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

Maria-Luiza Vladarean, Yura Malitsky, Volkan Cevher

Keywords: adaptive, primal-dual, convex-concave, composite optimization, local smoothness, local Lipschitz continuity

TL;DR: We propose a first-order primal-dual method with adaptivity to the local geometry of one of the objective's components.

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-Vũ 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.

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Code: https://github.com/mvladarean/adaptive_pda

Community Implementations: [ 1 code implementation](https://www.catalyzex.com/paper/arxiv:2110.15148/code)