Synthetic Principal Component Design: Fast Covariate Balancing with Synthetic Controls

TL;DR

This paper introduces a novel, efficient optimization algorithm for experimental design using synthetic controls, reducing the problem to phase synchronization and demonstrating superior empirical performance over random designs.

Contribution

It develops a globally convergent algorithm for experimental design via synthetic controls, with the first theoretical guarantees and practical effectiveness demonstrated on real datasets.

Findings

Algorithm surpasses random design in RMSE

Reduces experimental design to phase synchronization

Provides the first global optimality guarantee

Abstract

The optimal design of experiments typically involves solving an NP-hard combinatorial optimization problem. In this paper, we aim to develop a globally convergent and practically efficient optimization algorithm. Specifically, we consider a setting where the pre-treatment outcome data is available and the synthetic control estimator is invoked. 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 observed data. {Under this setting, we surprisingly observed that the optimal experimental design problem could be reduced to a so-called \textit{phase synchronization} problem.} We solve this problem via a normalized variant of the generalized power method with spectral initialization. On the theoretical side, we establish the first global optimality guarantee for experiment design when pre-treatment data is sampled from certain data-generating processes. Empirically, we conduct extensive experiments to demonstrate the effectiveness of our method on both the US Bureau of Labor Statistics and the Abadie-Diemond-Hainmueller California Smoking Data. In terms of the root mean square error, our algorithm surpasses the random design by a large margin.