In this work, we propose the {Randomized Coordinate Subgradient method} (RCS) for solving nonsmooth convex and nonsmooth nonconvex (nonsmooth weakly convex) optimization problems. RCS randomly selects one block coordinate to update at each iteration, making it more practical than updating all coordinates. We consider the linearly bounded subgradients assumption for the objective function, which is more general than the traditional Lipschitz continuity assumption, to account for practical scenarios. We then conduct thorough convergence analysis for RCS in both convex and nonconvex cases based on this generalized Lipschitz-type assumption. Specifically, we establish the $\widetilde{\mathcal{O}}(1/\sqrt{k})$ convergence rate in expectation and the $\tilde o(1/\sqrt{k})$ almost sure asymptotic convergence rate in terms of suboptimality gap when $f$ is nonsmooth convex. If $f$ further satisfies the global quadratic growth condition, the improved $\mathcal{O}(1/k)$ rate is shown in terms of the squared distance to the optimal solution set. For the case when $f$ is nonsmooth weakly convex and its subdifferential satisfies the global metric subregularity property, we derive the $\mathcal{O}(1/T^{1/4})$ iteration complexity in expectation, where $T$ is the total number of iterations. We also establish an asymptotic convergence result. To justify the global metric subregularity property utilized in the analysis, we establish this error bound condition for the concrete (real valued) robust phase retrieval problem, which is of independent interest. We provide a convergence lemma and the relationship between the global metric subregularity properties of a weakly convex function and its Moreau envelope, which are also of independent interest. Finally, we conduct several experiments to demonstrate the possible superiority of RCS over the subgradient method.

翻译：本文提出随机坐标次梯度方法（RCS）求解非光滑凸与非光滑非凸（非光滑弱凸）优化问题。RCS每次迭代随机选取一个块坐标进行更新，使其比全坐标更新更具实用性。我们采用目标函数的线性有界次梯度假设替代传统的Lipschitz连续性假设，以更贴合实际场景。基于该广义Lipschitz型假设，我们针对RCS在凸与非凸情形进行了严格的收敛性分析。具体而言，当$f$为非光滑凸函数时，我们建立了关于次优性差距的$\widetilde{\mathcal{O}}(1/\sqrt{k})$期望收敛速率和$\tilde o(1/\sqrt{k})$几乎必然渐近收敛速率；若$f$进一步满足全局二次增长条件，则证明关于到最优解集平方距离的$\mathcal{O}(1/k)$改进速率。当$f$为非光滑弱凸函数且其次微分满足全局度量次正则性性质时，我们推导出期望意义下的$\mathcal{O}(1/T^{1/4})$迭代复杂度，其中$T$为总迭代次数，并建立了渐近收敛结果。为验证分析中使用的全局度量次正则性性质，我们针对具体的（实值）鲁棒相位恢复问题建立了该误差界条件，该结果具有独立研究价值。我们给出了收敛引理及弱凸函数与其Moreau包络的全局度量次正则性性质之间的关系，这些结论同样具有独立研究价值。最后，通过系列实验展示了RCS相较于次梯度方法的潜在优越性。