NysADMM: faster composite convex optimization via low-rank approximation

TL;DR

NysADMM is a scalable algorithm that accelerates convex optimization by using low-rank approximations to improve ADMM efficiency, demonstrating significant speedups on real-world datasets.

Contribution

The paper introduces NysADMM, a novel low-rank approximation-based method that enhances ADMM's speed and scalability with strong theoretical guarantees.

Findings

NysADMM solves optimization problems in half the time of standard solvers.

It effectively handles applications like lasso, logistic regression, and SVMs.

The method outperforms traditional solvers across various statistical learning tasks.

Abstract

This paper develops a scalable new algorithm, called NysADMM, to minimize a smooth convex loss function with a convex regularizer. NysADMM accelerates the inexact Alternating Direction Method of Multipliers (ADMM) by constructing a preconditioner for the ADMM subproblem from a randomized low-rank Nystr\"om approximation. NysADMM comes with strong theoretical guarantees: it solves the ADMM subproblem in a constant number of iterations when the rank of the Nystr\"om approximation is the effective dimension of the subproblem regularized Gram matrix. In practice, ranks much smaller than the effective dimension can succeed, so NysADMM uses an adaptive strategy to choose the rank that enjoys analogous guarantees. Numerical experiments on real-world datasets demonstrate that NysADMM can solve important applications, such as the lasso, logistic regression, and support vector machines, in half the time (or less) required by standard solvers. The breadth of problems on which NysADMM beats standard solvers is a surprise: it suggests that ADMM is a dominant paradigm for numerical optimization across a wide range of statistical learning problems that are usually solved with bespoke methods.