  Goldstein's 1977 idealized iteration for minimizing a Lipschitz objective
fixes a distance - the step size - and relies on a certain approximate
subgradient. That "Goldstein subgradient" is the shortest convex combination of
objective gradients at points within that distance of the current iterate. A
recent implementable Goldstein-style algorithm allows a remarkable complexity
analysis (Zhang et al. 2020), and a more sophisticated variant (Davis and
Jiang, 2022) leverages typical objective geometry to force near-linear
convergence. To explore such methods, we introduce a new modulus, based on
Goldstein subgradients, that robustly measures the slope of a Lipschitz
function. We relate near-linear convergence of Goldstein-style methods to
linear growth of this modulus at minimizers. We illustrate the idea
computationally with a simple heuristic for Lipschitz minimization.
