Automatic differentiation of nonsmooth iterative algorithms

Jerome Bolte, Edouard Pauwels, Samuel Vaiter

Keywords: Fixed point solvers, nonsmooth algorithms, automatic differentiation, unrolling, piggy back differentiation, conservative gradients.

TL;DR: We describe the asymptotics of nonsmooth derivatives obtained by automatic differentiation of fixed point algorithms and show that they converge to classical derivatives in a generic sense.

Abstract: 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 contractivity 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.