We consider an accelerated proximal gradient algorithm for the composite optimization with “independent errors” (errors little related with historical information) for solving linear inverse problems. We present a new inexact version of FISTA algorithm considering deterministic and stochastic noises. We prove some convergence rates of the algorithm and we connect it with the current existing catalyst framework for many algorithms in machine learning. We show that a catalyst can be regarded as a special case of the FISTA algorithm where the smooth part of the function vanishes. Our framework gives a more generic formulation that provides convergence results for the deterministic and stochastic noise cases and also to the catalyst framework. Some of our results provide simpler alternative analysis of some existing results in literature, but they also extend the results to more generic situations.