  We consider randomized block coordinate stochastic mirror descent (RBSMD)
methods for solving high-dimensional stochastic optimization problems with
strongly convex objective functions. Our goal is to develop RBSMD schemes that
achieve a rate of convergence with a minimum constant factor with respect to
the choice of the stepsize sequence. To this end, we consider both subgradient
and gradient RBSMD methods addressing nonsmooth and smooth problems,
respectively. For each scheme, (i) we develop self-tuned stepsize rules
characterized in terms of problem parameters and algorithm settings; (ii) we
show that the non-averaging iterate generated by the underlying RBSMD method
converges to the optimal solution both in an almost sure and a mean sense;
(iii) we show that the mean squared error is minimized. When problem parameters
are unknown, we develop a unifying self-tuned update rule that can be applied
in both subgradient and gradient SMD methods, and show that for any arbitrary
and small enough initial stepsize, a suitably defined error bound is minimized.
We provide constant factor comparisons with standard SMD and RBSMD methods. Our
numerical experiments performed on an SVM model display that the self-tuned
schemes are significantly robust with respect to the choice of problem
parameters, and the initial stepsize.
