Coordinate Linear Variance Reduction for Generalized Linear Programming
arXiv:2111.01842
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 \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 -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.
arXiv comment39 pages, NeurIPS 2022
References in corpus (12)
- Distributionally Robust Logistic Regression
- Does Distributionally Robust Supervised Learning Give Robust Classifiers?
- Stochastic Primal-Dual Coordinate Method for Regularized Empirical Risk Minimization
- A Coordinate Descent Primal-Dual Algorithm with Large Step Size and Possibly Non Separable Functions
- Faster First-Order Primal-Dual Methods for Linear Programming using Restarts and Sharpness
- Variance Reduction for Matrix Games
- Stochastic Variance Reduction for Variational Inequality Methods
- Practical Large-Scale Linear Programming using Primal-Dual Hybrid Gradient
- A First-Order Algorithmic Framework for Wasserstein Distributionally Robust Logistic Regression
- Variance Reduction via Accelerated Dual Averaging for Finite-Sum Optimization
- Fast Distributionally Robust Learning with Variance Reduced Min-Max Optimization
- Random extrapolation for primal-dual coordinate descent