We initiate the study of nonsmooth optimization problems under bounded local subgradient variation, which postulates bounded difference between (sub)gradients in small local regions around points, in either average or maximum sense. The resulting class of objective functions encapsulates the classes of objective functions traditionally studied in optimization, which are defined based on either Lipschitz continuity of the objective or H\"{o}lder/Lipschitz continuity of its gradient. Further, the defined class contains functions that are neither Lipschitz continuous nor have a H\"{o}lder continuous gradient. When restricted to the traditional classes of optimization problems, the parameters defining the studied classes lead to more fine-grained complexity bounds, recovering traditional oracle complexity bounds in the worst case but generally leading to lower oracle complexity for functions that are not ``worst case.'' Some highlights of our results are that: (i) it is possible to obtain complexity results for both convex and nonconvex problems with the (local or global) Lipschitz constant being replaced by a constant of local subgradient variation and (ii) mean width of the subdifferential set around the optima plays a role in the complexity of nonsmooth optimization, particularly in parallel settings. A consequence of (ii) is that for any error parameter $\epsilon > 0$, parallel oracle complexity of nonsmooth Lipschitz convex optimization is lower than its sequential oracle complexity by a factor $\tilde{\Omega}\big(\frac{1}{\epsilon}\big)$ whenever the objective function is piecewise linear with polynomially many pieces in the input size. This is particularly surprising as existing parallel complexity lower bounds are based on such classes of functions. The seeming contradiction is resolved by considering the region in which the algorithm is allowed to query the objective.

翻译：我们开创性地研究了在有界局部次梯度变分条件下的非光滑优化问题，该条件假设在点附近的小局部区域内，（次）梯度在平均或最大意义下的差异有界。由此产生的目标函数类涵盖了传统优化中研究的目标函数类，这些函数类通常基于目标函数的Lipschitz连续性或其梯度的Hölder/Lipschitz连续性来定义。此外，所定义的函数类包含既非Lipschitz连续也非梯度Hölder连续的函数。当局限于传统优化问题类时，定义所研究类别的参数能带来更精细的复杂度界，在最坏情况下恢复传统的预言机复杂度界，但对于非“最坏情况”的函数通常导致更低的预言机复杂度。我们的部分亮点结果如下：(i) 对于凸和非凸问题，均有可能获得复杂度结果，其中（局部或全局）Lipschitz常数被局部次梯度变分常数所替代；(ii) 最优点附近次微分集的平均宽度在非光滑优化复杂度中发挥作用，尤其在并行设置中。(ii)的一个推论是：对于任意误差参数$\epsilon > 0$，只要目标函数是分段线性函数且分段数量关于输入大小为多项式级，非光滑Lipschitz凸优化的并行预言机复杂度比其串行预言机复杂度低一个因子$\tilde{\Omega}\big(\frac{1}{\epsilon}\big)$。这尤其令人惊讶，因为现有的并行复杂度下界正是基于此类函数。这一看似矛盾的结果通过考虑算法允许查询目标函数的区域而得以解决。