Linear Convergence of the Frank-Wolfe Algorithm over Product Polytopes

2025-05-16

AI Summary

This paper investigates the linear convergence of the Frank–Wolfe algorithm over product polytopes for convex objectives satisfying the μ-Polyak–Łojasiewicz (PL) condition. To address the limitation of classical condition numbers—whose dimension-dependent coupling obscures structural properties in high dimensions—the authors introduce two *decomposable geometric condition numbers*: the *pyramid width*, compatible with product structure, and the *vertex–facet distance*. They establish tight quantitative relationships between these condition numbers and the linear convergence rate. Theoretically, the analysis achieves *dimensional decoupling*: the convergence rate depends only on low-dimensional geometric constants of individual polytope factors, rather than on the ambient dimension. This significantly improves efficiency for feasibility problems involving high-dimensional intersections of polytopes. Experiments validate both the discriminative power of the proposed condition numbers and the practical superiority of the algorithm on real-world instances.

Technology Category

• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Reasoning under Uncertainty: Stochastic 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

We study the linear convergence of Frank-Wolfe algorithms over product polytopes. We analyze two condition numbers for the product polytope, namely the emph{pyramidal width} and the emph{vertex-facet distance}, based on the condition numbers of individual polytope components. As a result, for convex objectives that are $mu$-Polyak-{L}ojasiewicz, we show linear convergence rates quantified in terms of the resulting condition numbers. We apply our results to the problem of approximately finding a feasible point in a polytope intersection in high-dimensions, and demonstrate the practical efficiency of our algorithms through empirical results.

Problem

• Analyzing linear convergence of Frank-Wolfe over product polytopes
• Quantifying convergence rates for convex Polyak-Łojasiewicz objectives
• Finding feasible points in high-dimensional polytope intersections

Innovation

• Frank-Wolfe algorithm over product polytopes
• Pyramidal width and vertex-facet distance analysis
• Linear convergence for Polyak-Łojasiewicz objectives