Source-linked AI summary
Distributed convex optimization via continuous-time coordination algorithms with discrete-time communication
Solmaz S. Kia, Jorge Cortes, Sonia Martinez
TL;DR
The paper addresses distributed optimization of sums of local agent costs when continuous-time coordination must operate with discrete communication. It develops convergence-guaranteed continuous-time algorithms and communication schemes, including periodic and event-triggered implementations, under stated graph and convexity conditions.
Problem
Distributed optimization must combine local cost information across agents while using only neighbor exchanges and accommodating discrete communication constraints.
Method
The paper develops distributed continuous-time coordination algorithms and analyzes periodic, centralized event-triggered, and distributed asynchronous event-triggered communication implementations.
Results
The algorithms achieve exponential or asymptotic convergence under specified convexity and graph conditions, with event-triggered schemes that avoid Zeno behavior.
Takeaways & Limitations
The framework supports distributed network optimization with discrete communication while retaining convergence guarantees across several graph, convexity, and communication settings.
Takeaways & Limitations
The results rely on stated graph, convexity, smoothness, initialization, and communication assumptions, including strongly connected weight-balanced or connected graph settings.
Abstract
from arXiv · showhide
This paper proposes a novel class of distributed continuous-time coordination algorithms to solve network optimization problems whose cost function is a sum of local cost functions associated to the individual agents. We establish the exponential convergence of the proposed algorithm under (i) strongly connected and weight-balanced digraph topologies when the local costs are strongly convex with globally Lipschitz gradients, and (ii) connected graph topologies when the local costs are strongly convex with locally Lipschitz gradients. When the local cost functions are convex and the global cost function is strictly convex, we establish asymptotic convergence under connected graph topologies. We also characterize the algorithm's correctness under time-varying interaction topologies and study its privacy preservation properties. Motivated by practical considerations, we analyze the algorithm implementation with discrete-time communication. We provide an upper bound on the stepsize that guarantees exponential convergence over connected graphs for implementations with periodic communication. Building on this result, we design a provably-correct centralized event-triggered communication scheme that is free of Zeno behavior. Finally, we develop a distributed, asynchronous event-triggered communication scheme that is also free of Zeno with asymptotic convergence guarantees. Several simulations illustrate our results.
1 Introduction
The paper targets distributed convex optimization using local continuous-time dynamics with discrete neighbor communication. It combines continuous-time convergence analysis with communication-saving event-triggered strategies that avoid Zeno behavior.
- Distributed network optimization minimizes a sum of local convex costs, with applications including estimation, learning, and resource allocation.
- Each agent runs purely local continuous-time dynamics while communicating with neighbors at discrete instants.The design is motivated by combining continuous-time analysis with practical real-time communication constraints.
- Continuous-time distributed solvers facilitate classical stability analysis and can provide exponential convergence under regularity assumptions.
- Event-triggered communication methods trade computation and decision making for reduced communication, sensing, or actuator effort.
- The paper seeks continuous-time distributed dynamical systems with discrete communication while guaranteeing correctness and avoiding Zeno behavior.Zeno behavior is an infinite number of triggered events in finite time.
2 Preliminaries
This section establishes notation for vectors, matrices, norms, convexity, Lipschitz continuity, and graph Laplacians used in the distributed optimization analysis.
- The paper defines standard matrix, vector, aggregation, transpose, identity, and Kronecker-product notation for networked systems.Agent-local variables use superscripts, and aggregated network variables concatenate the agents’ vectors.
- Convexity and Lipschitz continuity: Strict convexity requires strictly increasing gradients along distinct points, while m-strong convexity imposes a quadratic lower bound with parameter m.
- Convexity and Lipschitz continuity: A function is M-Lipschitz when output differences are bounded by M times input differences.The section also states a gradient inequality for convex functions with M-Lipschitz gradients.
- Graph concepts: A weight-balanced digraph has matching weighted in-degree and out-degree at every node, although these degrees may differ between nodes.Every undirected graph is weight-balanced.
- Graph concepts: The graph Laplacian has a zero eigenvalue and all remaining eigenvalues have nonnegative real parts.For strongly connected weight-balanced digraphs, zero is a simple eigenvalue of both the Laplacian and its symmetric part.
- Graph concepts: For connected graphs, the eigenvalues of the symmetric Laplacian part equal the corresponding Laplacian eigenvalues.The paper extends the Laplacian and projection notation to variables of dimension d using Kronecker products.
3 Problem Definition
The paper formulates distributed optimization over a network whose global strictly convex objective is the sum of differentiable local costs. Agents must find the unique optimizer using local data and neighbor exchanges while preserving local-cost privacy.
- The network objective is f(x)=Σ_i f_i(x), where each agent i has a differentiable local cost f_i.
- The design goal is for every agent to solve min_x f(x) using only its own local data and information exchanged with neighbors.
- Feasibility and strict convexity imply a unique global optimizer x⋆.
- The paper studies conditions ensuring that no agent can obtain or reconstruct another agent’s local cost function.
4 Distributed Continuous-Time Algorithm for Convex Optimization
The paper introduces a distributed continuous-time coordination algorithm for network optimization and establishes convergence, privacy, and topology-dependent guarantees. The algorithm converges exponentially under several strong-convexity conditions, asymptotically under connected graphs with convex costs and a strictly convex global cost, and supports simplified initialization requirements.
