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：我们首次研究了在局部次梯度有界变差条件下的非光滑优化问题。该条件假设在点的局部小邻域内，(次)梯度之间的差异（在平均或最大意义下）是有界的。由此定义的目标函数类别涵盖了传统优化研究中基于目标函数Lipschitz连续性或其梯度H\"{o}lder/Lipschitz连续性定义的各类目标函数。此外，所定义的类别还包含既非Lipschitz连续又不具有H\"{o}lder连续梯度的函数。当限制在传统优化问题类别时，定义所研究类别的参数可导出更精细的复杂度界限：在最坏情况下恢复传统预言机复杂度界限，但对于非"最坏情况"函数通常能获得更低的预言机复杂度。我们研究结果的主要亮点包括：(i) 对于凸与非凸问题，均可用局部次梯度变差常数替代（局部或全局）Lipschitz常数来获得复杂度结果；(ii) 最优解附近次微分集合的平均宽度在非光滑优化复杂度（特别是并行设置中）起着关键作用。由(ii)导出的一个重要结论是：对于任意误差参数$\epsilon > 0$，当目标函数是输入规模中多项式分段数量的分段线性函数时，非光滑Lipschitz凸优化的并行预言机复杂度比其顺序预言机复杂度低$\tilde{\Omega}\big(\frac{1}{\epsilon}\big)$因子。这一发现尤为令人惊讶，因为现有的并行复杂度下界正是基于此类函数构建的。通过考虑算法允许查询目标函数的区域，这一表面矛盾得以解决。