  Additive noise models (ANMs) are an important setting studied in causal
inference. Most of the existing works on ANMs assume causal sufficiency, i.e.,
there are no unobserved confounders. This paper focuses on confounded ANMs,
where a set of treatment variables and a target variable are affected by an
unobserved confounder that follows a multivariate Gaussian distribution. We
introduce a novel approach for estimating the average causal effects (ACEs) of
any subset of the treatment variables on the outcome and demonstrate that a
small set of interventional distributions is sufficient to estimate all of
them. In addition, we propose a randomized algorithm that further reduces the
number of required interventions to poly-logarithmic in the number of nodes.
Finally, we demonstrate that these interventions are also sufficient to recover
the causal structure between the observed variables. This establishes that a
poly-logarithmic number of interventions is sufficient to infer the causal
effects of any subset of treatments on the outcome in confounded ANMs with high
probability, even when the causal structure between treatments is unknown. The
simulation results indicate that our method can accurately estimate all ACEs in
the finite-sample regime. We also demonstrate the practical significance of our
algorithm by evaluating it on semi-synthetic data.
