  Learning the underlying Bayesian Networks (BNs), represented by directed
acyclic graphs (DAGs), of the concerned events from purely-observational data
is a crucial part of evidential reasoning. This task remains challenging due to
the large and discrete search space. A recent flurry of developments followed
NOTEARS[1] recast this combinatorial problem into a continuous optimization
problem by leveraging an algebraic equality characterization of acyclicity.
However, the continuous optimization methods suffer from obtaining non-spare
graphs after the numerical optimization, which leads to the inflexibility to
rule out the potentially cycle-inducing edges or false discovery edges with
small values. To address this issue, in this paper, we develop a completely
data-driven DAG structure learning method without a predefined value to
post-threshold small values. We name our method NOTEARS with adaptive Lasso
(NOTEARS-AL), which is achieved by applying the adaptive penalty method to
ensure the sparsity of the estimated DAG. Moreover, we show that NOTEARS-AL
also inherits the oracle properties under some specific conditions. Extensive
experiments on both synthetic and a real-world dataset demonstrate that our
method consistently outperforms NOTEARS.
