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

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