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

TL;DR

This paper introduces an adaptive primal-dual algorithm for saddle point problems with locally Lipschitz continuous gradients, achieving optimal convergence rates and practical efficiency.

Contribution

It proposes a new adaptive Condat-Vu algorithm that adjusts stepsizes based on local smoothness and gradient norms, improving convergence for convex-concave problems.

Findings

Achieves an $ ext{O}(k^{-1})$ ergodic convergence rate.

Demonstrates linear convergence under strong convexity and full row rank conditions.

Numerical experiments confirm practical effectiveness.

Abstract

We consider the problem of finding a saddle point for the convex-concave objective $\min_x \max_y f(x) + \langle Ax, y\rangle - g^*(y)$, 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 $\mathcal{O}(k^{-1})$ 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.