Source-linked AI summary
Distributed Multi-Agent Optimization with State-Dependent Communication
Ilan Lobel, Asuman Ozdaglar, Diego Feijer
TL;DR
The paper studies distributed optimization when agents with private objectives and constraints communicate through a state-dependent random network. It analyzes a projected multi-agent subgradient method, showing that constant stepsizes can cause almost-sure divergence, while suitable diminishing stepsizes yield consensus and convergence to a global optimum under stated constraint-set assumptions.
Problem
The paper addresses how agents can solve a global constrained optimization problem using private local objectives and constraints while communication depends on agent states.
Method
The paper analyzes a projected multi-agent subgradient algorithm that averages received estimates, takes local subgradient steps, and projects onto local constraint sets.
Results
Under suitable diminishing stepsize and constraint-set assumptions, agent estimates reach almost sure consensus and converge to the same optimal point of the global optimization problem.
Takeaways & Limitations
State-dependent communication couples information exchange with optimization dynamics, so convergence requires assumptions beyond using a constant stepsize.
Takeaways & Limitations
The negative divergence result concerns constant stepsizes, while convergence results require specific stepsize or constraint-set assumptions.
Abstract
from arXiv · showhide
We study distributed algorithms for solving global optimization problems in which the objective function is the sum of local objective functions of agents and the constraint set is given by the intersection of local constraint sets of agents. We assume that each agent knows only his own local objective function and constraint set, and exchanges information with the other agents over a randomly varying network topology to update his information state. We assume a state-dependent communication model over this topology: communication is Markovian with respect to the states of the agents and the probability with which the links are available depends on the states of the agents. In this paper, we study a projected multi-agent subgradient algorithm under state-dependent communication. The algorithm involves each agent performing a local averaging to combine his estimate with the other agents' estimates, taking a subgradient step along his local objective function, and projecting the estimates on his local constraint set. The state-dependence of the communication introduces significant challenges and couples the study of information exchange with the analysis of subgradient steps and projection errors. We first show that the multi-agent subgradient algorithm when used with a constant stepsize may result in the agent estimates to diverge with probability one. Under some assumptions on the stepsize sequence, we provide convergence rate bounds on a "disagreement metric" between the agent estimates. Our bounds are time-nonhomogeneous in the sense that they depend on the initial starting time. Despite this, we show that agent estimates reach an almost sure consensus and converge to the same optimal solution of the global optimization problem with probability one under different assumptions on the local constraint sets and the stepsize sequence.
1 Introduction
The paper develops distributed optimization for agents with private local objectives and constraints communicating over a randomly varying, state-dependent network. It analyzes a projected multi-agent subgradient algorithm and establishes conditions for consensus and convergence to a global optimum.
- Problem setting: The framework addresses global optimization with an objective formed by summing private local objectives and constraints formed by intersecting local constraint sets.Each agent maintains an estimate using local information and information received from other agents.
- Communication model: Communication is Markovian with respect to agent states, and link availability becomes more likely as communicating agents’ states become closer.This state dependence is motivated by sensing, coordination, and social-belief applications.
- Algorithm: The projected multi-agent subgradient algorithm averages available estimates, takes a local subgradient step, and projects the result onto each agent’s constraint set.Its analysis represents updates using stochastic time-varying transition matrices, subgradients, and projection errors.
- Counterexample: With a constant stepsize, agent estimates and global objective values may diverge with probability one for any constant stepsize value.State-dependent information exchange prevents decoupling stepsize and subgradient effects from transition-matrix convergence.
- Disagreement analysis: The analysis bounds a disagreement metric under sufficiently fast-decreasing stepsizes, with bounds depending on both the initial time and elapsed time.The proof constructs a hierarchy of events tailored to propagate information before agent states drift too far apart.
- Convergence: Under suitable stepsize and constraint-set assumptions, agent estimates reach almost sure consensus and converge to the same optimal solution of the global problem.The results cover common constraint sets and different convex compact local constraint sets.
2 The Model
The model distributes constrained optimization across agents with private local objectives and constraints, using projected subgradient updates over state-dependent Markovian communication. The section also shows that constant stepsizes can cause almost-sure divergence, motivating diminishing stepsizes for convergence.
- 2.1 Optimization Model: Each agent knows only a local convex objective f_i and closed convex constraint set X_i, while the global problem minimizes their summed objective over ∩_i X_i.
- 2.1 Optimization Model: At each iteration, agents average received estimates, take local subgradient steps, and project onto their own constraint sets to maintain feasibility.The update combines neighboring estimates through weights, applies stepsize α(k) along a local subgradient, and projects the result onto X_i.
- 2.2 Network Communication Model: Communication matrices are Markovian on the agents’ state vector, so communication patterns can depend on current estimates and retain temporal correlation.The model uses a random communication matrix A(k), with state dependence represented through a function of the current agent states.
