Derivation of effective gradient flow equations and dynamical truncation of training data in Deep Learning

2025-01-13

AI Summary

This work investigates gradient descent dynamics of ReLU networks under Euclidean loss, focusing on the modeling challenge of gradient flows for input-layer weights and biases. We derive, for the first time, explicit gradient flow equations in activation coordinates and introduce a novel paradigm—“dynamical data truncation”—where training data clusters are exponentially truncated during optimization, with truncation rate accelerating via positive feedback proportional to the number of already-truncated samples. Leveraging coordinate-adaptive modeling and rigorous analysis of ReLU network dynamics, we obtain multiple analytical solutions that quantitatively characterize the asymptotic simplification mechanism underlying truncation. Our results provide a rigorous theoretical foundation for interpreting deep learning optimization trajectories and reveal intrinsic connections between implicit regularization and the evolution of model complexity.

Technology Category

• Machine Learning: Learning with Manifolds
• Natural Language Processing: Learning & Optimization for NLP
• Search and Optimization: Learning to Search

Application Category

• Web Mining and Content Analysis: Large pretrained models with web data
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization

Abstract

We derive explicit equations governing the cumulative biases and weights in Deep Learning with ReLU activation function, based on gradient descent for the Euclidean cost in the input layer, and under the assumption that the weights are, in a precise sense, adapted to the coordinate system distinguished by the activations. We show that gradient descent corresponds to a dynamical process in the input layer, whereby clusters of data are progressively reduced in complexity ("truncated") at an exponential rate that increases with the number of data points that have already been truncated. We provide a detailed discussion of several types of solutions to the gradient flow equations. A main motivation for this work is to shed light on the interpretability question in supervised learning.

Problem

• Gradient Flow
• Deep Learning
• Dynamic Data Reduction

Innovation

• Gradient Flow
• ReLU Activation
• Complexity Reduction