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Decentralized Inference for Spatial Data Using Low-Rank Models

Jianwei Shi, Sameh Abdulah, Ying Sun, Marc G. Genton

TL;DR

This work tackles scalable decentralized parameter inference for spatial low-rank models by replacing the non-decomposable log-likelihood with a variational ELBO objective. The ELBO decomposes into local and common components, enabling decentralized optimization via a block coordinate descent framework that leverages dynamic consensus averaging and multi-consensus. The authors establish local convexity of the negative ELBO near the true parameters and prove consistency and asymptotic normality of the estimators, along with convergence guarantees for the algorithm. Through extensive simulations and a Total Precipitable Water data application, the method demonstrates robustness to data partitioning, network structure, and varying ranks, while delivering competitive predictive performance and substantial computational advantages in distributed settings.

Abstract

Advancements in information technology have enabled the creation of massive spatial datasets, driving the need for scalable and efficient computational methodologies. While offering viable solutions, centralized frameworks are limited by vulnerabilities such as single-point failures and communication bottlenecks. This paper presents a decentralized framework tailored for parameter inference in spatial low-rank models to address these challenges. A key obstacle arises from the spatial dependence among observations, which prevents the log-likelihood from being expressed as a summation-a critical requirement for decentralized optimization approaches. To overcome this challenge, we propose a novel objective function leveraging the evidence lower bound, which facilitates the use of decentralized optimization techniques. Our approach employs a block descent method integrated with multi-consensus and dynamic consensus averaging for effective parameter optimization. We prove the convexity of the new objective function in the vicinity of the true parameters, ensuring the convergence of the proposed method. Additionally, we present the first theoretical results establishing the consistency and asymptotic normality of the estimator within the context of spatial low-rank models. Extensive simulations and real-world data experiments corroborate these theoretical findings, showcasing the robustness and scalability of the framework.

Decentralized Inference for Spatial Data Using Low-Rank Models

TL;DR

This work tackles scalable decentralized parameter inference for spatial low-rank models by replacing the non-decomposable log-likelihood with a variational ELBO objective. The ELBO decomposes into local and common components, enabling decentralized optimization via a block coordinate descent framework that leverages dynamic consensus averaging and multi-consensus. The authors establish local convexity of the negative ELBO near the true parameters and prove consistency and asymptotic normality of the estimators, along with convergence guarantees for the algorithm. Through extensive simulations and a Total Precipitable Water data application, the method demonstrates robustness to data partitioning, network structure, and varying ranks, while delivering competitive predictive performance and substantial computational advantages in distributed settings.

Abstract

Advancements in information technology have enabled the creation of massive spatial datasets, driving the need for scalable and efficient computational methodologies. While offering viable solutions, centralized frameworks are limited by vulnerabilities such as single-point failures and communication bottlenecks. This paper presents a decentralized framework tailored for parameter inference in spatial low-rank models to address these challenges. A key obstacle arises from the spatial dependence among observations, which prevents the log-likelihood from being expressed as a summation-a critical requirement for decentralized optimization approaches. To overcome this challenge, we propose a novel objective function leveraging the evidence lower bound, which facilitates the use of decentralized optimization techniques. Our approach employs a block descent method integrated with multi-consensus and dynamic consensus averaging for effective parameter optimization. We prove the convexity of the new objective function in the vicinity of the true parameters, ensuring the convergence of the proposed method. Additionally, we present the first theoretical results establishing the consistency and asymptotic normality of the estimator within the context of spatial low-rank models. Extensive simulations and real-world data experiments corroborate these theoretical findings, showcasing the robustness and scalability of the framework.

Paper Structure

This paper contains 36 sections, 22 theorems, 281 equations, 11 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Decentralized Optimization
  4. Weight Matrix
  5. Dynamic Consensus Averaging and Multi-Consensus
  6. Challenges in Spatial Low-Rank Models
  7. The Proposed Decentralized Method
  8. Theory
  9. Simulations
  10. Convergence under Various Scenarios
  11. Estimation Accuracy
  12. Evaluation of Prediction Accuracy, Computation Efficiency, and Smoothness Parameter Estimation
  13. Application to Total Precipitable Water Data
  14. Conclusion
  15. Analysis of Dynamic Average Consensus
  16. ...and 21 more sections

Key Result

Lemma 1

Let $\boldsymbol{y}^{t}=\boldsymbol{W} \boldsymbol{y}^{t-1}$ with $\boldsymbol{y}^{0} \in \mathbb{R}^{J}$, then $\boldsymbol{y}^{t} \rightarrow$$\frac{1}{J} \mathbf{1} \mathbf{1}^{\top} \boldsymbol{y}^{0}$ if and only if $\boldsymbol{W}$ is doubly stochastic and the mixing rate $\rho_{\boldsymbol{W}

Figures (11)

  • Figure 1: Decentralized block coordinate descent for spatial low-rank models.
  • Figure 2: Convergence of the decentralized method with different parameter settings.
  • Figure 3: Partitioned locations (upper panel) and convergence of the decentralized method (lower panel) under different data partitioning schemes
  • Figure 4: Convergence of the decentralized method: (a) with a network connection probability of 0.3; (b) with a network connection probability of 0.8; (c) under unequal local sample sizes.
  • Figure 5: (a) Prediction error. (b) Time comparison between distributed and non-distributed method. (c) Estimated value of the smooth parameter.
  • ...and 6 more figures

Theorems & Definitions (46)

  • Lemma 1: xiao2004fast
  • Remark 1: Initial points
  • Remark 2: Prediction
  • Remark 3: Cost analysis
  • Remark 4: Extension to multi-resolution models
  • Remark 5
  • Theorem 1: Local Convexity
  • Theorem 2: Consistency
  • Theorem 3: Asymptotic Normality
  • Remark 6
  • ...and 36 more