  In this paper we investigate the convergence of a recently popular class of
first-order primal-dual algorithms for saddle point problems under the presence
of errors occurring in the proximal maps and gradients. We study several types
of errors and show that, provided a sufficient decay of these errors, the same
convergence rates as for the error-free algorithm can be established. More
precisely, we prove the (optimal) $O(1/N)$ convergence to a saddle point in
finite dimensions for the class of non-smooth problems considered in this
paper, and prove a $O(1/N^2)$ or even linear $O(\theta^N)$ convergence rate if
either the primal or dual objective respectively both are strongly convex.
Moreover we show that also under a slower decay of errors we can establish
rates, however slower and directly depending on the decay of the errors. We
demonstrate the performance and practical use of the algorithms on the example
of nested algorithms and show how they can be used to split the global
objective more efficiently.
