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 distributed zeroth-order method equipped with a fractional-power powerball map. Unlike existing distributed zeroth-order methods that mainly refine gradient estimation or introduce primal--dual tracking, the proposed mechanism acts as a nonlinear feedback gain on the estimated gradient: it amplifies weak signals in flat regions and attenuates large stochastic estimates without adding transmitted states. Under standard smoothness, oracle-variance, and network-connectivity assumptions, ZOOM-PB achieves the leading nonconvex stationarity rate $\mathcal{O}(\sqrt{p/(nT)})$, where $p$ is the decision dimension, $n$ is the number of agents, and $T$ is the iteration horizon. Under the Polyak--{\L}ojasiewicz condition, it further attains the leading objective residual rate $\mathcal{O}(p/(nT))$. Thus the method preserves the known distributed ZO order while changing the finite-time behavior through a local nonlinear control gain. Simulations on black-box learning and sensor-driven UAV source seeking show faster empirical convergence in weak-signal regimes.