  We study a class of generalized linear programs (GLP) in a large-scale
setting, which includes simple, possibly nonsmooth convex regularizer and
simple convex set constraints. By reformulating (GLP) as an equivalent
convex-concave min-max problem, we show that the linear structure in the
problem can be used to design an efficient, scalable first-order algorithm, to
which we give the name \emph{Coordinate Linear Variance Reduction}
(\textsc{clvr}; pronounced "clever"). \textsc{clvr} yields improved complexity
results for (GLP) that depend on the max row norm of the linear constraint
matrix in (GLP) rather than the spectral norm. When the regularization terms
and constraints are separable, \textsc{clvr} admits an efficient lazy update
strategy that makes its complexity bounds scale with the number of nonzero
elements of the linear constraint matrix in (GLP) rather than the matrix
dimensions. On the other hand, for the special case of linear programs, by
exploiting sharpness, we propose a restart scheme for \textsc{clvr} to obtain
empirical linear convergence. Then we show that Distributionally Robust
Optimization (DRO) problems with ambiguity sets based on both $f$-divergence
and Wasserstein metrics can be reformulated as (GLPs) by introducing sparsely
connected auxiliary variables. We complement our theoretical guarantees with
numerical experiments that verify our algorithm's practical effectiveness, in
terms of wall-clock time and number of data passes.