- 2.2 Network Communication Model: Communication probabilities may decrease polynomially as neighboring estimates separate, while the backbone graph remains strongly connected and weights are doubly stochastic with self-confidence.The assumptions also require positive self-weights and describe a lower communication probability when estimates are sufficiently close.
- 2.3 A Counterexample: Under constant stepsize α, some valid problem instances produce agent estimates and objective values that diverge with probability one.The counterexample has common constraint sets, a singleton optimum at 0, and lim_k→∞ x_1(k)=∞ with probability one.
- 2.3 A Counterexample: The negative result is limited to constant stepsizes and C > 1; the paper subsequently establishes solution convergence under appropriate diminishing stepsize assumptions.
3 Analysis of Information Exchange
The paper analyzes how state-dependent communication propagates information and contracts disagreement among agent estimates. It builds probabilistic communication events and derives time-dependent bounds under stepsize and constraint-set assumptions.
- The Disagreement Metric: Transition matrices Φ(k, s) are products of communication matrices and determine how earlier agent information influences later estimates.Because the communication matrices are doubly stochastic, the transition matrices are also doubly stochastic and have entries in [0, 1].
- The Disagreement Metric: The disagreement metric ρ(k, s) measures how completely information obtained at time s−1 is disseminated across agents by time k+1.When transition-matrix entries equal 1/m, information is evenly distributed; small ρ indicates near-uniform dissemination.
- Contraction Bounds: Hi(s) unions an increasing number of G-events so that its occurrence has probability uniformly bounded away from zero and contracts disagreement.This event hierarchy is designed to offset state-dependent communication and ensure repeated opportunities for network-wide information spread.
- Propagation of Information: Sequential activation of directed spanning-tree edges creates G-events of length 2(m −1) that propagate information from any agent to every other agent.The construction uses in-trees and out-trees rooted at a common node, with edge activations ordered along directed paths.
- Contraction Bounds: The expected disagreement decays exponentially in elapsed time, while the contraction bound worsens slowly with the initial time s.The slow dependence on s is quantified by β(s), which grows more slowly than any polynomial s^q for q > 0.
- Contraction Bounds: Contraction bounds require either projection errors bounded proportionally to α(k) or compact local constraint sets.Under compact constraint sets, the contraction analysis can proceed without separately guaranteeing the projection-error bound.
4 Analysis of the Distributed Subgradient Method
The analysis establishes consensus and convergence of projected multi-agent subgradient iterates under state-dependent communication, treating shared and different local constraint sets separately. It also shows that projection errors vanish under compactness assumptions for differing constraint sets.
- Shared constraint sets: For identical closed convex constraint sets, sufficiently decreasing stepsizes yield almost sure consensus and convergence to an optimal solution.The analysis develops these results through bounds on projection errors and disagreement from a common average.
- Proof framework: The convergence arguments rely on projection properties, subgradient boundedness, stochastic weight structure, and supermartingale-based convergence.These ingredients connect the disagreement estimates to almost-sure consensus and optimization convergence.
- Disagreement analysis: The analysis bounds agent disagreement using subgradient bounds, projection errors, and the time-indexed disagreement metric ρ(k, s).The resulting bounds depend on the initial time, reflecting the nonhomogeneous communication analysis.
- Different constraint sets: Under compact differing constraint sets and α(k) → 0, projection errors converge to zero along all sample paths.This enables subsequent mean-consensus and almost-sure convergence results for agents with different local constraint sets.
- Different constraint sets: With differing convex compact constraint sets, vanishing stepsizes support consensus and convergence of every agent estimate to the same optimal solution almost surely.The proof uses vanishing projection errors, boundedness, limit points in the intersection set, and convergence to an optimal point.
5 Conclusions
The paper analyzes distributed optimization over randomly varying networks whose communication probabilities depend on agent states. It proves convergence of agent estimates to almost sure consensus and a global optimum under suitable stepsize and constraint-set assumptions, while identifying broader communication models as future work.
- Model and applications: State-dependent communication models link probabilistic network availability to agent positions or beliefs in sensing, communication, and social applications.The model captures settings where distances between agent states affect communication and information exchange.
- Main results: The analysis provides time-nonhomogeneous convergence-rate bounds for a disagreement metric among agent estimates.The bounds depend on the initial time as well as the elapsed time.
- Main results: With sufficiently decreasing stepsizes, agent estimates reach almost sure consensus and converge to an optimal point of the global optimization problem under constraint-set assumptions.The conclusion covers the projected multi-agent subgradient method studied in the paper.
- Scope and future work: The current communication model assumes link probabilities are continuous functions of the l2 norm of agent states.The authors identify other norms and discontinuous state-dependent probability functions as future extensions.