  We show that the perturbative expansion of general gauge theories can be
expressed in terms of gauge invariant variables to all orders in perturbations.
In this we generalize techniques developed in gauge invariant cosmological
perturbation theory, using Bardeen variables, by interpreting the passing over
to gauge invariant fields as a homotopy transfer of the strongly homotopy Lie
algebras encoding the gauge theory. This is illustrated for Yang-Mills theory,
gravity on flat and cosmological backgrounds and for the massless sector of
closed string theory. The perturbation lemma yields an algorithmic procedure to
determine the higher corrections of the gauge invariant variables and the
action in terms of these.
