  For strongly convex objectives that are smooth, the classical theory of
gradient descent ensures linear convergence relative to the number of gradient
evaluations. An analogous nonsmooth theory is challenging. Even when the
objective is smooth at every iterate, the corresponding local models are
unstable and the number of cutting planes invoked by traditional remedies is
difficult to bound, leading to convergences guarantees that are sublinear
relative to the cumulative number of gradient evaluations. We instead propose a
multipoint generalization of the gradient descent iteration for local
optimization. While designed with general objectives in mind, we are motivated
by a ``max-of-smooth'' model that captures the subdifferential dimension at
optimality. We prove linear convergence when the objective is itself
max-of-smooth, and experiments suggest a more general phenomenon.
