Unveiling the Training Dynamics of ReLU Networks through a Linear Lens

TL;DR

This paper introduces a linear analytical framework for understanding ReLU neural networks by examining input-dependent effective weights, revealing how training shapes class-specific representations and decision boundaries.

Contribution

It proposes a novel method to analyze ReLU networks as input-dependent linear models, providing insights into training dynamics and representation learning.

Findings

Effective weights converge for same-class samples during training

Effective weights diverge for different-class samples

Provides a new perspective on class boundary formation

Abstract

Deep neural networks, particularly those employing Rectified Linear Units (ReLU), are often perceived as complex, high-dimensional, non-linear systems. This complexity poses a significant challenge to understanding their internal learning mechanisms. In this work, we propose a novel analytical framework that recasts a multi-layer ReLU network into an equivalent single-layer linear model with input-dependent "effective weights". For any given input sample, the activation pattern of ReLU units creates a unique computational path, effectively zeroing out a subset of weights in the network. By composing the active weights across all layers, we can derive an effective weight matrix, $W_{\text{eff}}(x)$, that maps the input directly to the output for that specific sample. We posit that the evolution of these effective weights reveals fundamental principles of representation learning. Our work demonstrates that as training progresses, the effective weights corresponding to samples from the same class converge, while those from different classes diverge. By tracking the trajectories of these sample-wise effective weights, we provide a new lens through which to interpret the formation of class-specific decision boundaries and the emergence of semantic representations within the network.