Non-Euclidean Broximal Point Method: A Blueprint for Geometry-Aware Optimization
This work addresses the need for theoretical understanding of non-Euclidean optimization methods in deep learning, though it is incremental as it builds on an existing framework.
The paper extends the Broximal Point Method to non-Euclidean norms, showing that most convergence guarantees carry over, providing a theoretical framework for geometry-aware optimization algorithms.
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.