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Causal Customer Churn Analysis with Low-rank Tensor Block Hazard Model

Chenyin Gao, Zhiming Zhang, Shu Yang

TL;DR

The paper tackles causal churn analysis with multiple binary interventions by framing churn trajectories as a three-way tensor $\mathcal{Y}=(Y_{i,t,l})$ and imputing counterfactuals through a 1-bit tensor completion in a tensorized latent factor block hazard model. It combines a low-rank representation of the parameter tensor $\Theta$ with a clustering mechanism on the treatment mode to reduce the intervention space, and it employs IPTW with CBPS-based propensity scores alongside covariate-assisted loading for estimation. The authors establish non-asymptotic error bounds for both parameter estimation and clustering, and they validate the approach via synthetic experiments and a real bank churn dataset, demonstrating superior predictive performance and practical guidance for selecting homogeneous intervention groups. The framework supports identifying individualized optimal interventions to maximize retention and provides a scalable, theoretically-grounded approach to causal churn analysis with binary, time-to-event data.

Abstract

This study introduces an innovative method for analyzing the impact of various interventions on customer churn, using the potential outcomes framework. We present a new causal model, the tensorized latent factor block hazard model, which incorporates tensor completion methods for a principled causal analysis of customer churn. A crucial element of our approach is the formulation of a 1-bit tensor completion for the parameter tensor. This captures hidden customer characteristics and temporal elements from churn records, effectively addressing the binary nature of churn data and its time-monotonic trends. Our model also uniquely categorizes interventions by their similar impacts, enhancing the precision and practicality of implementing customer retention strategies. For computational efficiency, we apply a projected gradient descent algorithm combined with spectral clustering. We lay down the theoretical groundwork for our model, including its non-asymptotic properties. The efficacy and superiority of our model are further validated through comprehensive experiments on both simulated and real-world applications.

Causal Customer Churn Analysis with Low-rank Tensor Block Hazard Model

TL;DR

The paper tackles causal churn analysis with multiple binary interventions by framing churn trajectories as a three-way tensor and imputing counterfactuals through a 1-bit tensor completion in a tensorized latent factor block hazard model. It combines a low-rank representation of the parameter tensor with a clustering mechanism on the treatment mode to reduce the intervention space, and it employs IPTW with CBPS-based propensity scores alongside covariate-assisted loading for estimation. The authors establish non-asymptotic error bounds for both parameter estimation and clustering, and they validate the approach via synthetic experiments and a real bank churn dataset, demonstrating superior predictive performance and practical guidance for selecting homogeneous intervention groups. The framework supports identifying individualized optimal interventions to maximize retention and provides a scalable, theoretically-grounded approach to causal churn analysis with binary, time-to-event data.

Abstract

This study introduces an innovative method for analyzing the impact of various interventions on customer churn, using the potential outcomes framework. We present a new causal model, the tensorized latent factor block hazard model, which incorporates tensor completion methods for a principled causal analysis of customer churn. A crucial element of our approach is the formulation of a 1-bit tensor completion for the parameter tensor. This captures hidden customer characteristics and temporal elements from churn records, effectively addressing the binary nature of churn data and its time-monotonic trends. Our model also uniquely categorizes interventions by their similar impacts, enhancing the precision and practicality of implementing customer retention strategies. For computational efficiency, we apply a projected gradient descent algorithm combined with spectral clustering. We lay down the theoretical groundwork for our model, including its non-asymptotic properties. The efficacy and superiority of our model are further validated through comprehensive experiments on both simulated and real-world applications.
Paper Structure (34 sections, 9 theorems, 118 equations, 5 figures, 5 tables)

This paper contains 34 sections, 9 theorems, 118 equations, 5 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Basic setup
  4. Notation and preliminaries
  5. Potential outcomes and causal assumptions
  6. Tensorized hazard model
  7. Low-rank structure and treatment clustering
  8. Weighted likelihood estimation for $\Theta$
  9. Algorithm
  10. Propensity score estimation
  11. Projected gradient descent and spectral clustering
  12. Statistical theory
  13. Upper bound error for estimation
  14. Upper bound error for clustering
  15. Numerical studies
  16. ...and 19 more sections

Key Result

Theorem 5.1

Under Assumptions A1) to A4) and some regularity conditions, suppose $\mathcal{Y}$ is the binary tensor characterized by the parameter tensor $\Theta$ as model (eq:binary) with the link function $f(\cdot)$. Let $\widehat{\Theta}$ be the local maximizer of (eq:loss_1), there exist constants $c_{0}$,

Figures (5)

  • Figure 1: A new tensor representation of potential outcomes with three modes (customer $\times$ time $\times$ intervention).
  • Figure 2: Cumulative regret (top) and decision accuracy (bottom) of the proposed method and other competitors when $N=100,300,500,1000,2000$, $T=5,10$ and $k=2,3$.
  • Figure 3: Estimated intervention structure by the proposed model for the bank customer churn data. All interventions are clustered by blocks and ordered by their expected customer lifetimes.
  • Figure A1: Normalized mean squared error (left) and the misclassification error rate (right) when $T=5$ and $N=100,300,500,1000,2000$ over $100$ data replications.
  • Figure A2: Plot of the average estimated survival probabilities for all treatments when $N=1000,T=10$, $k=2$ (left) and $k=3$ (right) over $100$ data replications.

Theorems & Definitions (10)

  • Theorem 5.1
  • Corollary 5.2
  • Theorem 5.3
  • Lemma B.1
  • Lemma B.2
  • Lemma B.3
  • Lemma B.4
  • proof
  • Lemma B.5
  • Lemma B.6