assumption with linear fixed-effect models (also referred to an "interactive fixed-effect model"). These results are surprising, given that the optimal design of experiments, especially involving covariate matching, typically involves solving an NP-hard combinatorial optimization problem. Empirically, we apply our algorithm on US Bureau of Labor Statistics and the Abadie-Diemond-Hainmueller California Smoking Data. The experiments demonstrate that our algorithm surpasses the random design with a large margin in terms of the root mean square error.