Tight Stationarity Transfer and Repeatability for Inexact Proximal-Gradient Methods
Abstract
Proximal-gradient methods are widely used for composite optimization, but inexact proximal updates introduce a residual in the proximal optimality condition, complicating the analysis of how stationarity changes across a step. We study this question for composite objectives consisting of a smooth convex term and a possibly nonsmooth weakly convex term, focusing on the relationship between the directional-stationarity measures at the input and output points of an inexact proximal-gradient step. We first establish separate input and output bounds, but their combination can yield a conservative estimate because their worst-case residual directions may differ. By exploiting the magnitude and orientation of this shared residual, we establish tight joint bounds and derive an exact one-step stationarity-transfer factor that strictly improves the estimate from separate bounds in the short-step regime. We also identify the precise residual-tolerance threshold for finite uniform stationarity transfer. A further distinction emerges over multiple steps: endpoint worst cases can repeat over arbitrarily many iterations, whereas interior worst cases cannot persist across two consecutive steps, yielding strict two-step submultiplicativity. These results also provide principled tolerance-selection rules, descent estimates, and stationarity-complexity guarantees for inexact proximal-gradient methods.
Related
- Asymptotic Analysis of Gradient Mapping-type Stationarity Measure for the Sum of Nonconvex Nonsmooth Functions and Applications to Proximal Gradient-type Algorithms
- Randomized Branch Methods with Inexact Subproblems for Bouligand Stationarity in Linear and Quadratic Programs with Complementarity Constraints
- Inexact Proximal Point and Tseng Algorithms with Nonsummable Errors to Solve Monotone Inclusions
- Convergence Analysis of an Inexact Proximal Trust-Region Method
- An Inexact Halpern-accelerated Preconditioned Generalized Proximal Point Algorithm for the Maximal Monotone Inclusion Problem
BibTeX
@misc{mei2026tight,
title = {{Tight Stationarity Transfer and Repeatability for Inexact Proximal-Gradient Methods}},
author = {Shenghan Mei},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.38310},
eprint = {2610.38310},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.38310},
}
RIS
TY - UNPB
TI - Tight Stationarity Transfer and Repeatability for Inexact Proximal-Gradient Methods
AU - Shenghan Mei
PY - 2026
DA - 2026/10/28
PB - arXiv
DO - 10.48550/arxiv.2610.38310
UR - https://arxiv.org/abs/2610.38310
AB - Proximal-gradient methods are widely used for composite optimization, but inexact proximal updates introduce a residual in the proximal optimality condition, complicating the analysis of how stationarity changes across a step. We study this question for composite objectives consisting of a smooth convex term and a possibly nonsmooth weakly convex term, focusing on the relationship between the directional-stationarity measures at the input and output points of an inexact proximal-gradient step. We first establish separate input and output bounds, but their combination can yield a conservative estimate because their worst-case residual directions may differ. By exploiting the magnitude and orientation of this shared residual, we establish tight joint bounds and derive an exact one-step stationarity-transfer factor that strictly improves the estimate from separate bounds in the short-step regime. We also identify the precise residual-tolerance threshold for finite uniform stationarity transfer. A further distinction emerges over multiple steps: endpoint worst cases can repeat over arbitrarily many iterations, whereas interior worst cases cannot persist across two consecutive steps, yielding strict two-step submultiplicativity. These results also provide principled tolerance-selection rules, descent estimates, and stationarity-complexity guarantees for inexact proximal-gradient methods.
ER -