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In this paper we study the well-known greedy coordinate descent (GCD)algorithm to solve $\ell_1$-regularized problems and improve GCD by the twopopular strategies: Nesterov's acceleration and stochastic optimization.Firstly, we propose a new rule for greedy selection based on an $\ell_1$-normsquare approximation which is nontrivial to solve but convex, then an efficientalgorithm called "SOft ThreshOlding PrOjection (SOTOPO)" is proposed to exactlysolve the $\ell_1$-regularized $\ell_1$-norm square approximation problem,which is induced by the new rule. Based on the new rule and the SOTOPOalgorithm, the Nesterov's acceleration and stochastic optimization strategiesare then successfully applied to the GCD algorithm. The resulted algorithmcalled accelerated stochastic greedy coordinate descent (ASGCD) has the optimalconvergence rate $O(\sqrt{1/\epsilon})$, meanwhile, it reduces the iterationcomplexity of greedy selection up to a factor of sample size. Boththeoretically and empirically, we show that ASGCD has better performance forhigh-dimensional and dense problems with sparse solution.

Accelerated Stochastic Greedy Coordinate Descent by Soft Thresholding Projection onto Simplex