  Causal inference is known to be very challenging when only observational data
are available. Randomized experiments are often costly and impractical and in
instrumental variable regression the number of instruments has to exceed the
number of causal predictors. It was recently shown in Peters et al. [2016] that
causal inference for the full model is possible when data from distinct
observational environments are available, exploiting that the conditional
distribution of a response variable is invariant under the correct causal
model. Two shortcomings of such an approach are the high computational effort
for large-scale data and the assumed absence of hidden confounders. Here we
show that these two shortcomings can be addressed if one is willing to make a
more restrictive assumption on the type of interventions that generate
different environments. Thereby, we look at a different notion of invariance,
namely inner-product invariance. By avoiding a computationally cumbersome
reverse-engineering approach such as in Peters et al. [2016], it allows for
large-scale causal inference in linear structural equation models. We discuss
identifiability conditions for the causal parameter and derive asymptotic
confidence intervals in the low-dimensional setting. In the case of
non-identifiability we show that the solution set of causal Dantzig has
predictive guarantees under certain interventions. We derive finite-sample
bounds in the high-dimensional setting and investigate its performance on
simulated datasets.
