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LocalKMeans: Convergence of Lloyd's Algorithm with Distributed Local Iterations

Harsh Vardhan, Heng Zhu, Avishek Ghosh, Arya Mazumdar

TL;DR

This work develops LocalKMeans, a distributed Lloyd's algorithm that performs L local Lloyd iterations on each machine before syncing to reduce communication in a mixture of Gaussians setting. By adapting a virtual-iterate analysis to the non-convex Lloyd objective, it proves convergence guarantees for both the 2-cluster and K-cluster cases under suitable initialization and SNR conditions, yielding final misclustering bounds that include exp(-r^2), exp(-n), and d/n dependent terms. The method achieves substantial communication savings (a factor of L) while maintaining competitive clustering performance relative to centralized Lloyd's, at the cost of additional iterations, and demonstrates practical viability on synthetic and real datasets. Overall, the paper advances unsupervised distributed learning by providing rigorous convergence analysis for local-update schemes in clustering and showing tangible speedups in communication-constrained environments.

Abstract

In this paper, we analyze the classical $K$-means alternating-minimization algorithm, also known as Lloyd's algorithm (Lloyd, 1956), for a mixture of Gaussians in a data-distributed setting that incorporates local iteration steps. Assuming unlabeled data distributed across multiple machines, we propose an algorithm, LocalKMeans, that performs Lloyd's algorithm in parallel in the machines by running its iterations on local data, synchronizing only every $L$ of such local steps. We characterize the cost of these local iterations against the non-distributed setting, and show that the price paid for the local steps is a higher required signal-to-noise ratio. While local iterations were theoretically studied in the past for gradient-based learning methods, the analysis of unsupervised learning methods is more involved owing to the presence of latent variables, e.g. cluster identities, than that of an iterative gradient-based algorithm. To obtain our results, we adapt a virtual iterate method to work with a non-convex, non-smooth objective function, in conjunction with a tight statistical analysis of Lloyd steps.

LocalKMeans: Convergence of Lloyd's Algorithm with Distributed Local Iterations

TL;DR

This work develops LocalKMeans, a distributed Lloyd's algorithm that performs L local Lloyd iterations on each machine before syncing to reduce communication in a mixture of Gaussians setting. By adapting a virtual-iterate analysis to the non-convex Lloyd objective, it proves convergence guarantees for both the 2-cluster and K-cluster cases under suitable initialization and SNR conditions, yielding final misclustering bounds that include exp(-r^2), exp(-n), and d/n dependent terms. The method achieves substantial communication savings (a factor of L) while maintaining competitive clustering performance relative to centralized Lloyd's, at the cost of additional iterations, and demonstrates practical viability on synthetic and real datasets. Overall, the paper advances unsupervised distributed learning by providing rigorous convergence analysis for local-update schemes in clustering and showing tangible speedups in communication-constrained environments.

Abstract

In this paper, we analyze the classical -means alternating-minimization algorithm, also known as Lloyd's algorithm (Lloyd, 1956), for a mixture of Gaussians in a data-distributed setting that incorporates local iteration steps. Assuming unlabeled data distributed across multiple machines, we propose an algorithm, LocalKMeans, that performs Lloyd's algorithm in parallel in the machines by running its iterations on local data, synchronizing only every of such local steps. We characterize the cost of these local iterations against the non-distributed setting, and show that the price paid for the local steps is a higher required signal-to-noise ratio. While local iterations were theoretically studied in the past for gradient-based learning methods, the analysis of unsupervised learning methods is more involved owing to the presence of latent variables, e.g. cluster identities, than that of an iterative gradient-based algorithm. To obtain our results, we adapt a virtual iterate method to work with a non-convex, non-smooth objective function, in conjunction with a tight statistical analysis of Lloyd steps.

Paper Structure

This paper contains 64 sections, 23 theorems, 166 equations, 4 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Statistical Model
  3. Main Contributions
  4. Related Works
  5. LocalKMeans on $2$ symmetric clusters
  6. Proof Sketch
  7. Unrolling the recursion.
  8. Comparisons to Baselines
  9. LocalKMeans on K clusters
  10. Experiments
  11. Synthetic Data
  12. Real Datasets
  13. Conclusion
  14. Additional details for LocalKMeans on $K$-clusters
  15. Initialization requirement(Assumption \ref{['assumption:local_dev_k']}
  16. ...and 49 more sections

Key Result

Lemma 2.2

Suppose Assumption assumption:local_dev holds, and $r > c\cdot \max\{\sqrt{d/n},1\}$, $n > c' \log m$ for some constants $c,c'>0$. Let, $\Phi_1 \leq c" (\sqrt{\frac{\log(mn)}{mn}} + \frac{1}{r^2})$ and $\Phi_2 \leq c"(\frac{d}{nr^2} + \frac{1}{mn})$, for some constant $c">0$. Then $\forall t>1$ in A

Figures (4)

  • Figure 1: Performance of LocalKMeans on mixture of Gaussians with different initializations and SNRs. Horizontal Axis: Number of Total Iterations (T), Vertical Axis: Misclustering Ratio.
  • Figure 2: Misclusering ratio on real datasets.
  • Figure 3: KMeans objective on real datasets.
  • Figure 4: Performance of LocalKmeans on Gaussian Data with Different Factors. Horizontal Axis: Number of Total Iterations. Vertical Axis: Misclustering Ratio.

Theorems & Definitions (30)

  • Lemma 2.2: Single-step progress
  • Remark 2.3: Special case of centralized Lloyd's
  • Remark 2.4
  • Theorem 2.5: Final Error
  • Lemma 3.2: Single-step progress
  • Remark 3.3: Progress in $\Lambda^{(t)}$
  • Theorem 3.4: Final Error
  • Lemma C.1: Extension of Lemma 7.1 lu_statistical_2016
  • Lemma C.2: Extension of Lemma 7.2 lu_statistical_2016
  • Lemma C.3: Extension of Lemma 7.3 from lu_statistical_2016
  • ...and 20 more