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.

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@article{OJMO_2023__4__A6_0, author = {Olivier Fercoq}, title = {Quadratic error bound of the smoothed gap and the restarted averaged primal-dual hybrid gradient}, journal = {Open Journal of Mathematical Optimization}, eid = {6}, pages = {1--34}, publisher = {Universit\'e de Montpellier}, volume = {4}, year = {2023}, doi = {10.5802/ojmo.26}, language = {en}, url = {https://ojmo.centre-mersenne.org/articles/10.5802/ojmo.26/} }

TY - JOUR AU - Olivier Fercoq TI - Quadratic error bound of the smoothed gap and the restarted averaged primal-dual hybrid gradient JO - Open Journal of Mathematical Optimization PY - 2023 SP - 1 EP - 34 VL - 4 PB - Université de Montpellier UR - https://ojmo.centre-mersenne.org/articles/10.5802/ojmo.26/ DO - 10.5802/ojmo.26 LA - en ID - OJMO_2023__4__A6_0 ER -

%0 Journal Article %A Olivier Fercoq %T Quadratic error bound of the smoothed gap and the restarted averaged primal-dual hybrid gradient %J Open Journal of Mathematical Optimization %D 2023 %P 1-34 %V 4 %I Université de Montpellier %U https://ojmo.centre-mersenne.org/articles/10.5802/ojmo.26/ %R 10.5802/ojmo.26 %G en %F OJMO_2023__4__A6_0

Olivier Fercoq. Quadratic error bound of the smoothed gap and the restarted averaged primal-dual hybrid gradient. Open Journal of Mathematical Optimization, Volume 4 (2023), article no. 6, 34 p. doi : 10.5802/ojmo.26. https://ojmo.centre-mersenne.org/articles/10.5802/ojmo.26/

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