Nonlinear-Gain Distributed Zeroth-Order Optimization for Networked Black-Box Control

This letter studies distributed stochastic optimization over a peer-to-peer network when agents can query only zeroth-order function values. We propose ZOOM-PB, a coordinate-sampling method that blends each local ZO estimate with a fractional-power response while maintaining only a primal state. The raw estimate is retained as a linear anchor, and the nonlinear mixing weight is coupled to the optimization stepsize. This design is motivated by a basic obstruction: transforming heterogeneous or noisy local estimates before averaging can reverse the network direction. We bound that nonlinear residual directly from the raw oracle assumptions instead of imposing an aggregate-alignment condition. With a smooth stochastic-function oracle and a connected graph, ZOOM-PB attains the nonconvex stationarity order $\mathcal{O}(\sqrt{p/(nT)})$ and a Polyak--{\L}ojasiewicz statistical term of order $\mathcal{O}(p/(nT))$, after an explicit initialization transient. Numerical examples compare ZOOM-PB with seven distributed ZO baselines under matched query and message budgets.