An Improved Convergence Analysis of Cyclic Block Coordinate Descent-type Methods for Strongly Convex Minimization

The cyclic block coordinate descent-type (CBCD-type) methods have shown remarkable computational performance for solving strongly convex minimization problems. Typical applications includes many popular statistical machine learning methods such as elastic-net regression, ridge penalized logistic regression, and sparse additive regression. Existing optimization literature has shown that the CBCD-type methods attain iteration complexity of O(p · log(1/ )), where is a pre-specified accuracy of the objective value, and p is the number of blocks. However, such iteration complexity explicitly depends on p, and therefore is at least p times worse than those of gradient descent methods. To bridge this theoretical gap, we propose an improved convergence analysis for the CBCD-type methods. In particular, we first show that for a family of quadratic minimization problems, the iteration complexity of the CBCD-type methods matches that of the GD methods in term of dependency on p (up to a log p factor). Thus our complexity bounds are sharper than the existing bounds by at least a factor of p/ log p. We also provide a lower bound to confirm that our improved complexity bounds are tight (up to a log p factor) if the largest and smallest eigenvalues of the Hessian matrix do not scale with p. Finally, we generalize our analysis to other strongly convex minimization problems beyond quadratic ones.