Implicit Bias in Deep Linear Discriminant Analysis

2026-03-03

AI Summary

This work investigates the implicit regularization mechanism induced by the Deep Linear Discriminant Analysis (Deep LDA) objective during optimization, addressing a theoretical gap in understanding implicit biases in metric learning. By analyzing the gradient flow of the Deep LDA loss over an L-layer diagonal linear network, we uncover how, under balanced initialization, additive gradient updates are effectively transformed into multiplicative weight updates. We theoretically establish that this process inherently preserves a (2/L)-quasinorm conservation law, thereby forging the first explicit link between Deep LDA’s implicit regularization, network architecture, and the underlying optimization geometry. This insight offers a novel perspective on the generalization behavior of metric learning objectives.

Technology Category

Machine Learning: Deep Learning TheoryNatural Language Processing: Learning & Optimization for NLPSearch and Optimization: Learning to Search

Application Category

Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphsSearch and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for rankingWeb Mining and Content Analysis: Large pretrained models with web data

Abstract

While the Implicit Bias(or Implicit Regularization) of standard loss functions has been studied, the optimization geometry induced by discriminative metric-learning objectives remains largely unexplored.To the best of our knowledge, this paper presents an initial theoretical analysis of the implicit regularization induced by the Deep LDA,a scale invariant objective designed to minimize intraclass variance and maximize interclass distance. By analyzing the gradient flow of the loss on a L-layer diagonal linear network, we prove that under balanced initialization, the network architecture transforms standard additive gradient updates into multiplicative weight updates, which demonstrates an automatic conservation of the (2/L) quasi-norm.

Problem

• Implicit Bias
• Deep Linear Discriminant Analysis
• Implicit Regularization
• Metric Learning
• Optimization Geometry

Innovation

• Implicit Bias
• Deep LDA
• Gradient Flow
• Multiplicative Updates
• Quasi-norm Conservation