Quantitative Convergence of Quadratically Regularized Linear Programs

TL;DR

This paper provides a quantitative analysis of how solutions to quadratically regularized linear programs converge to the minimal-norm solution, including explicit thresholds and bounds, with applications to optimal transport.

Contribution

It offers the first explicit thresholds and bounds for convergence of quadratically regularized linear programs, enhancing understanding of their behavior in optimal transport.

Findings

Explicit regularization threshold for solution convergence

Bounds on suboptimality before threshold

Convergence rate in the large regularization regime

Abstract

Linear programs with quadratic regularization are attracting renewed interest due to their applications in optimal transport: unlike entropic regularization, the squared-norm penalty gives rise to sparse approximations of optimal transport couplings. It is well known that the solution of a quadratically regularized linear program over any polytope converges stationarily to the minimal-norm solution of the linear program when the regularization parameter tends to zero. However, that result is merely qualitative. Our main result quantifies the convergence by specifying the exact threshold for the regularization parameter, after which the regularized solution also solves the linear program. Moreover, we bound the suboptimality of the regularized solution before the threshold. These results are complemented by a convergence rate for the regime of large regularization. We apply our general results to the setting of optimal transport, where we shed light on how the threshold and suboptimality depend on the number of data points.