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This article is devoted to one particular case of using universal acceleratedproximal envelopes to obtain computationally efficient accelerated versions ofmethods used to solve various optimization problem setups. We propose aproximally accelerated coordinate descent method that achieves the efficientalgorithmic complexity of iteration and allows taking advantage of the datasparseness. It was considered an example of applying the proposed approach tooptimizing a SoftMax-like function, for which the described method allowingweaken the dependence of the computational complexity on the dimension $n$ in$\mathcal{O}(\sqrt{n})$ times and, in practice, demonstrates a fasterconvergence in comparison with standard methods. As an example of applying theproposed approach, it was shown a variant of obtaining on its basis someefficient methods for optimizing Markov Decision Processes (MDP) in a minimaxformulation with a Nesterov smoothed target functional.

On the Computational Efficiency of Catalyst Accelerated Coordinate Descent