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Zeroth-Order Feedback Optimization in Multi-Agent Systems: Tackling Coupled Constraints

Yingpeng Duan, Yujie Tang

TL;DR

This paper investigates distributed zeroth-order feedback optimization in multi-agent systems with coupled constraints, and employs two-point zeroth-order gradient estimation with delayed information to construct stochastic gradients and uses the constraint extrapolation technique and the averaging consensus framework to effectively handle the coupled constraints.

Abstract

This paper investigates distributed zeroth-order feedback optimization in multi-agent systems with coupled constraints, where each agent operates its local action vector and observes only zeroth-order information to minimize a global cost function subject to constraints in which the local actions are coupled. Specifically, we employ two-point zeroth-order gradient estimation with delayed information to construct stochastic gradients, and leverage the constraint extrapolation technique and the averaging consensus framework to effectively handle the coupled constraints. We also provide convergence rate and oracle complexity results for our algorithm, characterizing its computational efficiency and scalability by rigorous theoretical analysis. Numerical experiments are conducted to validate the algorithm's effectiveness.

Zeroth-Order Feedback Optimization in Multi-Agent Systems: Tackling Coupled Constraints

TL;DR

This paper investigates distributed zeroth-order feedback optimization in multi-agent systems with coupled constraints, and employs two-point zeroth-order gradient estimation with delayed information to construct stochastic gradients and uses the constraint extrapolation technique and the averaging consensus framework to effectively handle the coupled constraints.

Abstract

This paper investigates distributed zeroth-order feedback optimization in multi-agent systems with coupled constraints, where each agent operates its local action vector and observes only zeroth-order information to minimize a global cost function subject to constraints in which the local actions are coupled. Specifically, we employ two-point zeroth-order gradient estimation with delayed information to construct stochastic gradients, and leverage the constraint extrapolation technique and the averaging consensus framework to effectively handle the coupled constraints. We also provide convergence rate and oracle complexity results for our algorithm, characterizing its computational efficiency and scalability by rigorous theoretical analysis. Numerical experiments are conducted to validate the algorithm's effectiveness.

Paper Structure

This paper contains 12 sections, 13 theorems, 144 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Our Contributions
  3. Problem Formulation
  4. Algorithm Design
  5. Convergence Analysis
  6. Numerical Experiments
  7. Conclusion
  8. Supplementary Materials
  9. Some Auxiliary Lemmas
  10. A Critical Lemma for Establishing Convergence
  11. Building Blocks for Convergence Rate and Constraint Violation Analysis
  12. Bounds for Certain Quantities in Convergence Rate and Constraint Violation Analysis

Key Result

Theorem 1

Suppose Assumptions assumption_convex, assumption_smooth and assumption_lipschitz hold. Let $y_0^i = 0$ and $x_0^i = 0$, and set the parameters of Algorithm 1 as follows: $\theta_t=1$, $\gamma_t = 1$, $\frac{1}{\eta_t} = L_0 + L_{\max} + \frac{1}{\eta}$ and $\mu_t = \mu$ with where Then, we have and where $\bar{x}_T = \sum_{t=0}^{T-1} \gamma_t x_{t+1} / \sum_{t=0}^{T-1}\gamma_t$, and $[\cdot]_

Figures (3)

  • Figure 1: Convergence of Algorithm \ref{['alg:main']} on the numerical test case with constant step sizes $\eta_t=\mu_t=1/500$.
  • Figure 2: Convergence of Algorithm \ref{['alg:main']} on the numerical test case with constant step sizes $\eta_t=\mu_t=1/200$.
  • Figure 3: Convergence of Algorithm \ref{['alg:main']} on the numerical test case with diminishing step sizes $\eta_t=\mu_t=1/(\sqrt{t}+300)$.

Theorems & Definitions (26)

  • Remark 1
  • Remark 2
  • Theorem 1
  • Corollary 1
  • Remark 3
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • ...and 16 more