High-Probability Polynomial-Time Complexity of Restarted PDHG for Linear Programming

Abstract

The restarted primal-dual hybrid gradient method (rPDHG) is a first-order method recently known for its computational effectiveness in solving linear programming (LP) problems. Despite its impressive practical performance, the theoretical iteration bounds for rPDHG can be exponentially poor. To investigate this gap from a probabilistic perspective, we show that rPDHG achieves polynomial-time complexity in a high-probability sense, under assumptions on the probability distribution from which the data instance is generated. We consider both Gaussian and more general sub-Gaussian distribution models. For standard-form LP instances with $m$ constraints and $n$ variables, our bounds take a particularly simple form when $m$ is not too close to $n$: rPDHG iterates settle on the optimal basis in $\widetilde{O}\left(\tfrac{n^{2.5}m^{0.5}}{\delta}\right)$ iterations, followed by $O\left(\frac{n^{0.5}m^{0.5}}{\delta}\ln\big(\tfrac{1}{\varepsilon}\big)\right)$ iterations to compute an $\varepsilon $-optimal solution, with probability at least $1-\delta$ for $\delta$ that is not exponentially small. The Stage-I bound further improves to $\widetilde{O}\left(\frac{n^{2.5}}{\delta}\right)$ in the Gaussian distribution model. Experimental results confirm the tail behavior and the polynomial-time dependence on problem dimensions of the iteration counts. As an application of our probabilistic analysis, we explore how the disparity among the components of the optimal solution bears on the performance of rPDHG, and we provide guidelines for generating challenging LP instances.