Synthetic Combinations: A Causal Inference Framework for Combinatorial Interventions
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We consider a setting with 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 × 2^p causal parameters. Choosing combinations of interventions is a problem that naturally arises in many applications such as factorial design experiments, recommendation engines (e.g., showing a set of movies that maximizes engagement for users), combination therapies in medicine, selecting important features for ML models, etc. Running N × 2^p experiments to estimate the various parameters is 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 model that imposes latent structure across both units and combinations. We assume latent similarity across units (i.e., the potential outcomes matrix is rank r) and regularity in how combinations interact (i.e., the coefficients in the Fourier expansion of the potential outcomes is s sparse). We establish identification for all causal parameters despite unobserved confounding. We propose an estimation procedure, Synthetic Combinations, and establish finite-sample consistency under precise conditions on the observation pattern. Our results imply Synthetic Combinations consistently estimates unit-specific potential outcomes given poly(r) × (N + s^2p) observations. In comparison, previous methods that do not exploit structure across both units and combinations have sample complexity scaling as min(N × s^2p, r × (N + 2^p)). We use Synthetic Combinations to propose a data-efficient experimental design mechanism for combinatorial causal inference. We corroborate our theoretical findings with numerical simulations.
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