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 include 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 $O(p\cdot\log(1/\epsilon) where \epsilon$ 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 gradient descent methods in term of dependency on $p$ (up to a \log^2 p$ factor) Thus our complexity bounds are sharper than the existing bounds by at least a factor of $p/\log^2p$ We also provide a lower bound to confirm that our improved complexity bounds are tight (up to a \log^2 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
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