提出新型解析函数约束，解决可微图学习中的梯度消失问题。
Analytic DAG Constraints for Differentiable DAG Learning
- 用解析函数构造新型图结构约束，避免梯度消失。
- 理论证明该类函数在求导、相加、相乘下保持封闭性。
- 实验显示性能优于现有最优方法，适合因果推断研究者。
从观测数据中恢复潜在的有向无环图（DAG）结构是一项重大挑战，主要源于DAG约束优化问题的组合性质。近期研究发现，梯度消失是可微DAG学习中的主要障碍，并提出了多种DAG约束来缓解此问题。本文通过建立解析函数与DAG约束之间的理论联系，证明集合{f(x) = c₀ + ∑_{i=1}^∞ cᵢxⁱ | ∀i > 0, cᵢ > 0; r = lim_{i→∞} cᵢ/c_{i+1} > 0}中的解析函数可有效用于构建DAG约束。进一步证明该函数集在微分、求和、乘积等运算下保持封闭性，因此可通过已有约束组合生成新约束。基于这些性质，设计了一系列DAG约束并开发了高效评估算法。在多种设置下的实验表明，本方法优于先前的最先进对比方法。代码已开源：https://github.com/zzhang1987/AnalyticDAGLearning。
原文摘要 · Abstract (English)
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.
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