[Paper Review] Almost sure convergence of a randomized algorithm for relative localization in sensor networks

Chiara Ravazzi, Paolo Frasca+2 more|arXiv (Cornell University)|Mar 12, 2013

Distributed Control Multi-Agent Systems14 references3 citations

TL;DR

This paper proposes a fully distributed, asynchronous randomized gossip algorithm for relative localization in sensor networks. By leveraging ergodic theory, it proves that the time-average of node states almost surely converges to the least-squares solution, eliminating persistent oscillations without requiring a global clock or synchronization.

ABSTRACT

This paper regards the relative localization problem in sensor networks. We study a randomized algorithm, which is based on input-driven consensus dynamics and involves pairwise "gossip" communications and updates. Due to the randomness of the updates, the state of this algorithm ergodically oscillates around a limit value. Exploiting the ergodicity of the dynamics, we show that the time-average of the state almost surely converges to the least-squares solution of the localization problem. Remarkably, the computation of the time-average does not require the sensors to share any common clock. Hence, the proposed algorithm is fully distributed and asynchronous.

Motivation & Objective

• Address the challenge of distributed relative localization in sensor networks without centralized coordination or global synchronization.
• Develop a fully asynchronous algorithm that enables nodes to estimate relative positions using only local, noisy pairwise measurements.
• Overcome persistent random oscillations in standard gossip dynamics by introducing a time-average operation.
• Establish almost sure convergence of the time-averaged state to the optimal least-squares solution using ergodic theory.
• Enable practical deployment by eliminating the need for a global clock or iteration counter across nodes.

Proposed method

• Formulate the relative localization problem as a least-squares estimation task over a connected graph with noisy edge measurements.
• Design a randomized gossip algorithm where at each step, a random neighboring pair of nodes updates their states using a consensus-like rule.
• Model the state evolution as a Markov process driven by i.i.d. random matrices and noise vectors.
• Analyze the backward process using Lyapunov exponent conditions to establish almost sure convergence of the invariant distribution.
• Prove ergodicity of the forward process and show that the time-average of states converges almost surely to the expected value of the invariant distribution.
• Use the law of large numbers on a stationary version of the process to establish convergence of time-averaged estimates to the optimal solution.

Experimental results

Research questions

• RQ1 Can a fully distributed, asynchronous gossip algorithm achieve almost sure convergence to the optimal solution in relative localization?
• RQ2 How can persistent oscillations in randomized consensus dynamics be removed without global synchronization?
• RQ3 What conditions ensure the existence and uniqueness of an invariant distribution for the randomized update process?
• RQ4 Can time-averaging be used to achieve convergence without relying on a global iteration counter or clock?
• RQ5 How does ergodic theory enable convergence analysis in non-convergent, oscillatory randomized dynamics?

Key findings

• The time-average of the node states almost surely converges to the least-squares solution of the relative localization problem.
• The convergence is established using ergodic theory, without requiring a global clock or iteration counter.
• The invariant distribution of the Markov process governing the algorithm is unique and corresponds to the optimal solution.
• The expected value of the limiting state distribution equals the least-squares solution: E[x∞] = L†A⊤b.
• The proof relies on bounding the Lyapunov exponent of the product of random matrices, showing it is negative, which implies almost sure convergence of the backward process.
• The algorithm is fully distributed and asynchronous, with no need for coordination or synchronization among nodes.