  In this paper, we study zeroth-order algorithms for nonconvex-concave minimax
problems, which have attracted widely attention in machine learning, signal
processing and many other fields in recent years. We propose a zeroth-order
alternating randomized gradient projection (ZO-AGP) algorithm for smooth
nonconvex-concave minimax problems, and its iteration complexity to obtain an
$\varepsilon$-stationary point is bounded by $\mathcal{O}(\varepsilon^{-4})$,
and the number of function value estimation is bounded by
$\mathcal{O}(d_{x}+d_{y})$ per iteration. Moreover, we propose a zeroth-order
block alternating randomized proximal gradient algorithm (ZO-BAPG) for solving
block-wise nonsmooth nonconvex-concave minimax optimization problems, and the
iteration complexity to obtain an $\varepsilon$-stationary point is bounded by
$\mathcal{O}(\varepsilon^{-4})$ and the number of function value estimation per
iteration is bounded by $\mathcal{O}(K d_{x}+d_{y})$. To the best of our
knowledge, this is the first time that zeroth-order algorithms with iteration
complexity gurantee are developed for solving both general smooth and
block-wise nonsmooth nonconvex-concave minimax problems. Numerical results on
data poisoning attack problem and distributed nonconvex sparse principal
component analysis problem validate the efficiency of the proposed algorithms.
