Algorithm 1 ILSG algorithm

input: Dataset D Output: Optimal structure $G^{*}$ and its score BIC $\left(G^{*}\right)$  1: $G\leftarrow randomDAG\left(D\right)$  2: $G\leftarrow localsearch\left(G\right)$  3: $G^{*}\leftarrow emptygraph\left(D\right)$  4: while (termination condition is not met) do  5:   $G^{\prime }\leftarrow perturb\left(G\right)$  6:   $G^{\prime }\leftarrow localsearch\left(G^{\prime }\right)$  7:   $G\leftarrow compare(G,G^{\prime })$  8:   $G^{*}\leftarrow compare(G,G^{*})$  9:   if soft restart condition is met then 10:     $G\leftarrow randomDAG\left(D\right)$ 11:   end if 12: end while 13: return $G^{*}$, BIC $\left(G^{*}\right)$