A first-order primal-dual method with adaptivity to local smoothness

Abstract: We consider the problem of finding a saddle point for the convex-concave objective $$\min_x \max_y f(x) + \langle Ax, y\rangle - g<sup>*(y)$$, where $f$ is a convex function with locally Lipschitz gradient and $g$ is convex and possibly non-smooth. We propose an adaptive version of the Condat-V~u algorithm, which alternates between primal gradient steps and dual proximal steps. The method achieves stepsize adaptivity through a simple rule involving $|A|$ and the norm of recently computed gradients of $f$. Under standard assumptions, we prove an $\mathcal{O}(k<sup>{-1})$ ergodic convergence rate. Furthermore, when $f$ is also locally strongly convex and $A$ has full row rank we show that our method converges with a linear rate. Numerical experiments are provided for illustrating the practical performance of the algorithm.