  A regular partition $\mathcal{P}$ for a $3$-uniform hypergraph $H=(V,E)$
consists of a partition $V=V_1\cup \ldots \cup V_t$ and for each $ij\in
{[t]\choose 2}$, a partition $K_2[V_i,V_j]=P_{ij}^1\cup \ldots \cup
P_{ij}^{\ell}$, such that certain quasirandomness properties hold. The
complexity of $\mathcal{P}$ is the pair $(t,\ell)$. In this paper we show that
if a $3$-uniform hypergraph $H$ has $VC_2$-dimension at most $k$, then there is
a regular partition $\mathcal{P}$ for $H$ of complexity $(t,\ell)$, where
$\ell$ is bounded by a polynomial in the degree of regularity. This is a vast
improvement on the bound arising from the proof of this regularity lemma in
general, in which the bound generated for $\ell$ is of Wowzer type. This can be
seen as a higher arity analogue of the efficient regularity lemmas for graphs
and hypergraphs of bounded VC-dimension due to Alon-Fischer-Newman,
Lov\'{a}sz-Szegedy, and Fox-Pach-Suk.
