  In this paper we propose a class of randomized primal-dual methods to contend
with large-scale saddle point problems defined by a convex-concave function
$\mathcal{L}(\mathbf{x},y)\triangleq\sum_{i=1}^m
f_i(x_i)+\Phi(\mathbf{x},y)-h(y)$. We analyze the convergence rate of the
proposed method under mere convexity and strong convexity assumptions of
$\mathcal{L}$ in $\mathbf{x}$-variable. In particular, assuming
$\nabla_y\Phi(\cdot,\cdot)$ is Lipschitz and $\nabla_\mathbf{x}\Phi(\cdot,y)$
is coordinate-wise Lipschitz for any fixed $y$, the ergodic sequence generated
by the algorithm achieves the convergence rate of $\mathcal{O}(M/k)$ in the
expected primal-dual gap. Furthermore, assuming that $\mathcal{L}(\cdot,y)$ is
strongly convex for any $y$, and that $\Phi(\mathbf{x},\cdot)$ is affine for
any $\mathbf{x}$, the scheme enjoys a faster rate of $\mathcal{O}(M/k^2)$ in
terms of primal solution suboptimality. We implemented the proposed algorithmic
framework to solve kernel matrix learning problem, and tested it against other
state-of-the-art first-order methods
