  In this paper, we improve upon our previous work[24,22] and establish
convergence bounds on the objective function values of approximate
proximal-gradient descent (AxPGD), approximate accelerated proximal-gradient
descent (AxAPGD) and approximate proximal ADMM (AxWLM-ADMM) schemes. We
consider approximation errors that manifest rare extreme events and we
propagate their effects through iterations. We establish probabilistic
asymptotic and non-asymptotic convergence bounds as functions of the range
(upper/lower bounds) and variance of approximation errors. We use the derived
bound to assess AxPGD in a sparse model predictive control of a spacecraft
system and compare its accuracy with previously derived bounds.
