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Benign Overfitting and the Geometry of the Ridge Regression Solution in Binary Classification

Alexander Tsigler, Luiz F. O. Chamon, Spencer Frei, Peter L. Bartlett

TL;DR

The behavior of ridge regression in an overparameterized binary classification task is investigated and it is shown that ridge regression has qualitatively different behavior depending on the scale of the cluster mean vector and its interaction with the covariance matrix of the cluster distributions.

Abstract

In this work, we investigate the behavior of ridge regression in an overparameterized binary classification task. We assume examples are drawn from (anisotropic) class-conditional cluster distributions with opposing means and we allow for the training labels to have a constant level of label-flipping noise. We characterize the classification error achieved by ridge regression under the assumption that the covariance matrix of the cluster distribution has a high effective rank in the tail. We show that ridge regression has qualitatively different behavior depending on the scale of the cluster mean vector and its interaction with the covariance matrix of the cluster distributions. In regimes where the scale is very large, the conditions that allow for benign overfitting turn out to be the same as those for the regression task. We additionally provide insights into how the introduction of label noise affects the behavior of the minimum norm interpolator (MNI). The optimal classifier in this setting is a linear transformation of the cluster mean vector and in the noiseless setting the MNI approximately learns this transformation. On the other hand, the introduction of label noise can significantly change the geometry of the solution while preserving the same qualitative behavior.

Benign Overfitting and the Geometry of the Ridge Regression Solution in Binary Classification

TL;DR

The behavior of ridge regression in an overparameterized binary classification task is investigated and it is shown that ridge regression has qualitatively different behavior depending on the scale of the cluster mean vector and its interaction with the covariance matrix of the cluster distributions.

Abstract

In this work, we investigate the behavior of ridge regression in an overparameterized binary classification task. We assume examples are drawn from (anisotropic) class-conditional cluster distributions with opposing means and we allow for the training labels to have a constant level of label-flipping noise. We characterize the classification error achieved by ridge regression under the assumption that the covariance matrix of the cluster distribution has a high effective rank in the tail. We show that ridge regression has qualitatively different behavior depending on the scale of the cluster mean vector and its interaction with the covariance matrix of the cluster distributions. In regimes where the scale is very large, the conditions that allow for benign overfitting turn out to be the same as those for the regression task. We additionally provide insights into how the introduction of label noise affects the behavior of the minimum norm interpolator (MNI). The optimal classifier in this setting is a linear transformation of the cluster mean vector and in the noiseless setting the MNI approximately learns this transformation. On the other hand, the introduction of label noise can significantly change the geometry of the solution while preserving the same qualitative behavior.

Paper Structure

This paper contains 57 sections, 67 theorems, 402 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Binary classification problem setting
  3. Assumptions on the distribution of the covariates
  4. First result: recovering the geometry in the noiseless setting
  5. Second result: benign overfitting
  6. Comparison to the regression problem
  7. Related work
  8. Structure of the paper
  9. Additional notation
  10. Geometric picture for minimum norm interpolation
  11. MNI without label-flipping noise
  12. MNI with label-flipping noise and benign overfitting
  13. Assumptions on the data
  14. Gram matrix of the tails
  15. Algebraic assumptions
  16. ...and 42 more sections

Key Result

Proposition 1

The vectors ${{\boldsymbol w}_{\text{\tiny{MNI}}}}$ and ${\tilde{{\boldsymbol w}}_{\text{\tiny{MNI}}}}$ have the same direction, but different norms. They are related to each other as follows:

Figures (2)

  • Figure 1: Two possible relations between $\sqrt{V}$, $\sqrt{n}\Diamond$, and $N\sqrt{V}$ depending on $\|{\boldsymbol \mu}\|$.
  • Figure 2: Main bounds on the quantities of interest (up to a constant factor) as a function of $\|{\boldsymbol \mu}\|$. We write ${\boldsymbol w}_{\text{MNI}}^{(c)}$ to denote both ${{\boldsymbol w}_{\text{\tiny{MNI}}}}$ and ${{\boldsymbol w}^c_{\text{\tiny{MNI}}}}$ and ${\boldsymbol W} = ({\boldsymbol \Sigma} + n^{-1}\Lambda{\boldsymbol I}_p)^{-1}$.

Theorems & Definitions (71)

  • Proposition 1
  • Definition 2
  • Lemma 3: Lemma 16 from BO_ridge
  • Definition 4
  • Lemma 5
  • Proposition 6
  • Lemma 6: Relations between the main quantities
  • Theorem 7: Main lower bound
  • Theorem 8: Main upper bound
  • Definition 9
  • ...and 61 more