  We analyse the convergence of an approximate, fully inexact, ADMM algorithm
under additive, deterministic and probabilistic error models. We consider the
generalized ADMM scheme that is derived from generalized Lagrangian penalty
with additive (smoothing) adaptive-metric quadratic proximal perturbations. We
derive explicit deterministic and probabilistic convergence upper bounds for
the lower-C2 nonconvex case as well as the convex case under the Lipschitz
continuity condition. We also present more practical conditions on the proximal
errors under which convergence of the approximate ADMM to a suboptimal solution
is guaranteed with high probability. We consider statistically and
dynamically-unstructured conditional mean independent bounded error sequences.
We validate our results using both simulated and practical software and
algorithmic computational perturbations. We apply the proposed algorithm to a
synthetic LASSO and robust regression with k-support norm regularization
problems and test our proposed bounds under different computational noise
levels. Compared to classical convergence results, the adaptive probabilistic
bounds are more accurate in predicting the distance from the optimal set and
parasitic residual error under different sources of inaccuracies.
