  Duality places an important constraint on the renormalization group flows and
the phase diagrams. For self-dual theories, the self-duality can be promoted as
a symmetry, this leads to the multi-criticalities. This work investigates a
description of the deconfined quantum criticality, the $N_f=2$ QED$_3$, as an
example of self-dual theories and its multi-critical behavior under
perturbative deformations. The multi-criticality is described by the theory
with Gross-Neveu couplings and falls in a different universality class than the
standard deconfined quantum criticality. We systematically calculate the
scaling dimensions of various operators in the 3d quantum electrodynamics with
the Chern-Simons term and Gross-Neveu couplings by the large-$N$
renormalization group analysis. Specifically, we find certain non-relativistic
four-fermion interactions corresponding to the dimer-dimer interactions in the
lattice model will drive the deconfined quantum criticality to the first-order
transition, consistent with previous numerical studies.
