  Capturing the underlying structural causal relations represented by Directed
Acyclic Graphs (DAGs) has been a fundamental task in various AI disciplines.
Causal DAG learning via the continuous optimization framework has recently
achieved promising performance in terms of both accuracy and efficiency.
However, most methods make strong assumptions of homoscedastic noise, i.e.,
exogenous noises have equal variances across variables, observations, or even
both. The noises in real data usually violate both assumptions due to the
biases introduced by different data collection processes. To address the issue
of heteroscedastic noise, we introduce relaxed and implementable sufficient
conditions, proving the identifiability of a general class of SEM subject to
these conditions. Based on the identifiable general SEM, we propose a novel
formulation for DAG learning that accounts for the variation in noise variance
across variables and observations. We then propose an effective two-phase
iterative DAG learning algorithm to address the increasing optimization
difficulties and to learn a causal DAG from data with heteroscedastic variable
noise under varying variance. We show significant empirical gains of the
proposed approaches over state-of-the-art methods on both synthetic data and
real data.
