  Differentiation along algorithms, i.e., piggyback propagation of derivatives,
is now routinely used to differentiate iterative solvers in differentiable
programming. Asymptotics is well understood for many smooth problems but the
nondifferentiable case is hardly considered. Is there a limiting object for
nonsmooth piggyback automatic differentiation (AD)? Does it have any
variational meaning and can it be used effectively in machine learning? Is
there a connection with classical derivative? All these questions are addressed
under appropriate nonexpansivity conditions in the framework of conservative
derivatives which has proved useful in understanding nonsmooth AD. For
nonsmooth piggyback iterations, we characterize the attractor set of nonsmooth
piggyback iterations as a set-valued fixed point which remains in the
conservative framework. This has various consequences and in particular almost
everywhere convergence of classical derivatives. Our results are illustrated on
parametric convex optimization problems with forward-backward, Douglas-Rachford
and Alternating Direction of Multiplier algorithms as well as the Heavy-Ball
method.
