Trajectory-Restricted Optimization Conditions and Geometry-Aware Linear Convergence

2026-04-18

AI Summary

This work proposes a trajectory-restricted framework for linear convergence analysis that overcomes the conservatism of traditional first-order methods, whose guarantees often rely on global geometric conditions and worst-case constants. Instead of imposing regularity assumptions globally, our approach requires only local geometric properties—such as restricted Polyak–Łojasiewicz inequalities, error bounds, and quadratic growth—on the subset of the space actually traversed by the algorithm. We establish explicit relationships among the associated constants and show that, for piecewise polyhedral composite problems, once iterates enter a well-conditioned active manifold, convergence is governed by the restricted Hoffman constant of that manifold, yielding an improved effective condition number and faster local convergence. The results demonstrate that linear convergence fundamentally depends on the local geometry encountered along the algorithmic trajectory, rather than on global worst-case scenarios.

Technology Category

• Search and Optimization: Non-convex Optimization
• Machine Learning: Learning with Manifolds
• Constraint Satisfaction and Optimization: Constraint Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

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--Łojasiewicz 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.

Problem

• linear convergence
• trajectory-restricted
• geometric regularity
• Polyak–Łojasiewicz inequality
• Hoffman constant

Innovation

• trajectory-restricted convergence
• localized geometric regularity
• restricted Polyak–Łojasiewicz inequality
• Hoffman constant
• polyhedral composite optimization