- Algorithm: The proposed algorithm solves distributed minimization of a sum of local cost functions using continuous-time agent dynamics and discrete neighbor communication.Each agent uses local gradient information and exchanges neighboring agents’ x variables.
- Algorithm: Integral feedback corrects the mismatch between local gradients while agents seek agreement on the optimizer.The feedback variable v_i is driven by agent disagreement.
- Convergence: Exponential convergence holds on strongly connected, weight-balanced digraphs under strongly convex local costs with Lipschitz gradients, subject to parameter conditions.The result applies to the algorithm’s specified α, β, and φ conditions and preserves the initialization requirement on the v_i variables.
- Extensions: The analysis covers time-varying interaction topologies, privacy preservation, and initialization advantages, including a zero initialization that satisfies the required sum condition.A passive agent can reconstruct another agent’s local gradient only under specific neighbor, initialization, and adjacency-knowledge conditions.
- Convergence: Exponential convergence also holds over connected graphs with strongly convex costs and locally Lipschitz gradients for any α, β > 0.The guaranteed rate is not uniform unless the local gradients are globally Lipschitz.
- Convergence: For convex local costs and a strictly convex global cost, the algorithm achieves global asymptotic convergence over connected graphs.The proof uses a LaSalle invariance argument.
5 Continuous-time Evolution with Discrete-Time Communication
The paper adapts distributed continuous-time optimization to discrete communication, analyzing periodic, centralized event-triggered, and distributed asynchronous implementations. Under stated graph and cost-function conditions, these schemes provide convergence guarantees while addressing communication efficiency and Zeno behavior.
- Communication implementations: Discrete communication replaces the continuous-communication abstraction, with periodic and event-triggered schemes analyzed over connected graphs.Periodic communication uses synchronous intervals, while event-triggered communication ties transmission times to the network state.
- Periodic communication: Periodic communication guarantees exponential convergence when the synchronous interval satisfies the theorem’s upper bound.The result assumes strongly convex local costs with Lipschitz gradients and initialization satisfying Σ_i v_i(0)=0_d.
- Periodic communication: The periodic-communication proof bounds disagreement growth between transmissions so the Lyapunov function remains decreasing.The analysis studies q = ∥˜z_2:N∥/∥p∥ and selects τ so communication intervals below τ preserve the required Lyapunov decrease.
- Centralized event-triggered communication: A centralized event-triggered scheme avoids using the unavailable optimizer x⋆ by basing triggering on disagreement and communicated network variables.The resulting implementation uses ˜z_2:N and z_2:N rather than p, making the trigger independent of x⋆.
- Centralized event-triggered communication: Imposing a minimum inter-event time τ makes the centralized scheme Zeno-free while retaining exponential convergence.The lower bound on inter-execution times also permits bounds on communication energy.
- Distributed asynchronous communication: The distributed event-triggered scheme uses local variables and triggered neighbor states, yielding asynchronous, Zeno-free executions with asymptotic convergence guarantees.For each agent, inter-event times are lower bounded, and the resulting state converges toward x⋆ under the stated conditions.
6 Simulations
Simulations evaluate the continuous-time algorithm and its discrete-communication implementation on a 10-agent network with strongly convex local costs. The experiments include time-varying balanced digraphs, periodic communication, and comparisons with Euler discretization.
- Simulation setup: A 10-agent network is simulated using strongly convex local cost functions on R.The listed costs include exponential, quadratic, logarithmic, and mixed forms.
- Time-varying topology: The topology alternates every 2 seconds among three strongly connected, weight-balanced digraphs with unitary edge weights.Figure 2 shows one of the digraphs used in the simulations.
- Time-varying topology: Figure 1 shows convergence of algorithm (3) for different β values under the alternating topology.The simulations report convergence as guaranteed by Proposition 4.
- Time-varying topology: Larger β values produce faster convergence in the time-varying-topology simulations.This observation is stated in connection with Remark 2.
- Discrete communication: Figures 3 and 4 evaluate algorithm (18) with periodic communication and compare it with the Euler-discretized implementation of algorithm (3).The supplied captions identify the two performance-evaluation settings.
7 Conclusions
The paper presents continuous-time distributed coordination algorithms for network optimization and establishes convergence under several graph and cost-function conditions. It also studies discrete-time communication, including periodic and event-triggered schemes, with special attention to Zeno-free executions.
- Contributions: The algorithms solve strictly convex network optimization problems whose objective is a sum of local agent cost functions.This is the paper’s stated problem class.
- Continuous-time convergence: Strongly connected weight-balanced interactions yield exponential convergence for strongly convex costs with globally Lipschitz gradients, including dynamic networks preserving those properties.For connected undirected graphs, exponential convergence extends to locally Lipschitz gradients, while convex costs yield asymptotic convergence.
- Discrete communication: Periodic, centralized synchronous, and distributed asynchronous event-triggered communication schemes provide asymptotic convergence guarantees.The conclusion emphasizes the Zeno-free nature of the algorithm executions.
- Future work: Future work includes removing offline design-parameter computation and addressing disturbances, delays, asynchronous clocks, and broader coordination settings.The listed settings include constrained, time-varying, and online scenarios and networked games.