  We propose a new derivation from the full Loop Quantum Gravity (LQG) to the
Loop Quantum Cosmology (LQC) improved $\bar{\mu}$-scheme effective dynamics,
based on the reduced phase space formulation of LQG and a proposal of effective
Hamiltonian/action in the full LQG. A key step of our program is an improved
regularization of the full LQG Hamiltonian on a cubic lattice. The improved
Hamiltonian uses a set of "dressed holonomies" $h_\Delta(\mathfrak{s})$ which
not only depend on the connection $A$ but also depend on the length of the
curve $\mathfrak{s}$. With the improved Hamiltonian, we propose a quantum
effective action and derive a new set effective equations of motion (EOMs) for
the full LQG. Then we show that these new EOMs imply the $\bar{\mu}$-scheme
effective dynamics for both the homogeneous-isotropic and Bianchi-I cosmology,
and predict bounce and Planckian critical density. As a byproduct, although the
model is defined on a cubic lattice, we find that the improved effective
Hamiltonian of cosmology is invariant under the lattice refinement. The
cosmological effective dynamics, predictions of bounce and critical density are
results at the continuum limit.
