Differentiable causal discovery gains finite-sample guarantees: GLM-DAGs bound error for logistic and Poisson models
Synopsis
Causal structure learning built on differentiable acyclicity constraints has lacked finite-sample theory for nonlinear models. This work sets each node's conditional distribution to a generalized linear model, searches all topological orders through one continuous objective, and proves that global minimizers select the sparsest representation at the population level and attain an estimation error rate up to log factors at the sample level. Experiments examine structure recovery accuracy and runtime on logistic, Poisson and Gaussian DAGs.
Interpretation
Differentiable causal structure learning now has finite-sample statistical guarantees for nonlinear conditional distributions, covering the widely used generalized linear models such as logistic and Poisson. Finite-sample properties were previously established mainly for linear Gaussian, linear non-Gaussian and several non-differentiable methods, while differentiable approaches based on continuous optimization, though widely used for ease of implementation, had comparatively limited explicit finite-sample theory, especially beyond linear Gaussian models. A population identifiability result shows that under minimal signal and local curvature conditions the global objective selects the minimal-edge GLM-DAG representation; the finite-sample result further gives a high-probability bound for every empirical global minimizer.
The approach writes structure learning as a single continuous objective defined over all topological orders, made of a node-wise negative log-likelihood, an exact acyclicity constraint and a sparsity penalty. The sparsity penalty is a quasi-MCP, a nonconvex penalty that approximates counting directed edges more closely than a convex L1 surrogate while remaining compatible with end-to-end differentiable optimization. The acyclicity constraint uses a smooth function built from the matrix exponential, whose trace is zero exactly when the graph is acyclic; solving follows DAGMA's penalty-increase and warm-start procedure, compared against PC, FGES, MICODAG-CD and a negative control on logistic, Poisson and Gaussian DAGs.
Perspective
The conclusions target DAG models whose conditional distributions belong to the canonical-link generalized linear family and satisfy the stated covariance and Fisher-information regularity conditions; experiments generate Erdős–Rényi and scale-free graphs on synthetic data and measure recovery by structural Hamming distance at the CPDAG level.
The finite-sample bounds hold with high probability and depend on tuning constants and a minimal-signal condition; results in the unbounded case carry additional logarithmic factors. In the Poisson setting large means can cause numerical difficulties, so experiments concentrate on smaller graph sizes.