Core Concepts
Convergence of gradient flow in training deep neural networks.
Abstract
The article delves into the convergence of gradient flow for training deep neural networks, focusing on Residual Neural Networks (ResNets). It explores a mean-field model of infinitely deep and wide ResNet architectures parameterized by probability measures. The study aims to understand the optimization challenges posed by non-convexity and non-coercivity in training such architectures. By introducing a conditional Optimal Transport distance, the research demonstrates well-posedness and convergence results for gradient flow, offering insights into the dynamic formulation of this metric.
Abstract
Investigates convergence of gradient flow in training deep neural networks.
Explores mean-field model for infinitely deep and wide ResNets.
Proposes conditional Optimal Transport distance for training optimization.
Introduction
Significance of understanding neural network training dynamics.
Challenges in optimizing very deep architectures like ResNets.
Previous works on Multi-Layer Perceptrons (MLP) and convergence towards minimizers.
Mean-field Model of Neural Network
Parameterizing neural networks with probability measures.
Representation encompassing standard architectures like SHL perceptron and convolutional layers.
Supervised Learning Problem
Formulation for supervised learning task with data distribution and loss function.
Objective to minimize risk through NODE model parameterization.
Related Works and Contributions
Comparison with existing studies on NODEs, ResNets, and their convergence properties.
Contributions include proposing a model for infinitely deep ResNets with a consistent metric structure.
Metric Structure of Parameter Set
Definition of Conditional Optimal Transport distance as a modification of Wasserstein distance.
Dynamical Formulation of Conditional Optimal Transport
Analysis on absolutely continuous curves under Conditional OT metric space.
Characterization crucial for defining gradient flow equation in NODE model training.
Functional Properties
Study on mappings defined by mean-field models from functional analysis perspective.
Completeness
Proof that the metric space is complete based on Cauchy sequences property.
Comparison with W2 Distance
Relationship between Conditional OT distance d and Wasserstein distance W2 explained.
Dynamical Formulation Insights
Understanding continuity equations governing absolutely continuous curves under Conditional OT metric space.
Quotes
"If the number of features is finite but sufficiently large...the gradient flow converges towards a global minimizer."
"ResNet architecture has permitted the training of neural networks of almost arbitrary depth."