Sobolev acceleration for neural networks

2025-09-24

AI Summary

This work addresses the insufficient theoretical understanding of why Sobolev training accelerates convergence in ReLU networks. We establish the first rigorous theoretical framework under the student-teacher setting with Gaussian inputs, deriving exact expressions for the gradient and Hessian of the Sobolev loss. Our analysis reveals that incorporating target derivatives fundamentally improves the condition number of the loss landscape and enhances gradient signal quality, thereby accelerating gradient flow convergence. Crucially, we identify smoothness regularization and enriched gradient information as the core mechanisms underlying convergence acceleration. Extensive numerical experiments corroborate the theory: Sobolev training consistently improves both convergence speed and generalization across shallow and deep ReLU networks. This study provides the first interpretable theoretical foundation for Sobolev training and opens new avenues for designing efficient deep learning optimization methods grounded in functional-space regularization.

Technology Category

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

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

Sobolev training, which integrates target derivatives into the loss functions, has been shown to accelerate convergence and improve generalization compared to conventional $L^2$ training. However, the underlying mechanisms of this training method remain only partially understood. In this work, we present the first rigorous theoretical framework proving that Sobolev training accelerates the convergence of Rectified Linear Unit (ReLU) networks. Under a student-teacher framework with Gaussian inputs and shallow architectures, we derive exact formulas for population gradients and Hessians, and quantify the improvements in conditioning of the loss landscape and gradient-flow convergence rates. Extensive numerical experiments validate our theoretical findings and show that the benefits of Sobolev training extend to modern deep learning tasks.

Problem

• Understanding the mechanisms behind Sobolev training's acceleration effects
• Providing theoretical proof for convergence acceleration in ReLU networks
• Quantifying improvements in loss landscape conditioning and convergence rates

Innovation

• Integrates target derivatives into loss functions
• Proves accelerated convergence for ReLU networks
• Quantifies improved conditioning and convergence rates