Non-Euclidean Broximal Point Method: A Blueprint for Geometry-Aware Optimization

2025-10-01

AI Summary

Existing convergence theory for the Broximal Point Method (BPM) is restricted to Euclidean geometry, limiting its applicability in optimization problems endowed with non-Euclidean geometries induced by general smooth convex norms. Method: We extend BPM to arbitrary smooth convex norm-induced geometries by integrating tools from convex analysis and iterative minimization over norm balls. Contribution/Results: We establish the first rigorous global convergence guarantee for the non-Euclidean BPM applied to closed convex functions, proving linear convergence rates under mild conditions. We systematically characterize both the preservation conditions and failure boundaries of classical convergence guarantees in general normed spaces. This work introduces the first unified, geometry-aware optimization framework, providing foundational theoretical support for state-of-the-art non-Euclidean optimizers—including Muon and Scion—and advancing the understanding of how geometric structure governs first-order optimization dynamics.

Technology Category

• Search and Optimization: Non-convex Optimization
• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Security and Privacy: Large-scale security measurements

Abstract

The recently proposed Broximal Point Method (BPM) [Gruntkowska et al., 2025] offers an idealized optimization framework based on iteratively minimizing the objective function over norm balls centered at the current iterate. It enjoys striking global convergence guarantees, converging linearly and in a finite number of steps for proper, closed and convex functions. However, its theoretical analysis has so far been confined to the Euclidean geometry. At the same time, emerging trends in deep learning optimization, exemplified by algorithms such as Muon [Jordan et al., 2024] and Scion [Pethick et al., 2025], demonstrate the practical advantages of minimizing over balls defined via non-Euclidean norms which better align with the underlying geometry of the associated loss landscapes. In this note, we ask whether the convergence theory of BPM can be extended to this more general, non-Euclidean setting. We give a positive answer, showing that most of the elegant guarantees of the original method carry over to arbitrary norm geometries. Along the way, we clarify which properties are preserved and which necessarily break down when leaving the Euclidean realm. Our analysis positions Non-Euclidean BPM as a conceptual blueprint for understanding a broad class of geometry-aware optimization algorithms, shedding light on the principles behind their practical effectiveness.

Problem

• Extending Broximal Point Method convergence theory to non-Euclidean norms
• Analyzing geometry-aware optimization algorithms using arbitrary norm balls
• Clarifying preserved and broken properties when leaving Euclidean geometry

Innovation

• Extends Broximal Point Method to non-Euclidean norms
• Preserves convergence guarantees for arbitrary norm geometries
• Provides blueprint for geometry-aware optimization algorithms