Associated gradients: connection to conservative fields and application

In this paper, we show that the gradient associated with a representation of a locally Lipschitz piecewise-smooth function is a selection of a conservative field. Specifically, we prove that a set-valued map whose selections include the associated gradients has the chain rule property along Lipschitz curves. As a consequence, the Clarke subdifferential also satisfies the chain rule for this class of functions. Ultimately, this work reconciles two theoretical frameworks that address, in particular, nonsmooth automatic differentiation. As an important byproduct, it identifies a new class of path-differentiable functions. From an algorithmic perspective, under a boundedness assumption, we prove subsequential convergence of the stochastic subgradient method -- with dynamics driven by associated gradients -- to both conservative and Clarke critical points. In addition, the sequence of function values converges.