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Exponential convergence of a distributed divide-and-conquer algorithm for constrained convex optimization on networks

Nazar Emirov, Guohui Song, Qiyu Sun

TL;DR

This work presents a fully distributed divide-and-conquer (DAC) algorithm for constrained convex optimization on networks, achieving exponential convergence under smoothness, strong convexity, and locality assumptions, with graph polynomial-growth enabling contraction properties. The method decomposes the global problem into local subproblems around fusion centers, communicates only with neighboring centers, and updates the global solution through a structured aggregation. It extends to inequality constraints via a log-barrier formulation, preserving exponential convergence with rate dependent on barrier parameter and radius $R$. Theoretical guarantees are complemented by numerical experiments on $L_2$, quadratic, and entropy losses, demonstrating scalability to large networks. Overall, DAC offers a communication-efficient, scalable alternative to ALM/ADMM for large-scale constrained network optimization with provable convergence rates.

Abstract

We propose a divide-and-conquer (DAC) algorithm for constrained convex optimization over networks, where the global objective is the sum of local objectives attached to individual agents. The algorithm is fully distributed: each iteration solves local subproblems around selected fusion centers and coordinates only with neighboring fusion centers. Under standard assumptions of smoothness, strong convexity, and locality on the objective function, together with polynomial growth conditions on the underlying graph, we establish exponential convergence of the DAC iterations and derive explicit bounds for both exact and inexact local solvers. Numerical experiments on three representative losses ($L_2$ distance, quadratic, and entropy) confirm the theory and demonstrate scalability and effectiveness.

Exponential convergence of a distributed divide-and-conquer algorithm for constrained convex optimization on networks

TL;DR

This work presents a fully distributed divide-and-conquer (DAC) algorithm for constrained convex optimization on networks, achieving exponential convergence under smoothness, strong convexity, and locality assumptions, with graph polynomial-growth enabling contraction properties. The method decomposes the global problem into local subproblems around fusion centers, communicates only with neighboring centers, and updates the global solution through a structured aggregation. It extends to inequality constraints via a log-barrier formulation, preserving exponential convergence with rate dependent on barrier parameter and radius . Theoretical guarantees are complemented by numerical experiments on , quadratic, and entropy losses, demonstrating scalability to large networks. Overall, DAC offers a communication-efficient, scalable alternative to ALM/ADMM for large-scale constrained network optimization with provable convergence rates.

Abstract

We propose a divide-and-conquer (DAC) algorithm for constrained convex optimization over networks, where the global objective is the sum of local objectives attached to individual agents. The algorithm is fully distributed: each iteration solves local subproblems around selected fusion centers and coordinates only with neighboring fusion centers. Under standard assumptions of smoothness, strong convexity, and locality on the objective function, together with polynomial growth conditions on the underlying graph, we establish exponential convergence of the DAC iterations and derive explicit bounds for both exact and inexact local solvers. Numerical experiments on three representative losses ( distance, quadratic, and entropy) confirm the theory and demonstrate scalability and effectiveness.

Paper Structure

This paper contains 17 sections, 6 theorems, 110 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem setting
  3. Graphs with polynomial growth property
  4. Objective functions
  5. Fusion centers for distributed and decentralized implementation
  6. Constraints
  7. Divide-and-conquer algorithm and exponential convergence
  8. Constrained optimization problems without inequality constraints
  9. Constrained optimization problems with inequality constraints
  10. Numerical simulations
  11. $L_{2}$ distance loss
  12. Quadratic loss
  13. Entropy loss
  14. Proofs
  15. Proof of Theorem \ref{['maintheorem1.thm']}
  16. ...and 2 more sections

Key Result

Theorem 3.1

Suppose that Assumptions PolynomialGrowth.assumption, assump:f, assump:J and assump:A hold. Consider the constrained convex optimization problem convexoptimizationlinearconstraint.def and denote its unique minimizer by ${\bf x}^*$. Set and where $d:=d(\mathcal{G})$ and $D_1(\mathcal{G})$ are Beurling dimension and density of the graph ${\mathcal{G}}$ respectively, $m, c_1, c_2$ and $L_1$ are co

Figures (3)

  • Figure 1: Approximation error $\left\lVert {\bf x}^{(n)} - {\bf x}^* \right\rVert_2$ for the orthogonal projection problem.
  • Figure 2: Approximation error $\left\lVert {\bf x}^{(n)} - {\bf x}^* \right\rVert_2$ for the constrained quadratic optimization problem.
  • Figure 3: Approximation error $\lVert {\bf x}^{(n)} - {\bf x}^* \rVert_2$ for the entropy maximization problem.

Theorems & Definitions (10)

  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Lemma 5.1
  • proof : Proof of Lemma \ref{['maintheorem1.lem1']}
  • Lemma 5.2
  • proof
  • Lemma 5.3
  • proof
  • proof : Proof of Theorem \ref{['maintheorem1.thm']}