  This paper investigates in which cases continuous optimization for directed
acyclic graph (DAG) structure learning can and cannot perform well and why this
happens, and suggests possible directions to make the search procedure more
reliable. Reisach et al. (2021) suggested that the remarkable performance of
several continuous structure learning approaches is primarily driven by a high
agreement between the order of increasing marginal variances and the
topological order, and demonstrated that these approaches do not perform well
after data standardization. We analyze this phenomenon for continuous
approaches assuming equal and non-equal noise variances, and show that the
statement may not hold in either case by providing counterexamples,
justifications, and possible alternative explanations. We further demonstrate
that nonconvexity may be a main concern especially for the non-equal noise
variances formulation, while recent advances in continuous structure learning
fail to achieve improvement in this case. Our findings suggest that future
works should take into account the non-equal noise variances formulation to
handle more general settings and for a more comprehensive empirical evaluation.
Lastly, we provide insights into other aspects of the search procedure,
including thresholding and sparsity, and show that they play an important role
in the final solutions.
