Learning Coefficient Heterogeneity over Networks: A Distributed Spanning-Tree-Based Fused-Lasso Regression

Abstract

Identifying the latent cluster structure based on model heterogeneity is a fundamental but challenging task arises in many machine learning applications. In this article, we study the clustered coefficient regression problem in the distributed network systems, where the data are locally collected and held by nodes. Our work aims to improve the regression estimation efficiency by aggregating the neighbors’ information while also identifying the cluster membership for nodes. To achieve efficient estimation and clustering, we develop a distributed spanning-tree-based fused-lasso regression (DTFLR) approach. In particular, we propose an adaptive spanning-tree-based fusion penalty for the low-complexity clustered coefficient regression. We show that our proposed estimator satisfies statistical oracle properties. Additionally, to solve the problem parallelly, we design a distributed generalized alternating direction method of multiplier algorithm, which has a simple node-based implementation scheme and enjoys a linear convergence rate. Collectively, our results in this article contribute to the theories of low-complexity clustered coefficient regression and distributed optimization over networks. Thorough numerical experiments and real-world data analysis are conducted to verify our theoretical results, which show that our approach outperforms existing works in terms of estimation accuracy, computation speed, and communication costs. Supplementary materials for this article are available online.

Keywords:

• Adaptive fused-lasso
• Distributed generalized ADMM
• Minimum spanning tree
• MODIS
• Sensornets

Supplementary Materials

The supplementary materials contain detailed derivations for the proposed algorithm, proofs for the theoretical results, as well as the additional numerical simulations. The implementation code is also provided in the supplementary materials.

Disclosure Statement

The authors report there are no competing interests to declare.

Notes

1 Our algorithms and results in this article can easily be extended to cases with datasets of unbalanced sizes.

2 The O and Θ notation are the Bachmann–Landau notations Knuth (1976). $a_n=O(b_n)$ denotes that there exist positive constants C and n₀ with $\left|a_n\right|\le C{b}_n$ for all $n\ge n_0$; $a_n=\Theta (b_n)$ denotes that there exist positive constants C, $C\prime$ and n₀ with $C\prime \le a_n\le C{b}_n$ for all $n\ge n_0$.

3 http://www.sensornets.csiro.au/

4 https://www.usgs.gov/centers/eros/science/national-land-cover-database

Additional information

Funding

This work has been supported in part by NSF grants CAREER CNS-2110259, CNS-2112471, CCF-1934884, CCF-2110252, and a Google Faculty Research Award.