  The ability to answer causal questions is crucial in many domains, as causal
inference allows one to understand the impact of interventions. In many
applications, only a single intervention is possible at a given time. However,
in some important areas, multiple interventions are concurrently applied.
Disentangling the effects of single interventions from jointly applied
interventions is a challenging task -- especially as simultaneously applied
interventions can interact. This problem is made harder still by unobserved
confounders, which influence both treatments and outcome. We address this
challenge by aiming to learn the effect of a single-intervention from both
observational data and sets of interventions. We prove that this is not
generally possible, but provide identification proofs demonstrating that it can
be achieved under non-linear continuous structural causal models with additive,
multivariate Gaussian noise -- even when unobserved confounders are present.
Importantly, we show how to incorporate observed covariates and learn
heterogeneous treatment effects. Based on the identifiability proofs, we
provide an algorithm that learns the causal model parameters by pooling data
from different regimes and jointly maximizing the combined likelihood. The
effectiveness of our method is empirically demonstrated on both synthetic and
real-world data.
