Primal-dual forward-reflected splitting with linesearch for distributed min-max problems under local Lipschitz smoothness
Abstract
We consider a network of finitely many agents aiming to cooperatively solve a convex-concave min-max problem combining nonsmooth terms with a smooth saddle function whose partial gradients are only locally Lipschitz continuous. Without requiring prior knowledge of the local Lipschitz moduli of the partial gradients, we propose a distributed proximal-reflected-gradient algorithm which incorporates a distributed backtracking procedure to adjust the stepsizes at each iteration. The algorithm is derived by reformulating the min-max problem as a composite monotone inclusion involving the sum of a maximally monotone operator, a linearly composed maximally monotone operator, and a single-valued locally Lipschitz continuous monotone operator. To solve this inclusion, we introduce a novel primal-dual forward-reflected splitting method with linesearch and establish its convergence using a Lyapunov-based analysis. In addition, when specialised to the min-max formulation, we derive an 𝒪(1/N) ergodic convergence rate for a consensus-penalised gap function. Finally, applying this splitting framework with mixing matrices yields our distributed method for consensus min-max optimisation.
Related
- An Adaptive Linesearch-free Method for Monotone Variational Inequalities under Local Lipschitz Continuity
- A Prediction--Correction Analysis of Two-Way Block Splitting in Distributed Learning
- Distributed Formation Model Predictive Control Under Communication Denial: An Edge-Splitting ADMM with Centralized-Equivalence Gating and Confidence-Decayed Coupling
- A Stochastic LineseArch Method (SLAM) for Minimizing Expectation Residuals: Weakening Boundedness of Gradient
- Linesearch-Free Nonlinearly Preconditioned GD with Semiautomatic Geometry Design
BibTeX
@misc{atenas2026primaldual,
title = {{Primal-dual forward-reflected splitting with linesearch for distributed min-max problems under local Lipschitz smoothness}},
author = {Felipe Atenas and Minh N. Dao and Matthew K. Tam and David Torregrosa-Belén},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.23518},
eprint = {2610.23518},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.23518},
}
RIS
TY - UNPB
TI - Primal-dual forward-reflected splitting with linesearch for distributed min-max problems under local Lipschitz smoothness
AU - Felipe Atenas
AU - Minh N. Dao
AU - Matthew K. Tam
AU - David Torregrosa-Belén
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2610.23518
UR - https://arxiv.org/abs/2610.23518
AB - We consider a network of finitely many agents aiming to cooperatively solve a convex-concave min-max problem combining nonsmooth terms with a smooth saddle function whose partial gradients are only locally Lipschitz continuous. Without requiring prior knowledge of the local Lipschitz moduli of the partial gradients, we propose a distributed proximal-reflected-gradient algorithm which incorporates a distributed backtracking procedure to adjust the stepsizes at each iteration. The algorithm is derived by reformulating the min-max problem as a composite monotone inclusion involving the sum of a maximally monotone operator, a linearly composed maximally monotone operator, and a single-valued locally Lipschitz continuous monotone operator. To solve this inclusion, we introduce a novel primal-dual forward-reflected splitting method with linesearch and establish its convergence using a Lyapunov-based analysis. In addition, when specialised to the min-max formulation, we derive an 𝒪(1/N) ergodic convergence rate for a consensus-penalised gap function. Finally, applying this splitting framework with mixing matrices yields our distributed method for consensus min-max optimisation.
ER -