Analytic DAG Constraints for Differentiable DAG Learning
Zhen Zhang Ignavier Ng Dong Gong Yuhang Liu Mingming Gong Biwei Huang Kun Zhang Anton van den Hengel Javen Qinfeng Shi
Recovering the underlying Directed Acyclic Graph (DAG)
  structures from observational data presents a formidable challenge, partly due
  to the combinatorial nature of the DAG-constrained optimization
  problem. Recently, researchers have identified gradient vanishing as
  one of the primary obstacles in differentiable DAG learning and have
  proposed several DAG constraints to mitigate this issue. By developing
  the necessary theory to establish a connection between analytic
  functions and DAG constraints, we demonstrate that analytic functions
  from the set $\\{f(x) = c_0 + \\sum_{i=1}^{\infty}c_ix^i | \\forall i > 0, c_i > 0; r = \\lim_{i\\rightarrow \\infty}c_{i}/c_{i+1} > 0\\}$ can be employed to
  formulate effective DAG constraints. Furthermore, we establish that
  this set of functions is closed under several functional operators,
  including differentiation, summation, and
  multiplication. Consequently, these operators can be leveraged to
  create novel DAG constraints based on existing ones. Using these
  properties, we design a series of DAG constraints and develop an
  efficient algorithm to evaluate them. Experiments
   in various settings demonstrate that our DAG constraints
  outperform previous state-of-the-art comparators. Our implementation is available at https://github.com/zzhang1987/AnalyticDAGLearning.
DAG, Causal Discovery, Structural Learning