Unified Ergodic Primal-Dual Gap Rates with Unhalved Primal Stepsizes

We study ergodic primal-dual gap rates for first-order primal-dual methods applied to $$\underset{x}{\min }f(x)+g(x)+h(Ax),$$ where $f$ is smooth and convex, $g$ and $h$ are proper, closed, convex functions, and $A$ is linear. Standard gap-rate proofs often impose the halved smooth-stepsize condition $\tau \le 1\mathrm{/}L$, even though the corresponding convergence theory allows the larger range $\tau <2\mathrm{/}L$. We introduce a residual-to-gap transfer principle: positive residual terms in the one-step gap inequality are controlled by the decrease of a Lyapunov function. This yields $O(1\mathrm{/}K)$ ergodic primal-dual gap bounds with the unhalved primal stepsize $\tau <2\mathrm{/}L$ for Condat–Vũ, PD3O, AFBA/PDDY, and PAPC/PDFP${}^2$O, under their algorithm-dependent product conditions. We also give a two-dimensional counterexample showing that the fully separated rectangle $\tau <2\mathrm{/}L$, $\tau \eta \mathrm{\parallel }A{\mathrm{\parallel }}^2<4\mathrm{/}3$ cannot hold in the general three-function setting.