On Inhomogeneous Infinite Products of Stochastic Matrices and Applications
CoRR(2024)
摘要
With the growth of magnitude of multi-agent networks, distributed
optimization holds considerable significance within complex systems.
Convergence, a pivotal goal in this domain, is contingent upon the analysis of
infinite products of stochastic matrices (IPSMs). In this work, convergence
properties of inhomogeneous IPSMs are investigated. The convergence rate of
inhomogeneous IPSMs towards an absolute probability sequence π is derived.
We also show that the convergence rate is nearly exponential, which coincides
with existing results on ergodic chains. The methodology employed relies on
delineating the interrelations among Sarymsakov matrices, scrambling matrices,
and positive-column matrices. Based on the theoretical results on inhomogeneous
IPSMs, we propose a decentralized projected subgradient method for time-varying
multi-agent systems with graph-related stretches in (sub)gradient descent
directions. The convergence of the proposed method is established for convex
objective functions, and extended to non-convex objectives that satisfy
Polyak-Lojasiewicz conditions. To corroborate the theoretical findings, we
conduct numerical simulations, aligning the outcomes with the established
theoretical framework.
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