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We analyze the coordinate descent method with a new coordinate selection strategy, called volume sampling. This strategy prescribes selecting subsets of variables of certain size proportionally to the determinants of principal submatrices of the matrix, which bounds the curvature of the objective function. In the particular case when the size of the subsets equals one, volume sampling coincides with the well-known strategy of sampling coordinates proportionally to their Lipschitz constants. For the coordinate descent with volume sampling, we establish the convergence rates for both convex and strongly convex problems. Our theoretical results show that, by increasing the size of the subsets, it is possible to accelerate the method up to the factor which depends on the spectral gap between the corresponding largest eigenvalues of the curvature matrix. Several numerical experiments confirm our theoretical conclusions. [ABSTRACT FROM AUTHOR]

In this paper we prove a new complexity bound for a variant of the accelerated coordinate descent method [Yu. Nesterov, SIAM J. Optim., 22 (2012), pp. 341{362]. We show that this method often outperforms the standard fast gradient methods (FGM [Yu. Nesterov, Dokl. Akad. Nauk SSSR, 269 (1983), pp. 542{547; Math. Program. (A), 103 (2005), pp. 127{152]) on optimization problems with dense data. In many important situations, the computational expenses of oracle and method itself at each iteration of our scheme are perfectly balanced (both depend linearly on dimensions of the problem). As application examples, we consider unconstrained convex quadratic minimization and the problems arising in the smoothing technique [Nesterov, Math. Program. (A), 103 (2005), pp. 127{152]. On some special problem instances, the provable acceleration factor with respect to FGM can reach the square root of the number of variables. Our theoretical conclusions are confirmed by numerical experiments. [ABSTRACT FROM AUTHOR]