Coordinate-type subgradient methods for addressing nonsmooth optimization problems are relatively underexplored due to the set-valued nature of the subdifferential. In this work, our study focuses on nonsmooth composite optimization problems, encompassing a wide class of convex and weakly convex (nonconvex nonsmooth) problems. By utilizing the chain rule of the composite structure properly, we introduce the Randomized Coordinate Subgradient method (RCS) for tackling this problem class. To the best of our knowledge, this is the first coordinate subgradient method for solving general nonsmooth composite optimization problems. In theory, 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 convergence analysis for RCS in both convex and weakly convex 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 the suboptimality gap when $f$ is convex. For the case when $f$ is weakly convex and its subdifferential satisfies the global metric subregularity property, we derive the $\mathcal{O}(\varepsilon^{-4})$ iteration complexity in expectation. 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. We also provide a convergence lemma and the relationship between the global metric subregularity properties of a weakly convex function and its Moreau envelope. Finally, we conduct several experiments to demonstrate the possible superiority of RCS over the subgradient method.

翻译：由于次微分的集值特性，用于处理非光滑优化问题的坐标类次梯度方法相对未被充分探索。本研究聚焦于非光滑复合优化问题，涵盖一大类凸函数和弱凸函数（非凸非光滑）问题。通过恰当利用复合结构的链式法则，我们提出了随机坐标次梯度方法（RCS）以解决此类问题。据我们所知，这是首个用于求解一般非光滑复合优化问题的坐标次梯度方法。在理论层面，我们考虑了目标函数的线性有界次梯度假设（比传统Lipschitz连续性假设更一般化），以适应实际场景。基于这一广义Lipschitz型假设，我们分别在凸函数和弱凸函数情形下对RCS进行了收敛性分析。具体而言，在$f$为凸函数时，我们建立了关于次优性差距的期望$\widetilde{\mathcal{O}}(1/\sqrt{k})$收敛速率和几乎必然渐近$\tilde o(1/\sqrt{k})$收敛速率；在$f$为弱凸函数且其次微分满足全局度量次正则性条件时，我们推导出期望$\mathcal{O}(\varepsilon^{-4})$迭代复杂度，并建立了渐近收敛性结果。为验证分析中使用的全局度量次正则性条件，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。我们还给出了收敛性引理，并揭示了弱凸函数的全局度量次正则性与其Moreau包络之间的关系。最后，通过多项实验展示了RCS相对于次梯度方法的潜在优越性。