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In this paper, we design first-order primal-dual algorithms to address theconvex-concave saddle point problem, which is formulated as $\min_{x}\max_{y}f(x) + \langle Kx, y \rangle - g(y)$. Notably, both functions $f$ and $g$exhibit a composite structure, combining ``nonsmooth'' + ``smooth'' components.When both $f$ and $g$ are convex, we first propose a first-order primal-dualalgorithm that incorporates inertial extrapolation, and prove that it achievesa non-ergodic $\mathcal{O}(1/k)$ convergence rate, where $k$ represents thenumber of iterations. In the case where $g$ is strongly convex, we furtherpresent two accelerated first-order primal-dual algorithms by employingNesterov's extrapolation technique, and show that these algorithms exhibit anon-ergodic convergence rate of $\mathcal{O}(1/k^2)$. Additionally, we considera first-order primal-dual algorithm that demonstrates non-ergodic linearconvergence when both $f$ and $g$ are strongly convex.

Non-ergodic convergence rates of first-order primal-dual algorithms for saddle point problems