Learning Temporal Causal Structure via Smooth Differentiable Optimization

2026-06-02

AI Summary

This work addresses the challenge of causal discovery in multivariate time series involving instantaneous effects by proposing an end-to-end differentiable method that embeds acyclicity constraints directly into the model parameterization. The approach leverages a Gumbel–Sinkhorn differentiable sorting operator to learn a topological ordering of variables and enforces triangularity on the instantaneous coefficient matrix, thereby transforming the hard acyclicity constraint into a differentiable form amenable to gradient-based optimization. Integrated within a structural vector autoregression (SVAR) framework, the model enables unified end-to-end training. Evaluated on three real-world benchmarks, the method significantly outperforms twelve baseline approaches in both accuracy and computational efficiency, achieving state-of-the-art performance while demonstrating exceptional scalability—accelerating inference by over sixfold on large-scale datasets.

Abstract

Causal discovery with instantaneous effects in multivariate time series is challenging, as the instantaneous structure must be acyclic. Prior methods enforce this by either separating instantaneous and lagged estimation into multi-stage pipelines or imposing algebraic acyclicity constraints via complex augmented Lagrangian optimization, both of which incur high computational cost. In this work, we propose a different approach: we learn a differentiable permutation of variables using the Gumbel--Sinkhorn operator and triangularize the instantaneous coefficient matrix of a Structural Vector Autoregressive (SVAR) model in the learned order. This converts acyclicity from a hard constraint into a parameterization and keeps it valid throughout optimization. In doing so, our method enables unified, continuous optimization with gradient-based learning, leading to improved efficiency in time--series causal discovery. Across three real-world benchmarks, our method achieves the best overall performance compared with 12 baselines in both discovery accuracy and efficiency. On the large-scale benchmark, it further demonstrates strong scalability, achieving more than a 6x speedup over competing methods.

Problem

• causal discovery
• instantaneous effects
• acyclicity
• time series
• SVAR

Innovation

• differentiable permutation
• acyclicity constraint
• Gumbel-Sinkhorn operator
• Structural Vector Autoregressive (SVAR)
• gradient-based optimization