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Minimizing Chebyshev Prototype Risk Magically Mitigates the Perils of Overfitting

Nathaniel Dean, Dilip Sarkar

TL;DR

The paper addresses overfitting in overparameterized DNNs by deriving a Chebyshev-based bound (CPR) that ties intra-class feature covariance to inter-class prototype separation. It introduces the explicit CPR (exCPR) loss, a multi-component objective that minimizes class-prototype aligned covariance while reducing prototype similarity, all with log-linear-time computations. Empirical results on CIFAR-10/100 and STL-10 show exCPR improves generalization across architectures and training subsets, with scalable computation and strong theoretical backing via Lemmas and Corollaries. The work provides a principled, prototype-focused regularization framework that accelerates covariance reduction and stabilizes generalization in deep networks.

Abstract

Overparameterized deep neural networks (DNNs), if not sufficiently regularized, are susceptible to overfitting their training examples and not generalizing well to test data. To discourage overfitting, researchers have developed multicomponent loss functions that reduce intra-class feature correlation and maximize inter-class feature distance in one or more layers of the network. By analyzing the penultimate feature layer activations output by a DNN's feature extraction section prior to the linear classifier, we find that modified forms of the intra-class feature covariance and inter-class prototype separation are key components of a fundamental Chebyshev upper bound on the probability of misclassification, which we designate the Chebyshev Prototype Risk (CPR). While previous approaches' covariance loss terms scale quadratically with the number of network features, our CPR bound indicates that an approximate covariance loss in log-linear time is sufficient to reduce the bound and is scalable to large architectures. We implement the terms of the CPR bound into our Explicit CPR (exCPR) loss function and observe from empirical results on multiple datasets and network architectures that our training algorithm reduces overfitting and improves upon previous approaches in many settings. Our code is available at https://github.com/Deano1718/Regularization_exCPR .

Minimizing Chebyshev Prototype Risk Magically Mitigates the Perils of Overfitting

TL;DR

The paper addresses overfitting in overparameterized DNNs by deriving a Chebyshev-based bound (CPR) that ties intra-class feature covariance to inter-class prototype separation. It introduces the explicit CPR (exCPR) loss, a multi-component objective that minimizes class-prototype aligned covariance while reducing prototype similarity, all with log-linear-time computations. Empirical results on CIFAR-10/100 and STL-10 show exCPR improves generalization across architectures and training subsets, with scalable computation and strong theoretical backing via Lemmas and Corollaries. The work provides a principled, prototype-focused regularization framework that accelerates covariance reduction and stabilizes generalization in deep networks.

Abstract

Overparameterized deep neural networks (DNNs), if not sufficiently regularized, are susceptible to overfitting their training examples and not generalizing well to test data. To discourage overfitting, researchers have developed multicomponent loss functions that reduce intra-class feature correlation and maximize inter-class feature distance in one or more layers of the network. By analyzing the penultimate feature layer activations output by a DNN's feature extraction section prior to the linear classifier, we find that modified forms of the intra-class feature covariance and inter-class prototype separation are key components of a fundamental Chebyshev upper bound on the probability of misclassification, which we designate the Chebyshev Prototype Risk (CPR). While previous approaches' covariance loss terms scale quadratically with the number of network features, our CPR bound indicates that an approximate covariance loss in log-linear time is sufficient to reduce the bound and is scalable to large architectures. We implement the terms of the CPR bound into our Explicit CPR (exCPR) loss function and observe from empirical results on multiple datasets and network architectures that our training algorithm reduces overfitting and improves upon previous approaches in many settings. Our code is available at https://github.com/Deano1718/Regularization_exCPR .
Paper Structure (25 sections, 5 theorems, 45 equations, 2 figures, 6 tables, 1 algorithm)

This paper contains 25 sections, 5 theorems, 45 equations, 2 figures, 6 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminary Notations
  3. Related Work
  4. DeCov
  5. OrthoReg
  6. Squentropy
  7. Theoretical Framework
  8. Chebyshev Prototype Risk (CPR)
  9. Approach and Algorithm
  10. Computation of $\mathcal{L}_{proto}$
  11. Computation of $\mathcal{L}_{CS}$
  12. Computation of $\mathcal{L}_{cov}$
  13. Loss Summary
  14. Empirical Results
  15. Key Evaluation Method
  16. ...and 10 more sections

Key Result

Lemma 3.2

Given a sufficiently trained classifier with low empirical risk, $f({\bm{x}},\theta)$, a fixed prototype feature vector ${\bm{p}}_k$ of dimension $J$, which is the mean feature vector of a corresponding class $k$, a prototype dissimilarity value $DS_k$, a feature vector ${\bm{v}}_k$ for a randomly d where $\mathbf{1} \in [1]^J$ is the ones vector.

Figures (2)

  • Figure 1: Neural network split into a feature extractor and classifier (last fully connected layer) acting on an input ${\bm{x}}$. Class prototypes are learned feature vectors that comprehensively represent each class' trained features.
  • Figure 2: Illustration of sorting and shifting strategy to optimize prototype weighted feature covariance in $\mathcal{O}(JlogJ)$ time. Black line represents indices of class prototype sorted by its activation values; blue lines are example activations greater than prototype at corresponding index and green activations are smaller. The value of $\nu$ can be selected to target positive (1), negative (-1), or both (0) signs of feature covariance.

Theorems & Definitions (11)

  • Definition 3.1
  • Lemma 3.2
  • Corollary 3.3
  • Definition 3.4
  • Lemma 4.1
  • Corollary 4.2
  • proof
  • proof
  • proof
  • Corollary 6.1
  • ...and 1 more