ZO-COSMO: Index-Free One-Hop Mixing for Decentralized Zeroth-Order Optimization

Sparse communication in decentralized zeroth-order learning requires compatible peer-state coordinates. We characterize this one-hop condition and develop textsfZO-COSMO, coupling two-query estimation with average-preserving masked consensus using q values per active link. Global supports serve all-neighbor mixing; matching updates require agreement only within each pair. We derive a sharp contraction-per-scalar bound within the matching class and convergence guarantees for the core and sparse-momentum updates. At fixed matching, exact moment identities characterize how shared directions preserve gradient-heterogeneity cancellation and redistribute estimation error and disagreement. Mechanism experiments cover unequal curvatures, noise, and sparse momentum. Further tests span 64 synthetic agents and eight logical Qwen LoRA workers. At matched payload budgets, Qwen2-7B QNLI gains 3.65 accuracy points over explicit-index Rand-k; edge-local updates gain 3.42 and 2.53 points over all-neighbor mixing on eight-worker complete and ring graphs. A matched-first-step ablation gives a 3.92-point momentum benefit. Seed-aware and same-matching controls distinguish encoding, scheduling, and query correlation.