相关梯度：与保守场的联系及应用
Piecewise smooth functions and conservative fields: calculus for nonsmooth nonconvex optimization beyond stratification
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中文总结 AI 辅助
研究局部Lipschitz、分段光滑函数相关梯度与保守场的联系，证明相关集合沿Lipschitz曲线有链式法则性质，调和两个非光滑自动微分理论框架，给出随机次梯度方法在有界性假设下的收敛性。
中文摘要 AI 辅助
在本文中，我们证明了与局部Lipschitz、分段光滑函数的一种表示相关联的梯度是保守场的一种选择。实际上，我们证明了一个包含这些相关梯度的明确定义的集合沿Lipschitz曲线具有链式法则性质。因此，作为其凸包子集的Clarke次微分对于这些函数也继承了该性质。这项工作调和了两个特别处理非光滑自动微分的理论框架。作为副产品，它还证明了一类新的路径可微函数。从算法角度，在有界性假设下，我们给出了随机次梯度方法迭代到保守和Clarke临界点的子序列收敛性以及函数值的收敛性，其中使用相关梯度来驱动动态。
英文摘要
In this paper, we show that the piecewise gradient associated with a representation of a continuous piecewise-$C^{p}$ function is a selection of a conservative field. Specifically, we prove that a set-valued map whose selections include the piecewise gradients, also called associated gradients, has the chain rule property along Lipschitz curves. As a consequence, continuous piecewise smooth functions are path differentiable and their Clarke subdifferentials satisfy the chain rule property. These results establish a connection between a representation-based calculus for nonsmooth automatic differentiation and the conservative-field framework. From an algorithmic perspective, we show that bounded iterates of the stochastic piecewise-gradient method converge to the set of conservative critical points. With an additional Lebesgue-null interface condition, a generic stepsize scaling, and a generic initialization, this result holds for the Clarke critical set. Finally, we demonstrate that, even with the interface condition, the graph of a Lipschitz continuous and piecewise-$C^{\infty}$ function need not admit a $C^1$ stratification satisfying Whitney's condition (a). Thus, the analysis developed here does not fall within the setting of Whitney stratifiable functions (e.g., semialgebraic functions).
