Coordinate Linear Variance Reduction for Generalized Linear Programming

Chaobing Song, Cheuk Yin Lin, Stephen Wright, Jelena Diakonikolas

Keywords: Linear Programming, Variance Reduction, Min-max optimization, Distributionally Robust Optimization

Abstract: 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 Coordinate Linear Variance Reduction (CLVR; pronounced ``clever''). 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, 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 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.

TL;DR: We provide a novel variance reduced primal-dual algorithm for generalized linear programs with improved theoretical and empirical performance among primal-dual methods and that is competitive with off-the-shelf solvers on considered datasets.

Community Implementations: [ 1 code implementation](https://www.catalyzex.com/paper/arxiv:2111.01842/code)