# Synthetic Principal Component Design: Fast Covariate Balancing with Synthetic Controls

arXiv: https://arxiv.org/abs/2211.15241 · HTML: https://mlsanity.com/paper/2211.15241/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 58 references that are in the map.
- First posted on arXiv 28 Nov 2022.

## Builds on (most cited first, 24 of 24)

- [An Introduction to Matrix Concentration Inequalities](https://mlsanity.com/paper/1501.01571/) (2015)
- [Phase Retrieval via Wirtinger Flow: Theory and Algorithms](https://mlsanity.com/paper/1407.1065/) (2014)
- [Spectral Methods for Data Science: A Statistical Perspective](https://mlsanity.com/paper/2012.08496/) (2020)
- [Solving Random Quadratic Systems of Equations Is Nearly as Easy as Solving Linear Systems](https://mlsanity.com/paper/1505.05114/) (2015)
- [Synthetic Difference in Differences](https://mlsanity.com/paper/1812.09970/) (2018)
- [Covariate Balancing Propensity Score by Tailored Loss Functions](https://mlsanity.com/paper/1601.05890/) (2016)
- [Near-optimal bounds for phase synchronization](https://mlsanity.com/paper/1703.06605/) (2017)
- [Nonconvex phase synchronization](https://mlsanity.com/paper/1601.06114/) (2016)
- [Optimization Techniques on Riemannian Manifolds](https://mlsanity.com/paper/1407.5965/) (2014)
- [Robust Synthetic Control](https://mlsanity.com/paper/1711.06940/) (2017)
- [Tightness of the maximum likelihood semidefinite relaxation for angular synchronization](https://mlsanity.com/paper/1411.3272/) (2014)
- [Riemannian preconditioning](https://mlsanity.com/paper/1405.6055/) (2014)
- [On the Estimation Performance and Convergence Rate of the Generalized Power Method for Phase Synchronization](https://mlsanity.com/paper/1603.00211/) (2016)
- [Sync-Rank: Robust Ranking, Constrained Ranking and Rank Aggregation via Eigenvector and Semidefinite Programming Synchronization](https://mlsanity.com/paper/1504.01070/) (2015)
- [Balancing Covariates in Randomized Experiments with the Gram-Schmidt Walk Design](https://mlsanity.com/paper/1911.03071/) (2019)
- [On the Assumptions of Synthetic Control Methods](https://mlsanity.com/paper/2112.05671/) (2021)
- [The Gram-Schmidt Walk: A Cure for the Banaszczyk Blues](https://mlsanity.com/paper/1708.01079/) (2017)
- [Improvements on SCORE, Especially for Weak Signals](https://mlsanity.com/paper/1811.05927/) (2018)
- [Non-Convex Exact Community Recovery in Stochastic Block Model](https://mlsanity.com/paper/2006.15843/) (2020)
- [Synthetic Controls for Experimental Design](https://mlsanity.com/paper/2108.02196/) (2021)
- [Synthetic Design: An Optimization Approach to Experimental Design with Synthetic Controls](https://mlsanity.com/paper/2112.00278/) (2021)
- [An Algorithm for Koml'os Conjecture Matching Banaszczyk's bound](https://mlsanity.com/paper/1605.02882/) (2016)
- [Linear Convergence of a Proximal Alternating Minimization Method with Extrapolation for ℓ1-Norm Principal Component Analysis](https://mlsanity.com/paper/2107.07107/) (2021)
- [On the Optimality of Randomization in Experimental Design: How to Randomize for Minimax Variance and Design-Based Inference](https://mlsanity.com/paper/2005.03151/) (2020)

## Built on by (most cited first, 1 of 1)

- [Spectral Truncation in Synthetic Control](https://mlsanity.com/paper/2607.25074/) (2026)

## Abstract (arXiv)

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 textitphase 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.