Learning causal structure from observational data has attracted much attention, and it is notoriously challenging to find the underlying structure in the presence of confounders (hidden direct common causes of two variables).

CelebA is a well-known large-scale face dataset. Same as previous works [41, 58], we employ this dataset to predict the color of the human hair as "blond" or "not blond".

However, some recent studies haverecognized that most ofthese approaches failtoimprovethe performance over empirical risk minimization especially when applied to overparameterized neural networks.

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We thank all the reviewers for their insightful and encouraging comments. Per your suggestion, we will update the appendix by adding more explanations about the proof ideas. Similarly, we can extend convergence results in Theorem 4 in Appendix from FS to IFS. We expect that the approach developed in this paper will fuel this future investigation. We will update this into the revision.

Interest in this problem derives from its applications to Bayesian inference and differential privacy.