Synthetic Combinations: A Causal Inference Framework for Combinatorial Interventions

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).