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Cardinality-Regularized Hawkes-Granger Model

Tsuyoshi Idé, Georgios Kollias, Dzung T. Phan, Naoki Abe

TL;DR

This work addresses the problem of learning sparse Granger causality in Hawkes processes, where conventional likelihood-based approaches suffer from a pathological singularity that prevents true sparsity. It introduces a cardinality-regularized MM framework, termed L0 Hawkes, that enforces sparsity via an $\\ell_0$ penalty on the triggering matrix $\\mathsf{A}$ and yields semi-analytic instance- and type-level causal diagnoses through instance triggering probabilities $q_{n,i}$. By combining MM with Jensen bounds and an $\\epsilon$-sparsity strategy, the method achieves robust, interpretable sparse graphs while maintaining tractable optimization with complexity $\\mathcal{O}(N^2+D^2)$. The authors validate the approach on synthetic benchmarks and real-world data from power grids and cloud data centers, showing that L0 Hawkes achieves higher break-even accuracy and sparser, more meaningful causal structures than neural Granger or conventional sparse Hawkes models. Overall, the paper provides a principled, scalable framework for precise instance-wise causality in event data, with practical impact for prioritizing and diagnosing failures in complex systems.

Abstract

We propose a new sparse Granger-causal learning framework for temporal event data. We focus on a specific class of point processes called the Hawkes process. We begin by pointing out that most of the existing sparse causal learning algorithms for the Hawkes process suffer from a singularity in maximum likelihood estimation. As a result, their sparse solutions can appear only as numerical artifacts. In this paper, we propose a mathematically well-defined sparse causal learning framework based on a cardinality-regularized Hawkes process, which remedies the pathological issues of existing approaches. We leverage the proposed algorithm for the task of instance-wise causal event analysis, where sparsity plays a critical role. We validate the proposed framework with two real use-cases, one from the power grid and the other from the cloud data center management domain.

Cardinality-Regularized Hawkes-Granger Model

TL;DR

This work addresses the problem of learning sparse Granger causality in Hawkes processes, where conventional likelihood-based approaches suffer from a pathological singularity that prevents true sparsity. It introduces a cardinality-regularized MM framework, termed L0 Hawkes, that enforces sparsity via an penalty on the triggering matrix and yields semi-analytic instance- and type-level causal diagnoses through instance triggering probabilities . By combining MM with Jensen bounds and an -sparsity strategy, the method achieves robust, interpretable sparse graphs while maintaining tractable optimization with complexity . The authors validate the approach on synthetic benchmarks and real-world data from power grids and cloud data centers, showing that L0 Hawkes achieves higher break-even accuracy and sparser, more meaningful causal structures than neural Granger or conventional sparse Hawkes models. Overall, the paper provides a principled, scalable framework for precise instance-wise causality in event data, with practical impact for prioritizing and diagnosing failures in complex systems.

Abstract

We propose a new sparse Granger-causal learning framework for temporal event data. We focus on a specific class of point processes called the Hawkes process. We begin by pointing out that most of the existing sparse causal learning algorithms for the Hawkes process suffer from a singularity in maximum likelihood estimation. As a result, their sparse solutions can appear only as numerical artifacts. In this paper, we propose a mathematically well-defined sparse causal learning framework based on a cardinality-regularized Hawkes process, which remedies the pathological issues of existing approaches. We leverage the proposed algorithm for the task of instance-wise causal event analysis, where sparsity plays a critical role. We validate the proposed framework with two real use-cases, one from the power grid and the other from the cloud data center management domain.
Paper Structure (21 sections, 1 theorem, 38 equations, 9 figures, 1 table, 1 algorithm)

This paper contains 21 sections, 1 theorem, 38 equations, 9 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related work
  3. Preliminaries
  4. Problem setting
  5. Likelihood and intensity function
  6. Cardinality-Regularized Hawkes-Granger Model
  7. Intensity function and Granger causality
  8. Cardinality-regularized minorization-maximization framework
  9. Sparse Causal Learning via Cardinality Regularization
  10. Algorithm summary
  11. Experiments
  12. Concluding remarks
  13. Acknowledgement
  14. Solutions for Baseline Intensity and Decay Parameters
  15. Experimental Details
  16. ...and 6 more sections

Key Result

Theorem 1

For $p \geq 1$, the problem $\max_{\bm{x}}\left\{ \sum_{m}\Psi_m(x_m)- \tau \| \bm{x}\|_p \right\}$ is convex and has a unique solution. Let $\bm{x}^{**}$ be the solution. The solution cannot be sparse, i.e., $x^{**}_m \neq 0$ for $\forall m$, if $g_m >0$.

Figures (9)

  • Figure 1: The Hawkes-Granger model allows two different levels of causal analysis: (a) instance-wise and (b) type-wise, in which well-defined sparsity is essential for causal diagnosis. (c) Example of five-variate point process data, where each '$|$' represents an event instance.
  • Figure 2: Illustration of Hawkes model in Eq. \ref{['eq:Hawkes-indensity-func-general']}, showing $\lambda_d(t\mid \mathcal{H}_4)$ as an example. The $d_1$- and $d_3$-types are not causally related to $d$.
  • Figure 3: Three cases in Eq. \ref{['eq:Psi-l0objective']}: (a) $x_m^* > \epsilon$ and (b) two possibilities when $x_m^* \leq \epsilon$.
  • Figure 4: TN (red) and TP (blue) accuracies as a function of log regularization strength.
  • Figure 5: Comparison of $\bm{x}^*$ (flattened $\mathsf{A}$ in each row) computed with 100 different $\tau$ values.
  • ...and 4 more figures

Theorems & Definitions (2)

  • Definition 1: Hawkes process and Granger non-causality eichler2017graphical
  • Theorem 1