  The cyclic block coordinate descent-type (CBCD-type) methods, which performs
iterative updates for a few coordinates (a block) simultaneously throughout the
procedure, 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 for strongly convex minimization, the
CBCD-type methods attain iteration complexity of
$\mathcal{O}(p\log(1/\epsilon))$, 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 the complexity $\mathcal{O}(\log(1/\epsilon))$ of gradient
descent (GD) 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
$\mathcal{O}(\log^2(p)\cdot\log(1/\epsilon))$ of the CBCD-type methods matches
that of the GD 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 $p/\log^2(p)$. We also provide a lower bound to confirm that our
improved complexity bounds are tight (up to a $\log^2 (p)$ factor), under the
assumption that 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.
