On the Benefits of Weight Normalization for Overparameterized Matrix Sensing

2025-10-01

AI Summary

This work addresses the weak theoretical foundation of weight normalization (WN) in overparameterized matrix sensing. We propose a novel framework integrating generalized weight normalization with Riemannian optimization. Theoretically, we establish—for the first time—how WN leverages overparameterization to accelerate optimization: under suitable initialization, the algorithm achieves linear convergence, reducing iteration complexity from polynomial to linear and improving sample complexity polynomially. A key contribution is the rigorous proof that WN upgrades convergence from sublinear to linear—i.e., exponential acceleration—with the acceleration magnitude increasing significantly with the degree of overparameterization. This result provides new insights into implicit regularization and optimization dynamics in deep learning.

Technology Category

• Machine Learning: Optimization
• Search and Optimization: Non-convex Optimization
• Natural Language Processing: Learning & Optimization for NLP

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

While normalization techniques are widely used in deep learning, their theoretical understanding remains relatively limited. In this work, we establish the benefits of (generalized) weight normalization (WN) applied to the overparameterized matrix sensing problem. We prove that WN with Riemannian optimization achieves linear convergence, yielding an exponential speedup over standard methods that do not use WN. Our analysis further demonstrates that both iteration and sample complexity improve polynomially as the level of overparameterization increases. To the best of our knowledge, this work provides the first characterization of how WN leverages overparameterization for faster convergence in matrix sensing.

Problem

• Analyzes weight normalization benefits in overparameterized matrix sensing problems
• Proves weight normalization enables linear convergence with Riemannian optimization
• Demonstrates improved iteration and sample complexity through overparameterization

Innovation

• Weight normalization with Riemannian optimization achieves linear convergence
• Iteration complexity improves polynomially with overparameterization
• Sample complexity improves polynomially with overparameterization