Derivative-free Alternating Projection Algorithms for General Nonconvex-Concave Minimax Problems
This addresses optimization challenges in machine learning and signal processing by providing the first zeroth-order methods with guaranteed complexity for these problems, though it is incremental as it extends existing gradient-based approaches to derivative-free settings.
The paper tackles nonconvex-concave minimax problems by proposing zeroth-order algorithms, achieving an iteration complexity of O(ε⁻⁴) for ε-stationary points and reducing function evaluations to O(d_x + d_y) per iteration for smooth cases and O(K d_x + d_y) for block-wise nonsmooth cases.
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.