On the Computational Efficiency of Catalyst Accelerated Coordinate Descent

TL;DR

This paper introduces a proximally accelerated coordinate descent method that improves computational efficiency and leverages data sparsity, demonstrated on SoftMax-like functions and Markov Decision Processes.

Contribution

It proposes a novel accelerated coordinate descent algorithm using universal proximal envelopes, reducing complexity dependence on data dimension and enhancing practical convergence.

Findings

Achieves $ ilde{O}(rac{1}{ oot n})$ complexity for SoftMax optimization

Demonstrates faster convergence than standard methods in experiments

Provides efficient algorithms for MDP optimization in a minimax setting

Abstract

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