The bulk locality in the constructive holographic renormalization group
requires miraculous cancellations among various local renormalization group
functions. The cancellation is not only from the properties of the spectrum but
from more detailed aspects of operator product expansions in relation to
conformal anomaly. It is remarkable that one-loop computation of the universal
local renormalization group functions in the weakly coupled limit of the
$\mathcal{N}=4$ super Yang-Mills theory fulfils the necessary condition for the
cancellation in the strongly coupled limit in its $SL(2,\mathbf{Z})$ duality
invariant form. From the consistency between the quantum renormalization group
and the holographic renormalization group, we determine some unexplored local
renormalization group functions (e.g. diffusive term in the beta function for
the gauge coupling constant) in the strongly coupled limit of the planar
$\mathcal{N}=4$ super Yang-Mills theory.