, the Algebraic Birkhoff Decomposition in Eq. () applies to give the renormalization $\phi_{+}$ from which the renormalized MZVs can be derived when $\vep$ goes to zero, as in the QFT case. However, there is an important difference from the QFT renormalization: in order to equip the regularized MZVs and the corresponding Hopf algebra with a suitable algebra structure that reflects the quasi-shuffle relation of the regularized MZVs, an extra parameter vector $\vec{r}$ has to be introduced in the regularized sums in addition to $\vep$. Thus the renormalized MZVs at $\vec{s}$ from the Algebraic Birkhoff Decomposition depend on $\vec{r}$, resulting in the [**renormalized directional MZVs**]{} $\zeta(\wvec{\vec{s}}{\vec{r}})$ in Definition .

This dependency on $\vec{r}$ is removed in the following Section  in a consistent manner, giving the [**renormalized MZVs**]{} $\gzeta(\vec{s})$ in Definition . Our main result Theorem  shows that the renormalized MZVs satisfy the quasi-shuffle (or stuffle) relation, and include as special cases the MZVs defined either by convergence, by analytic continuation, or by regularization in the sense of Ihara-Kaneko-Zagier . Parts of the proof are postponed to Section  and Section . Here is the hierarchy of MZVs introduced in this paper: $$\begin{aligned}
& \mbox{formal MZVs}\ \zeta(\vec{s}) \to \mbox{directional regularized MZVs}\ Z(\wvec{\vec{s}}{\vec{r}};\vep) 