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On the Convergence of a Federated Expectation-Maximization Algorithm

Zhixu Tao, Rajita Chandak, Sanjeev Kulkarni

TL;DR

The paper analyzes the convergence of a federated EM algorithm for a Federated Mixture of $K$ Linear Regressions (FMLR) under data heterogeneity across clients. It develops population- and empirical-EM theory that holds across all regimes of the number of clients $m$ and per-client samples $n$, establishing a SNR threshold $Ω(√K)$ for uniform contraction and showing that, in many regimes, the EM algorithm converges in a constant number of iterations as $m$ grows relative to $n$. It also proves minimax-optimality in the SNR scaling and reveals that larger cluster separation $Δ_{\max}$ does not universally improve convergence. Complemented by synthetic experiments, the results demonstrate that heterogeneity can even accelerate convergence of iterative federated algorithms, offering practical guidance for FL deployments with heterogeneous data.

Abstract

Data heterogeneity has been a long-standing bottleneck in studying the convergence rates of Federated Learning algorithms. In order to better understand the issue of data heterogeneity, we study the convergence rate of the Expectation-Maximization (EM) algorithm for the Federated Mixture of $K$ Linear Regressions model (FMLR). We completely characterize the convergence rate of the EM algorithm under all regimes of $m/n$ where $m$ is the number of clients and $n$ is the number of data points per client. We show that with a signal-to-noise-ratio (SNR) of order $Ω(\sqrt{K})$, the well-initialized EM algorithm converges within the minimax distance of the ground truth under all regimes. Interestingly, we identify that when the number of clients grows reasonably with respect to the number of data points per client, the EM algorithm only requires a constant number of iterations to converge. We perform experiments on synthetic data to illustrate our results. In line with our theoretical findings, the simulations show that rather than being a bottleneck, data heterogeneity can accelerate the convergence of iterative federated algorithms.

On the Convergence of a Federated Expectation-Maximization Algorithm

TL;DR

The paper analyzes the convergence of a federated EM algorithm for a Federated Mixture of Linear Regressions (FMLR) under data heterogeneity across clients. It develops population- and empirical-EM theory that holds across all regimes of the number of clients and per-client samples , establishing a SNR threshold for uniform contraction and showing that, in many regimes, the EM algorithm converges in a constant number of iterations as grows relative to . It also proves minimax-optimality in the SNR scaling and reveals that larger cluster separation does not universally improve convergence. Complemented by synthetic experiments, the results demonstrate that heterogeneity can even accelerate convergence of iterative federated algorithms, offering practical guidance for FL deployments with heterogeneous data.

Abstract

Data heterogeneity has been a long-standing bottleneck in studying the convergence rates of Federated Learning algorithms. In order to better understand the issue of data heterogeneity, we study the convergence rate of the Expectation-Maximization (EM) algorithm for the Federated Mixture of Linear Regressions model (FMLR). We completely characterize the convergence rate of the EM algorithm under all regimes of where is the number of clients and is the number of data points per client. We show that with a signal-to-noise-ratio (SNR) of order , the well-initialized EM algorithm converges within the minimax distance of the ground truth under all regimes. Interestingly, we identify that when the number of clients grows reasonably with respect to the number of data points per client, the EM algorithm only requires a constant number of iterations to converge. We perform experiments on synthetic data to illustrate our results. In line with our theoretical findings, the simulations show that rather than being a bottleneck, data heterogeneity can accelerate the convergence of iterative federated algorithms.
Paper Structure (19 sections, 14 theorems, 115 equations, 5 figures, 1 algorithm)

This paper contains 19 sections, 14 theorems, 115 equations, 5 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Our contributions
  3. Related Work
  4. Problem Setup and EM Algorithm
  5. Notation
  6. The FMLR model
  7. EM Algorithm
  8. Main Results
  9. Experiments
  10. Conclusions and Discussions
  11. Proofs for Section \ref{['sec:setup']}
  12. Proof of Proposition \ref{['prop:population_EM']}
  13. Proof of Proposition \ref{['prop:empirical_EM']}
  14. Proofs for Section \ref{['sec:results']}
  15. Proof of Theorem \ref{['theorem.1']}
  16. ...and 4 more sections

Key Result

Proposition 2

Suppose Assumption as:dgp holds and $\{(x_i, y_i)\}_{i=1}^n$ are generated by the FMLR model as given in Algorithm alg:fmlr. Then for each $k \in [K]$, one iteration of the population EM, given the current estimates $\boldsymbol{\theta}$, is given by

Figures (5)

  • Figure 1: Effect of number of data points $n$
  • Figure 2: Effect of number of clusters $K$
  • Figure 3: Effect of dimension $d$
  • Figure 4: Effect of SNR
  • Figure 5: Effect of $\Delta_{\max}$

Theorems & Definitions (14)

  • Proposition 2: Population EM
  • Proposition 3: Empirical EM
  • Theorem 5: Uniform consistency
  • Theorem 6: Empirical uniform consistency
  • Corollary 7
  • Lemma 8
  • Proposition 9
  • Lemma 10
  • Lemma 11
  • Lemma 12
  • ...and 4 more