Determine the sharpness of the full spectral dependence

Determine whether the full spectral dependence, including the squared spectral-gap penalty in the convergence bound for sparse one-step partial consensus, is sharp for the ZO-COSMO decentralized zeroth-order optimization method.

Background

The convergence analysis of ZO-COSMO produces a network-dependent penalty involving the factor $(1- ho^2)^{-2}$, where $ho$ is the spectral radius of the centered mixing matrix. The appendix proves that the sparse activation factor $q/d$ is unavoidable for one-step partial coordinate mixing, but the argument does not establish that the complete spectral dependence is optimal.

Resolving this issue would require either a matching lower bound or a sharper analysis that improves the spectral dependence. This would distinguish intrinsic communication and topology limitations from artifacts introduced by Young’s inequality and the treatment of disagreement injection.