The subgradient method is one of the most fundamental algorithmic schemes for nonsmooth optimization. The existing complexity and convergence results for this algorithm are mainly derived for Lipschitz continuous objective functions. In this work, we first extend the typical complexity results for the subgradient method to convex and weakly convex minimization without assuming Lipschitz continuity. Specifically, we establish $\mathcal{O}(1/\sqrt{T})$ bound in terms of the suboptimality gap ``$f(x) - f^*$'' for convex case and $\mathcal{O}(1/{T}^{1/4})$ bound in terms of the gradient of the Moreau envelope function for weakly convex case. Furthermore, we provide convergence results for non-Lipschitz convex and weakly convex objective functions using proper diminishing rules on the step sizes. In particular, when $f$ is convex, we show $\mathcal{O}(\log(k)/\sqrt{k})$ rate of convergence in terms of the suboptimality gap. With an additional quadratic growth condition, the rate is improved to $\mathcal{O}(1/k)$ in terms of the squared distance to the optimal solution set. When $f$ is weakly convex, asymptotic convergence is derived. The central idea is that the dynamics of properly chosen step sizes rule fully controls the movement of the subgradient method, which leads to boundedness of the iterates, and then a trajectory-based analysis can be conducted to establish the desired results. To further illustrate the wide applicability of our framework, we extend the complexity results to the truncated subgradient, the stochastic subgradient, the incremental subgradient, and the proximal subgradient methods for non-Lipschitz functions.

翻译：次梯度方法是解决非光滑优化问题最基础的算法框架之一。现有关于该算法的复杂度与收敛性结果主要针对Lipschitz连续目标函数推导。本文首先将次梯度方法的典型复杂度结果扩展到无需假设Lipschitz连续性的凸优化与弱凸优化问题。具体而言，针对凸情形建立了关于次优性间隙“$f(x) - f^*$”的$\mathcal{O}(1/\sqrt{T})$界，针对弱凸情形建立了关于Moreau包络函数梯度的$\mathcal{O}(1/{T}^{1/4})$界。此外，通过采用适当递减的步长规则，我们为非Lipschitz凸函数与弱凸目标函数提供了收敛性结果。特别地，当$f$为凸函数时，我们证明了关于次优性间隙的$\mathcal{O}(\log(k)/\sqrt{k})$收敛速率。在附加二次增长条件的情况下，关于到最优解集平方距离的收敛速率改进至$\mathcal{O}(1/k)$。当$f$为弱凸函数时，推导了渐近收敛性。核心思想在于：适当选择的步长规则动态完全控制了次梯度方法的迭代轨迹，从而保证迭代点有界性，进而可通过基于轨迹的分析方法建立所需结果。为进一步展示本文框架的广泛适用性，我们将复杂度结果扩展到适用于非Lipschitz函数的截断次梯度法、随机次梯度法、增量次梯度法以及近端次梯度法。