Smooth, Sparse, and Stable: Finite-Time Exact Skeleton Recovery via Smoothed Proximal Gradients

TL;DR

This work addresses the gap between continuous optimization and discrete DAG structure in causal discovery by introducing the Hybrid-Order Acyclicity Constraint (AHOC) and optimizing it with Smoothed Proximal Gradient (SPG-AHOC). It proves a Finite-Time Oracle Property via Topological Locking, ensuring exact DAG support recovery in finite iterations despite smoothing. The authors provide a rigorous axiomatic analysis, local statistical guarantees under Local Restricted Strong Convexity, and demonstrate state-of-the-art performance on synthetic benchmarks and real data (Sachs) with exact sparsity. They also reveal a fundamental Stability-Sensitivity trade-off and discuss limitations under extreme geometry, offering directions for future work such as dynamic smoothing and scalability improvements. The approach combines theoretical completeness with practical effectiveness, delivering exact structural recovery without post-hoc thresholding and improving robustness and efficiency in causal graph learning.

Abstract

Continuous optimization has significantly advanced causal discovery, yet existing methods (e.g., NOTEARS) generally guarantee only asymptotic convergence to a stationary point. This often yields dense weighted matrices that require arbitrary post-hoc thresholding to recover a DAG. This gap between continuous optimization and discrete graph structures remains a fundamental challenge. In this paper, we bridge this gap by proposing the Hybrid-Order Acyclicity Constraint (AHOC) and optimizing it via the Smoothed Proximal Gradient (SPG-AHOC). Leveraging the Manifold Identification Property of proximal algorithms, we provide a rigorous theoretical guarantee: the Finite-Time Oracle Property. We prove that under standard identifiability assumptions, SPG-AHOC recovers the exact DAG support (structure) in finite iterations, even when optimizing a smoothed approximation. This result eliminates structural ambiguity, as our algorithm returns graphs with exact zero entries without heuristic truncation. Empirically, SPG-AHOC achieves state-of-the-art accuracy and strongly corroborates the finite-time identification theory.

Figures (8)

• Figure 1: Visualizing Structural Constraint Instability (SCI).Left (Red/Green): Standard constraints suffer from either vanishing gradients at the origin (Type I) or exploding gradients near the boundary (Type II). Right (Blue): Our AHOC formulation maintains a stable, non-vanishing gradient signal near zero due to the smoothed $L_1$-Synergy (Axiom 2), while preventing explosion via normalization (Axiom 3).
• Figure 2: Mechanism of Finite-Time Identification. A phase space trajectory comparison. Red (Standard Methods): Continuous optimization without proximal operators asymptotically approaches the axes ($w_1=0$) but never touches them—a phenomenon akin to Zeno's Paradox—necessitating arbitrary thresholding. Blue (SPG-AHOC): The proximal operator leverages Topological Locking, strictly "snapping" the trajectory onto the axis (the sparse manifold) in finite steps once it enters the regularization band, ensuring exact zero.
• Figure 3: Scalability Comparison (Log Scale). SPG-AHOC (Blue) demonstrates superior efficiency in the $d \le 200$ regime, achieving a 1.8x speedup over SOTA DAGMA and a 4.8x speedup over NOTEARS at $d=50$. While DAGMA is often considered faster, SPG-AHOC provides comparable or superior speeds in mid-dimensions.
• Figure 4: Visualizing the Stabilizer Effect (Exp 7). Optimization trajectory (first outer loop) vs. $\lambda_1$. Small $\lambda_1$ values (e.g., 0.1) lead to a rapid increase in $\left\lVert\mathbf{W}\right\rVert_F$, causing the optimizer to drift into non-convex regions and eventually fail. Larger $\lambda_1$ values (e.g., 1.0, 2.0) effectively constrain $\left\lVert\mathbf{W}\right\rVert_F$ near zero initially, satisfying the stability condition and enabling convergence.
• Figure 5: Experiment 1 (Type II SCI): Gradient Norm vs. Spectral Radius. Comparison of gradient behaviors as the matrix approaches a cycle ($\rho \to 1$). Standard constraints explode, whereas AHOC and AAC remain bounded (Axiom 3).

Theorems & Definitions (32)

• Definition 2.4: Standard Hadamard Constraint Class $\mathcal{H}$
• Theorem 2.5: Refined Incompatibility Theorem
• Theorem 2.6: Axiomatic Analysis of AAC
• Definition 2.7: AHOC and S-AHOC Constraints
• Theorem 2.8: Axiomatic Analysis of AHOC/S-AHOC
• Remark 2.9: Theoretical Barrier for Proximal-LS
• Remark 2.10: The Logic of Exactness: Why Smoothing Doesn't Break Sparsity
• Remark 2.11: Mechanism: Topological Locking
• Theorem 2.12: Convergence of SPG-AHOC Inner Loop
• Remark 2.13: Convergence Rate