# Synthetic Combinations: A Causal Inference Framework for Combinatorial Interventions

arXiv: https://arxiv.org/abs/2303.14226 · HTML: https://mlsanity.com/paper/2303.14226/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 48 references that are in the map.
- First posted on arXiv 24 Mar 2023.

## Builds on (most cited first, 10 of 10)

- [Causal Matrix Completion](https://mlsanity.com/paper/2109.15154/) (2021)
- [Low-Rank Matrix Completion: A Contemporary Survey](https://mlsanity.com/paper/1907.11705/) (2019)
- [Missing Not at Random in Matrix Completion: The Effectiveness of Estimating Missingness Probabilities Under a Low Nuclear Norm Assumption](https://mlsanity.com/paper/1910.12774/) (2019)
- [On Robustness of Principal Component Regression](https://mlsanity.com/paper/1902.10920/) (2019)
- [Large Scale Prediction with Decision Trees](https://mlsanity.com/paper/2104.13881/) (2021)
- [Synthetic Interventions](https://mlsanity.com/paper/2006.07691/) (2020)
- [Causal Inference with Corrupted Data: Measurement Error, Missing Values, Discretization, and Differential Privacy](https://mlsanity.com/paper/2107.02780/) (2021)
- [Estimation and Inference with Trees and Forests in High Dimensions](https://mlsanity.com/paper/2007.03210/) (2020)
- [Sparse learning with CART](https://mlsanity.com/paper/2006.04266/) (2020)
- [Regression-based causal inference with factorial experiments: estimands, model specifications, and design-based properties](https://mlsanity.com/paper/2101.02400/) (2021)

## Built on by (most cited first, 12 of 12)

- [Multi-Armed Bandits with Network Interference](https://mlsanity.com/paper/2405.18621/) (2024)
- [Intervention Generalization: A View from Factor Graph Models](https://mlsanity.com/paper/2306.04027/) (2023)
- [Adaptive Principal Component Regression with Applications to Panel Data](https://mlsanity.com/paper/2307.01357/) (2023)
- [Causal Imputation for Counterfactual SCMs: Bridging Graphs and Latent Factor Models](https://mlsanity.com/paper/2402.14777/) (2024)
- [Structured Learning of Compositional Sequential Interventions](https://mlsanity.com/paper/2406.05745/) (2024)
- [A Causal Inference Framework for Data Rich Environments](https://mlsanity.com/paper/2504.01702/) (2025)
- [Doubly Robust Fusion of Many Treatments for Policy Learning](https://mlsanity.com/paper/2505.08092/) (2025)
- [Concept Modulation Models: A Unified Framework for Identifiability and Extrapolation](https://mlsanity.com/paper/2606.18509/) (2026)
- [Extrapolation Guarantees for Perturbation Modeling Under the Additive Latent Shift Assumption](https://mlsanity.com/paper/2504.18522/) (2025)
- [Probabilistic Factorial Experimental Design for Combinatorial Interventions](https://mlsanity.com/paper/2506.03363/) (2025)
- [Causal Matrix Completion under Multiple Treatments via Mixed Synthetic Nearest Neighbors](https://mlsanity.com/paper/2603.11942/) (2026)
- [Are Good Generators Good Decision-Makers? Policy Learning for General Interventions via Retargeted Counterfactual Generation](https://mlsanity.com/paper/2606.07399/) (2026)

## Abstract (arXiv)

Consider a setting where there are N heterogeneous units and p interventions. Our goal is to learn unit-specific potential outcomes for any combination of these p interventions, i.e., N × 2p causal parameters. Choosing a combination of interventions is a problem that naturally arises in a variety of applications such as factorial design experiments, recommendation engines, combination therapies in medicine, conjoint analysis, etc. Running N × 2p experiments to estimate the various parameters is likely expensive and/or infeasible as N and p grow. Further, with observational data there is likely confounding, i.e., whether or not a unit is seen under a combination is correlated with its potential outcome under that combination. To address these challenges, we propose a novel latent factor model that imposes structure across units (i.e., the matrix of potential outcomes is approximately rank r), and combinations of interventions (i.e., the coefficients in the Fourier expansion of the potential outcomes is approximately s sparse). We establish identification for all N × 2p parameters despite unobserved confounding. We propose an estimation procedure, Synthetic Combinations, and establish it is finite-sample consistent and asymptotically normal under precise conditions on the observation pattern. Our results imply consistent estimation given poly(r) × left( N + s2pright) observations, while previous methods have sample complexity scaling as min(N × s2p,   poly(r) × (N + 2p)). We use Synthetic Combinations to propose a data-efficient experimental design. Empirically, Synthetic Combinations outperforms competing approaches on a real-world dataset on movie recommendations. Lastly, we extend our analysis to do causal inference where the intervention is a permutation over p items (e.g., rankings).