  We provide an algebraic framework to describe renormalization in regularity
structures based on multi-indices for a large class of semi-linear stochastic
PDEs. This framework is ``top-down", in the sense that we postulate the form of
the counterterm and use the renormalized equation to build a canonical smooth
model for it. The core of the construction is a generalization of the Hopf
algebra of derivations in [LOT23], which is extended beyond the structure group
to describe the model equation via an exponential map: This allows to implement
a renormalization procedure which resembles the preparation map approach in our
context.
