Proximal Algorithms are known to be popular in solving non-smooth convex loss minimization framework due to their low iteration costs and good performance. Convergence rate analysis is an essential part in the process of designing new proximal methods. In this paper, we present a viscosity-approximation-based proximal gradient algorithm and prove its linear convergence rate. We also present its accelerated variant and discuss the condition for the improved convergence rate. These algorithms are applied to solve the problem of multiclass image classification problem. CIFAR10, a popular publicly available benchmark real image classification dataset is used to experimentally validate our theoretical proofs, and the classification performances are compared with that of the state-of-the-art algorithms. To the best of our knowledge, it is the first time that the viscosity-approximation concept is applied to a multiclass classification problem.