Establish matching lower bounds for S and I coupling rates

Establish matching lower bounds for the convergence-rate upper bounds of the S coupling, which shares supports but uses independent signs across matched edges, and the I coupling, which uses independent supports and signs across matched edges, in the decentralized zeroth-order matching framework.

Background

The paper derives a correlation-dependent convergence upper bound for three matching-based couplings: G, which shares supports and signs across pairs; S, which shares supports but uses independent signs; and I, which uses independent supports and signs. The analysis shows different rates for these couplings under heterogeneity, but an upper bound alone does not determine whether the stated dependence on the coupling parameters is tight.

A matching lower bound would clarify whether the convergence differences predicted for S and I reflect inherent limitations of these coupling strategies or merely slack in the proof. Such a result would also help assess the sharpness of the paper’s theoretical comparison between correlated and independent sparse directions.