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Average-case optimization analysis for distributed consensus algorithms on regular graphs

Nhat Trung Nguyen, Alexander Rogozin, Alexander Gasnikov

TL;DR

This paper deriving the optimal method for consensus on regular graphs, showing its relation to the Heavy Ball method, analyzing its asymptotic convergence rate, and comparing it to various first-order methods through numerical experiments.

Abstract

The consensus problem in distributed computing involves a network of agents aiming to compute the average of their initial vectors through local communication, represented by an undirected graph. This paper focuses on the studying of this problem using an average-case analysis approach, particularly over regular graphs. Traditional algorithms for solving the consensus problem often rely on worst-case performance evaluation scenarios, which may not reflect typical performance in real-world applications. Instead, we apply average-case analysis, focusing on the expected spectral distribution of eigenvalues to obtain a more realistic view of performance. Key contributions include deriving the optimal method for consensus on regular graphs, showing its relation to the Heavy Ball method, analyzing its asymptotic convergence rate, and comparing it to various first-order methods through numerical experiments.

Average-case optimization analysis for distributed consensus algorithms on regular graphs

TL;DR

This paper deriving the optimal method for consensus on regular graphs, showing its relation to the Heavy Ball method, analyzing its asymptotic convergence rate, and comparing it to various first-order methods through numerical experiments.

Abstract

The consensus problem in distributed computing involves a network of agents aiming to compute the average of their initial vectors through local communication, represented by an undirected graph. This paper focuses on the studying of this problem using an average-case analysis approach, particularly over regular graphs. Traditional algorithms for solving the consensus problem often rely on worst-case performance evaluation scenarios, which may not reflect typical performance in real-world applications. Instead, we apply average-case analysis, focusing on the expected spectral distribution of eigenvalues to obtain a more realistic view of performance. Key contributions include deriving the optimal method for consensus on regular graphs, showing its relation to the Heavy Ball method, analyzing its asymptotic convergence rate, and comparing it to various first-order methods through numerical experiments.
Paper Structure (26 sections, 19 theorems, 97 equations, 4 figures, 5 algorithms)

This paper contains 26 sections, 19 theorems, 97 equations, 4 figures, 5 algorithms.

Table of Contents

  1. Introduction
  2. Average-case optimization analysis
  3. Contributions
  4. Consensus Problem
  5. Polynomial-Based Iterative Methods
  6. From First-Order Methods to Polynomials
  7. Gradient Descent
  8. Gradient Descent with Momentum
  9. Framework for average-case analysis
  10. Average-case analysis and expected error
  11. Optimal method
  12. Optimal method for regular graph
  13. Spectrum of regular graph
  14. Optimal method
  15. Expected errors
  16. ...and 11 more sections

Key Result

Lemma 1

Suppose that the sequence $\{x_t\}_{t=1}^{\infty}$ is generated by a first-order algorithm of the kind in eq:first_order_methods_quad, starting with $x_0$. If $\lim\limits_{t\to\infty} x_t = x_*$ and $f(x_*) = 0$, then $x_* = \frac{1}{n} \mathbf{1} \mathbf{1}^T x_0$.

Figures (4)

  • Figure 1: Regular graphs with $n=20$, $k=3$.
  • Figure 2: Spectrum of regular graph with $n=5000$, $k=3$.
  • Figure 3: Comparison of convergence speeds of algorithms on regular graphs. The top row shows the spectral distribution of the gossip matrix for $5000$-vertex regular graphs with degree parameters $k = 3$, $8$, and $15$. The bottom row displays the normalized distance to consensus of each algorithm at each iteration, with the y-axis scaled logarithmically. In these experiments, each node vector size is set to $d = 1000$.
  • Figure 4: Convergence speeds of the optimal method and the conjugate gradient method

Theorems & Definitions (30)

  • Definition 1
  • Lemma 1
  • proof
  • Lemma 2
  • Theorem 3.1: berthier2020acceleratedpedregosa2021residualmomentumpaquette2023halting
  • Definition 2: Empirical/Expected spectral distribution
  • Theorem 4.1: pedregosa2020acceleration
  • Definition 3
  • Lemma 3: Three-term recurrence, pedregosa2020acceleration, fischer2011polynomial
  • Theorem 4.2: Theorem 2.1 from pedregosa2020acceleration
  • ...and 20 more