In the algorithm Intersort, Chevalley et al. (2024) proposed a score-based method to discover the causal order of variables in a Directed Acyclic Graph (DAG) model, leveraging interventional data to outperform existing methods. However, as a score-based method over the permutahedron, Intersort is computationally expensive and non-differentiable, limiting its ability to be utilised in problems involving large-scale datasets, such as those in genomics and climate models, or to be integrated into end-to-end gradient-based learning frameworks. We address this limitation by reformulating Intersort using differentiable sorting and ranking techniques. Our approach enables scalable and differentiable optimization of causal orderings, allowing the continuous score function to be incorporated as a regularizer in downstream tasks. Empirical results demonstrate that causal discovery algorithms benefit significantly from regularizing on the causal order, underscoring the effectiveness of our method. Our work opens the door to efficiently incorporating regularization for causal order into the training of differentiable models and thereby addresses a long-standing limitation of purely associational supervised learning.

翻译：在Intersort算法中，Chevalley等人（2024）提出了一种基于评分的方法来发现有向无环图（DAG）模型中变量的因果序，该方法利用干预数据以超越现有方法。然而，作为在排列多面体上的评分方法，Intersort计算成本高昂且不可微分，这限制了其在大规模数据集（如基因组学和气候模型中的数据）问题中的应用，也无法将其集成到端到端的基于梯度的学习框架中。我们通过使用可微分的排序和排名技术重新表述Intersort来解决这一局限性。我们的方法实现了因果序的可扩展且可微分的优化，使得连续评分函数能够作为正则化项融入下游任务中。实证结果表明，因果发现算法通过正则化因果序显著受益，这印证了我们方法的有效性。我们的工作为将因果序正则化高效融入可微分模型的训练开辟了道路，从而解决了纯关联性监督学习长期存在的局限性。