  We analyse the convergence of the proximal gradient algorithm for convex
composite problems in the presence of gradient and proximal computational
inaccuracies. We derive new tighter deterministic and probabilistic bounds that
we use to verify a simulated (MPC) and a synthetic (LASSO) optimization
problems solved on a reduced-precision machine in combination with an
inaccurate proximal operator. We also show how the probabilistic bounds are
more robust for algorithm verification and more accurate for application
performance guarantees. Under some statistical assumptions, we also prove that
some cumulative error terms follow a martingale property. And conforming to
observations, e.g., in \cite{schmidt2011convergence}, we also show how the
acceleration of the algorithm amplifies the gradient and proximal computational
errors.
