There has been a growing interest in causal learning in recent years. Commonly used representations of causal structures, including Bayesian networks and structural equation models (SEM), take the form of directed acyclic graphs (DAGs). We provide a novel mixed-integer quadratic programming formulation and associated algorithm that identifies DAGs on up to 50 vertices, where these are identifiable. We call this method ExDAG, which stands for Exact learning of DAGs. Although there is a superexponential number of constraints that prevent the formation of cycles, the algorithm adds constraints violated by solutions found, rather than imposing all constraints in each continuous-valued relaxation. Our empirical results show that ExDAG outperforms local state-of-the-art solvers in terms of precision and outperforms state-of-the-art global solvers with respect to scaling, when considering Gaussian noise. We also provide validation with respect to other noise distributions.

翻译：近年来，因果学习领域引起了日益广泛的关注。因果结构的常用表示方法，包括贝叶斯网络和结构方程模型（SEM），均采用有向无环图（DAG）的形式。本文提出了一种新颖的混合整数二次规划模型及相应算法，该算法能够在可识别的情况下，对多达50个顶点的DAG进行精确学习。我们将此方法命名为ExDAG，即“有向无环图的精确学习”。尽管存在超指数级数量的约束条件用于防止环的形成，但该算法仅针对已发现解中违反的约束进行添加，而非在每次连续值松弛中施加全部约束。实验结果表明，在高斯噪声条件下，ExDAG在精度上优于当前局部最优求解器，在可扩展性方面优于当前全局最优求解器。我们还针对其他噪声分布提供了验证结果。