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

TL;DR

This paper derives explicit gradient flow equations for deep learning with ReLU activations, revealing how data complexity reduces exponentially during training, which enhances interpretability in supervised learning.

Contribution

It introduces a novel derivation of gradient flow equations and demonstrates a dynamical truncation process of data complexity in deep learning models.

Findings

Gradient descent corresponds to a dynamical process reducing data complexity.

Data clusters are truncated exponentially faster with more data points.

The equations provide insights into interpretability of supervised learning.

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.