Associated gradients: connection to conservative fields and application

In this paper, we demonstrate that the gradient associated to a representation of a locally Lipschitz, piecewise-smooth function is a selection of a conservative field.

Indeed, we prove that a well-defined set, containing these associated gradients, has the chain rule property along Lipschitz curves.

Consequently, the Clarke subdifferential, as a subset of its convex hull, also inherits this property for these functions.

Ultimately, this work reconciles two theoretical frameworks that address notably nonsmooth automatic differentiation.

As a byproduct, it also proves a new class of path differentiable functions.

From an algorithmic perspective, under a boundedness assumption, we provide the subsequence convergence of the iterates to both conservative and Clarke critical points, as well as the convergence of the function values, for the stochastic subgradient method, where associated gradients are used to drive the dynamic.