Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Decentralized Online Regularized Learning Over Random Time-Varying Graphs

Xiwei Zhang, Tao Li, Xiaozheng Fu

TL;DR

This work develops a decentralized online regularized linear regression algorithm operating over random time-varying graphs. By leveraging a nonnegative supermartingale framework and a novel sample path spatio-temporal persistence of excitation condition, the authors establish almost sure convergence of all node estimates to the true parameter $x_0$ under broad, non-i.i.d. settings. They further derive a regret bound of $ ext{Regret}_{ ext{LMS}}(i,T)= obreak\mathcal{O}(T^{1- au} obreak\ln T)$ with $ au\nobreak\in(0.5,1)$ and provide non-asymptotic rates for the estimation error, extending prior results to random, time-varying digraphs with both additive and multiplicative communication noises. The results offer practical guarantees for distributed learning in networks with uncertain connectivity and data, and suggest directions for optimizing regularization and handling more complex noise models. Overall, the paper advances understanding of how regularization and persistent excitation interact to ensure reliable decentralized online learning in dynamic networks.

Abstract

We study the decentralized online regularized linear regression algorithm over random time-varying graphs. At each time step, every node runs an online estimation algorithm consisting of an innovation term processing its own new measurement, a consensus term taking a weighted sum of estimations of its own and its neighbors with additive and multiplicative communication noises and a regularization term preventing over-fitting. It is not required that the regression matrices and graphs satisfy special statistical assumptions such as mutual independence, spatio-temporal independence or stationarity. We develop the nonnegative supermartingale inequality of the estimation error, and prove that the estimations of all nodes converge to the unknown true parameter vector almost surely if the algorithm gains, graphs and regression matrices jointly satisfy the sample path spatio-temporal persistence of excitation condition. Especially, this condition holds by choosing appropriate algorithm gains if the graphs are uniformly conditionally jointly connected and conditionally balanced, and the regression models of all nodes are uniformly conditionally spatio-temporally jointly observable, under which the algorithm converges in mean square and almost surely. In addition, we prove that the regret upper bound is $O(T^{1-τ}\ln T)$, where $τ\in (0.5,1)$ is a constant depending on the algorithm gains.

Decentralized Online Regularized Learning Over Random Time-Varying Graphs

TL;DR

This work develops a decentralized online regularized linear regression algorithm operating over random time-varying graphs. By leveraging a nonnegative supermartingale framework and a novel sample path spatio-temporal persistence of excitation condition, the authors establish almost sure convergence of all node estimates to the true parameter under broad, non-i.i.d. settings. They further derive a regret bound of with and provide non-asymptotic rates for the estimation error, extending prior results to random, time-varying digraphs with both additive and multiplicative communication noises. The results offer practical guarantees for distributed learning in networks with uncertain connectivity and data, and suggest directions for optimizing regularization and handling more complex noise models. Overall, the paper advances understanding of how regularization and persistent excitation interact to ensure reliable decentralized online learning in dynamic networks.

Abstract

We study the decentralized online regularized linear regression algorithm over random time-varying graphs. At each time step, every node runs an online estimation algorithm consisting of an innovation term processing its own new measurement, a consensus term taking a weighted sum of estimations of its own and its neighbors with additive and multiplicative communication noises and a regularization term preventing over-fitting. It is not required that the regression matrices and graphs satisfy special statistical assumptions such as mutual independence, spatio-temporal independence or stationarity. We develop the nonnegative supermartingale inequality of the estimation error, and prove that the estimations of all nodes converge to the unknown true parameter vector almost surely if the algorithm gains, graphs and regression matrices jointly satisfy the sample path spatio-temporal persistence of excitation condition. Especially, this condition holds by choosing appropriate algorithm gains if the graphs are uniformly conditionally jointly connected and conditionally balanced, and the regression models of all nodes are uniformly conditionally spatio-temporally jointly observable, under which the algorithm converges in mean square and almost surely. In addition, we prove that the regret upper bound is , where is a constant depending on the algorithm gains.
Paper Structure (8 sections, 11 theorems, 166 equations, 4 figures, 2 tables)

This paper contains 8 sections, 11 theorems, 166 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Decentralized Regularized Linear Regression
  3. Main Results
  4. Numerical Example
  5. Conclusions
  6. Proofs of Theorems \ref{['ATheorem1']}-\ref{['Atheorem4']} and Verification of Example \ref{['Aexample333']}
  7. Proofs of Lemmas \ref{['Alemma4']}-\ref{['Alemma5']}
  8. Key Lemmas

Key Result

Lemma 1

For the algorithm (Aalgorithm), if Assumptions (A1)-(A2) hold, the algorithm gains $a(k)$, $b(k)$ and $\lambda(k)$ monotonically decrease to zero, and there exists a positive integer $h$ and a positive constant $\rho_0$, such that $\sup_{k\ge 0}(\|\mathcal{L}_{\mathcal{G}(k)}\|+(\mathbb E[\|{\mathca where $\{\Omega(k),k\ge 0\}$ and $\{\Gamma(k),k\ge 0\}$ are nonnegative deterministic real sequence

Figures (4)

  • Figure 1: The sample paths of $R(k)$ with different settings.
  • Figure 2: The sample paths of estimation errors with different settings.
  • Figure 3: Mean value of norms of $10$ nodes' states.
  • Figure 4: Mean square estimation errors of nodes for Setting III.

Theorems & Definitions (31)

  • Remark 1
  • Remark 2
  • Remark 2
  • Example 1
  • Remark 3
  • Remark 4
  • Remark 5
  • Lemma 1
  • Remark 6
  • Theorem 1
  • ...and 21 more