On the Infimal Sub-differential Size of Primal-Dual Hybrid Gradient Method and Beyond

Primal-dual hybrid gradient method (PDHG, a.k.a. Chambolle and Pock method) is a well-studied algorithm for minimax optimization problems with a bilinear interaction term. Recently, PDHG is used as the base algorithm for a new LP solver PDLP that aims to solve large LP instances by taking advantage of modern computing resources, such as GPU and distributed system. Most of the previous convergence results of PDHG are either on duality gap or on distance to the optimal solution set, which are usually hard to compute during the solving process. In this paper, we propose a new progress metric for analyzing PDHG, which we dub infimal sub-differential size (IDS), by utilizing the geometry of PDHG iterates. IDS is a natural extension of the gradient norm of smooth problems to non-smooth problems, and it is tied with KKT error in the case of LP. Compared to traditional progress metrics for PDHG, IDS always has a finite value and can be computed only using information of the current solution. We show that IDS monotonically decays, and it has an $\mathcal O(\frac{1}{k})$ sublinear rate for solving convex-concave primal-dual problems, and it has a linear convergence rate if the problem further satisfies a regularity condition that is satisfied by applications such as linear programming, quadratic programming, TV-denoising model, etc. The simplicity of our analysis and the monotonic decay of IDS suggest that IDS is a natural progress metric to analyze PDHG. As a by-product of our analysis, we show that the primal-dual gap has $\mathcal O(\frac{1}{\sqrt{k}})$ convergence rate for the last iteration of PDHG for convex-concave problems. The analysis and results on PDHG can be directly generalized to other primal-dual algorithms, for example, proximal point method (PPM), alternating direction method of multipliers (ADMM) and linearized alternating direction method of multipliers (l-ADMM).