# Quadratic error bound of the smoothed gap and the restarted averaged primal-dual hybrid gradient

arXiv: https://arxiv.org/abs/2206.03041 · HTML: https://mlsanity.com/paper/2206.03041/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Jun 2022.

## Built on by (most cited first, 2 of 2)

- [PDHG-Unrolled Learning-to-Optimize Method for Large-Scale Linear Programming](https://mlsanity.com/paper/2406.01908/) (2024)
- [Restarts subject to approximate sharpness: A parameter-free and optimal scheme for first-order methods](https://mlsanity.com/paper/2301.02268/) (2023)

## Abstract (arXiv)

We study the linear convergence of the primal-dual hybrid gradient method. After a review of current analyses, we show that they do not explain properly the behavior of the algorithm, even on the most simple problems. We thus introduce the quadratic error bound of the smoothed gap, a new regularity assumption that holds for a wide class of optimization problems. Equipped with this tool, we manage to prove tighter convergence rates. Then, we show that averaging and restarting the primal-dual hybrid gradient allows us to leverage better the regularity constant. Numerical experiments on linear and quadratic programs, ridge regression and image denoising illustrate the findings of the paper.