  Graphical models have become a very popular tool for representing
dependencies within a large set of variables and are key for representing
causal structures. We provide results for uniform inference on high-dimensional
graphical models with the number of target parameters $d$ being possible much
larger than sample size. This is in particular important when certain features
or structures of a causal model should be recovered. Our results highlight how
in high-dimensional settings graphical models can be estimated and recovered
with modern machine learning methods in complex data sets. To construct
simultaneous confidence regions on many target parameters, sufficiently fast
estimation rates of the nuisance functions are crucial. In this context, we
establish uniform estimation rates and sparsity guarantees of the square-root
estimator in a random design under approximate sparsity conditions that might
be of independent interest for related problems in high-dimensions. We also
demonstrate in a comprehensive simulation study that our procedure has good
small sample properties.
