High-Probability Polynomial-Time Complexity of Restarted PDHG for Linear Programming
\(\)The restarted primal-dual hybrid gradient method (rPDHG) is a first-order method that has recently received significant attention for its computational effectiveness in solving linear program (LP) problems. Despite its impressive practical performance, the theoretical iteration bounds for rPDHG can be exponentially poor. To shrink this gap between theory and practice, 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 not only Gaussian distribution models but also sub-Gaussian distribution models as well. For standard-form LP instances with \(m\) linear constraints and \(n\) decision variables, we prove that rPDHG iterates settle on the optimal basis in \(\widetilde{O}\left(\frac{n^{2.5}m^{0.5}}{\delta}\right)\) iterations, followed by \(O\left(\frac{n^{0.5}m^{0.5}}{\delta}\ln\big(\frac{1}{\varepsilon}\big)\right)\) iterations to compute an \(\varepsilon\)-optimal solution. These bounds hold with probability at least \(1-\delta\) for \(\delta\) that is not exponentially small. The first-stage 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 test instance.