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.

翻译：我们研究了一类大规模场景下的广义线性规划（GLP），其包含简单（可能非光滑）凸正则项与简单凸集约束。通过将（GLP）重构为等价的凸-凹极小极大问题，我们证明可利用问题中的线性结构设计一种高效、可扩展的一阶算法，并命名为"坐标线性方差缩减"（CLVR；发音为"clever"）。CLVR为（GLP）提供了改进的复杂度结果，其复杂度依赖于（GLP）中线性约束矩阵的最大行范数而非谱范数。当正则项与约束可分离时，CLVR可采用高效的惰性更新策略，使得复杂度上界与（GLP）中线性约束矩阵的非零元素数量成比例，而非矩阵维度。另一方面，针对线性规划这一特例，通过利用问题的尖锐性，我们为CLVR提出了一种重启方案以实现经验线性收敛。随后，我们证明基于f-散度和Wasserstein度量的模糊集下的分布鲁棒优化（DRO）问题可通过引入稀疏连接的辅助变量重构为（GLP）。我们通过数值实验验证了算法在实际运行时间与数据遍历次数方面的有效性，对理论保证进行了补充。