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A Unifying System Theory Framework for Distributed Optimization and Games

Guido Carnevale, Nicola Mimmo, Giuseppe Notarstefano

TL;DR

This work develops a unifying, system-theoretic framework for distributed optimization and games by extracting a global aggregation $\alpha(\chi)$ from a centralized method and emulating it with a consensus-based proxy. Interpreting the interconnection as a Singularly Perturbed (SP) system, it provides a general convergence theorem that shows distributed schemes inherit centralized convergence properties, with exponential/linear convergence in the unique-solution case. A key contribution is a novel distributed algorithm for constraint-coupled problems that achieves linear convergence, derived from a centralized augmented-Lagrangian approach and implemented via dynamic average consensus. The framework thus enables systematic design and analysis of a wide range of distributed algorithms across optimization and game-theoretic scenarios, including constraint coupling and aggregative structures, with practical convergence guarantees.

Abstract

This paper introduces a systematic methodological framework to design and analyze distributed algorithms for optimization and games over networks. Starting from a centralized method, we identify an aggregation function involving all the decision variables (e.g., a global cost gradient or constraint) and introduce a distributed consensus-oriented scheme to asymptotically approximate the unavailable information at each agent. Then, we delineate the proper methodology for intertwining the identified building blocks, i.e., the optimization-oriented method and the consensus-oriented one. The key intuition is to interpret the obtained interconnection as a singularly perturbed system. We rely on this interpretation to provide sufficient conditions for the building blocks to be successfully connected into a distributed scheme exhibiting the convergence guarantees of the centralized algorithm. Finally, we show the potential of our approach by developing a new distributed scheme for constraint-coupled problems with a linear convergence rate.

A Unifying System Theory Framework for Distributed Optimization and Games

TL;DR

This work develops a unifying, system-theoretic framework for distributed optimization and games by extracting a global aggregation from a centralized method and emulating it with a consensus-based proxy. Interpreting the interconnection as a Singularly Perturbed (SP) system, it provides a general convergence theorem that shows distributed schemes inherit centralized convergence properties, with exponential/linear convergence in the unique-solution case. A key contribution is a novel distributed algorithm for constraint-coupled problems that achieves linear convergence, derived from a centralized augmented-Lagrangian approach and implemented via dynamic average consensus. The framework thus enables systematic design and analysis of a wide range of distributed algorithms across optimization and game-theoretic scenarios, including constraint coupling and aggregative structures, with practical convergence guarantees.

Abstract

This paper introduces a systematic methodological framework to design and analyze distributed algorithms for optimization and games over networks. Starting from a centralized method, we identify an aggregation function involving all the decision variables (e.g., a global cost gradient or constraint) and introduce a distributed consensus-oriented scheme to asymptotically approximate the unavailable information at each agent. Then, we delineate the proper methodology for intertwining the identified building blocks, i.e., the optimization-oriented method and the consensus-oriented one. The key intuition is to interpret the obtained interconnection as a singularly perturbed system. We rely on this interpretation to provide sufficient conditions for the building blocks to be successfully connected into a distributed scheme exhibiting the convergence guarantees of the centralized algorithm. Finally, we show the potential of our approach by developing a new distributed scheme for constraint-coupled problems with a linear convergence rate.
Paper Structure (15 sections, 4 theorems, 124 equations, 4 figures, 1 algorithm)

This paper contains 15 sections, 4 theorems, 124 equations, 4 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Formulation and Preliminaries
  3. Unifying Theory for distributed algorithms: Systematic Design and Formal Guarantees
  4. Centralized Meta-Algorithm for Optimization and Games
  5. Naive Double Loop Distributed Algorithm
  6. Distributed Meta-Algorithm for Optimization and Games
  7. Building Blocks for Distributed Optimization and Games Setups
  8. Centralized Methods for Optimization and Games
  9. Dynamic Average Consensus
  10. Novel Distributed Algorithm for Constraint-Coupled Optimization
  11. Distributed Algorithm Design
  12. Numerical Simulations
  13. Conclusions
  14. A LaSalle's Invariance Principle for SP Systems
  15. Composite Dynamic Average Consensus

Key Result

Theorem 1

Consider eq:global_update and let Assumptions ass:centralized_equilibria, ass:centralized_convergence, ass:orthogonality, ass:equilibria_consensus, and ass:tracking hold for a suitable $\mathcal{T}$ as in eq:change_of_variables. Then, there exists $\bar{\delta} \in (0,1)$, such that, for all $\delta If further Assumption ass:additional_assumption holds true, then there exist $r_1, r_2, r_3 > 0$ su

Figures (4)

  • Figure 1: Centralized (left) and distributed (right) architectures.
  • Figure 2: Block diagram of \ref{['eq:global_update']}.
  • Figure 3: Algorithm \ref{['eq:new_distributed_algorithm']}: evolution over iterations $t$ of the quantities $\left \|\chi^t - \chi^\star \right \|$ and $\left \|\mathcal{T}_\perp(\mathrm{z}^t) - \mathrm{z}_\perp^{\text{eq}}(\chi^t) \right \|$.
  • Figure 4: Comparison between the centralized method \ref{['eq:parallel_for_cc']} and its distributed counterpart with different values of $\delta$.

Theorems & Definitions (14)

  • Definition 1: GNE FacchineiKanzowGNE2010
  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • ...and 4 more