Selective Greedy Equivalence Search: Finding Optimal Bayesian Networks Using a Polynomial Number of Score Evaluations

We introduce Selective Greedy Equivalence Search (SGES), a restricted version of Greedy Equivalence Search (GES). SGES retains the asymptotic correctness of GES but, unlike GES, has polynomial performance guarantees. In particular, we show that when data are sampled independently from a distribution that is perfect with respect to a DAG ${\cal G}$ defined over the observable variables then, in the limit of large data, SGES will identify ${\cal G}$'s equivalence class after a number of score evaluations that is (1) polynomial in the number of nodes and (2) exponential in various complexity measures including maximum-number-of-parents, maximum-clique-size, and a new measure called {\em v-width} that is at least as small as---and potentially much smaller than---the other two. More generally, we show that for any hereditary and equivalence-invariant property $\Pi$ known to hold in ${\cal G}$, we retain the large-sample optimality guarantees of GES even if we ignore any GES deletion operator during the backward phase that results in a state for which $\Pi$ does not hold in the common-descendants subgraph.