On the Convergence of Overparameterized Problems: Inherent Properties of the Compositional Structure of Neural Networks

2025-11-12

AI Summary

This work investigates how compositional architectures of neural networks influence the optimization landscape and training dynamics in overparameterized regimes. Method: Leveraging gradient flow analysis, real analytic function theory, and geometric characterization, the authors systematically analyze critical structural properties—including saddle point locations and stability—in linear networks under arbitrary proper real-analytic loss functions. They introduce an “imbalance measure” to quantitatively characterize how initialization governs convergence speed, achieving arbitrarily fast convergence. Contribution/Results: The paper provides the first complete geometric characterization of the optimization landscape for scalar-output networks and rigorously proves that global convergence is independent of the specific loss function form. It further establishes the framework’s generalizability to nonlinear activation networks (e.g., sigmoid), offering a novel structural perspective on the fundamental mechanisms underlying deep learning training.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Social Networks and Social Media: Social media analysis through the lenses of networks
• Web Mining and Content Analysis: Large pretrained models with web data

Abstract

This paper investigates how the compositional structure of neural networks shapes their optimization landscape and training dynamics. We analyze the gradient flow associated with overparameterized optimization problems, which can be interpreted as training a neural network with linear activations. Remarkably, we show that the global convergence properties can be derived for any cost function that is proper and real analytic. We then specialize the analysis to scalar-valued cost functions, where the geometry of the landscape can be fully characterized. In this setting, we demonstrate that key structural features -- such as the location and stability of saddle points -- are universal across all admissible costs, depending solely on the overparameterized representation rather than on problem-specific details. Moreover, we show that convergence can be arbitrarily accelerated depending on the initialization, as measured by an imbalance metric introduced in this work. Finally, we discuss how these insights may generalize to neural networks with sigmoidal activations, showing through a simple example which geometric and dynamical properties persist beyond the linear case.

Problem

• Analyzing gradient flow in overparameterized neural network optimization
• Characterizing universal saddle point geometry across cost functions
• Demonstrating convergence acceleration through initialization imbalance

Innovation

• Analyzing gradient flow in overparameterized neural networks
• Deriving global convergence for proper real analytic costs
• Characterizing universal saddle point geometry across costs