In this paper, we consider a stochastic distributed nonconvex optimization problem with the cost function being distributed over $n$ agents having access only to zeroth-order (ZO) information of the cost. This problem has various machine learning applications. As a solution, we propose two distributed ZO algorithms, in which at each iteration each agent samples the local stochastic ZO oracle at two points with an adaptive smoothing parameter. We show that the proposed algorithms achieve the linear speedup convergence rate $mathcal{O}(sqrt{p/(nT)})$ for smooth cost functions and $mathcal{O}(p/(nT))$ convergence rate when the global cost function additionally satisfies the Polyak--Lojasiewicz (P--L) condition, where $p$ and $T$ are the dimension of the decision variable and the total number of iterations, respectively. To the best of our knowledge, this is the first linear speedup result for distributed ZO algorithms, which enables systematic processing performance improvements by adding more agents. We also show that the proposed algorithms converge linearly when considering deterministic centralized optimization problems under the P--L condition. We demonstrate through numerical experiments the efficiency of our algorithms on generating adversarial examples from deep neural networks in comparison with baseline and recently proposed centralized and distributed ZO algorithms.
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