Trajectory-Restricted Optimization Conditions and Geometry-Aware Linear Convergence

TL;DR

This paper introduces a localized geometric framework for analyzing linear convergence of first-order methods, emphasizing the importance of the geometry encountered along optimization trajectories rather than worst-case global conditions.

Contribution

It develops restricted variants of classical inequalities that depend on the geometry of the regions traversed, providing new insights into local convergence behavior.

Findings

Convergence rates depend on geometric quantities along the trajectory.

For polyhedral problems, convergence is controlled by restricted Hoffman constants.

Effective condition numbers improve once iterates enter well-conditioned regions.

Abstract

Linear convergence of first-order methods is typically characterized by global optimization conditions whose constants reflect worst-case geometry of the ambient space. In high-dimensional or structured problems, these global constants can be arbitrarily conservative and fail to capture the geometry actually encountered by optimization trajectories. In this paper, we develop a trajectory-restricted framework for linear convergence based on localized geometric regularity. We introduce restricted variants of the Polyak--{\L}ojasiewicz inequality, error bound, and quadratic growth conditions that are required to hold only on subsets of the domain. We show that classical convergence guarantees extend under these localized conditions, and in key cases, we develop new arguments that yield explicit relationships between the corresponding constants. The resulting rates are governed by geometric quantities associated with the regions traversed by the algorithm. For polyhedral composite problems, we prove that convergence is controlled by restricted Hoffman constants corresponding to the active polyhedral faces visited along the trajectory. Once the iterates enter a well-conditioned face, the effective condition number improves accordingly. Our work provides a geometric quantification for fast local convergence after active-set or manifold identification and more broadly suggests that linear convergence is fundamentally governed by the geometry of the subsets explored by the algorithm, rather than by worst-case global conditioning.