Convergence Analysis of Distributed Optimization: A Dissipativity Framework

TL;DR

The paper tackles convergence analysis for distributed optimization over networks by introducing a system-theoretic framework based on incremental dissipativity and contraction. It treats distributed algorithms as interconnections of local LTI blocks with nonlinear oracles, and provides a step-by-step LMI-based procedure to certify contraction with rate $\gamma\in(0,1)$, independent of network topology. By encoding oracle properties as sector bounds and aggregating local dissipativity through a network interconnection, the approach handles heterogeneous algorithms and yields a practical convergence test via semidefinite programming. Numerical comparisons with classical distributed gradient descent demonstrate that the dissipativity framework not only recovers standard conditions but can also identify larger feasible regions, highlighting potential for heterogeneity-aware algorithm design in distributed optimization systems.

Abstract

We develop a system-theoretic framework for the structured analysis of distributed optimization algorithms. We model such algorithms as a network of interacting dynamical systems and derive tests for convergence based on incremental dissipativity and contraction theory. This approach yields a step-by-step analysis pipeline independent of the network structure, with conditions expressed as linear matrix inequalities. In addition, a numerical comparison with traditional analysis methods is presented, in the context of distributed gradient descent.

Figures (5)

• Figure 1: Block diagram of a distributed optimization algorithm modelled as a collection of LTI systems $\Sigma_i$ in feedback interconnection with their oracles $\varphi_i$ and coupled via $u=My$.
• Figure 2: Communication graph used in the example.
• Figure 3: Grid search over $\rho$ and $\eta$, with $\mu=0.05$ and $K = 1$, using the communication graph in Fig. \ref{['fig:graph']}. Blue squares show where the LMIs \ref{['eq:dgd-local-LMI']} and \ref{['eq:Laplacian-global-LMI']} are feasible, while the gray area is where \ref{['eq:eta-bound']} hold.
• Figure 4: Grid search over $\rho_i$ and $\eta_i$, as in Fig. \ref{['fig:grid-search-basic']}, for single agents, with all other step sizes fixed at $\rho_i=0.35$, $\eta_i=0.025$.
• Figure 5: Average logarithmic error with 1000 uniformly random initial conditions from $(-25, 25)^4$. The red curve shows $\rho_i=0.35$ and $\eta_i=0.025$ for $i=1,2,3,4$, and the blue curve corresponds changing $\rho_1$ to $1.05$ and $\eta_1$ to $0.075$.

Theorems & Definitions (10)

• Example 1
• Definition 1
• Definition 2
• Definition 3
• Lemma 1
• proof
• Theorem 1
• proof
• Theorem 2
• proof