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Regularized Linear Regression for Binary Classification
Machine Learning · Submitted 3 Nov 2023
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim In particular, we demonstrate that ridge regression invariably improves the classification error.
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The abstract, sentence by sentence
Regularized linear regression is a promising approach for binary classification problems in which the training set has noisy labels since the regularization term can help to avoid interpolating the mislabeled data points. In this paper we provide a systematic study of the effects of the regularization strength on the performance of linear classifiers that are trained to solve binary classification problems by minimizing a regularized least-squares objective. We consider the over-parametrized regime and assume that the classes are generated from a Gaussian Mixture Model (GMM) where a fraction $c<\frac{1}{2}$ of the training data is mislabeled. Under these assumptions, we rigorously analyze the classification errors resulting from the application of ridge, $\ell_1$, and $\ell_\infty$ regression. In particular, we demonstrate that ridge regression invariably improves the classification error. We prove that $\ell_1$ regularization induces sparsity and observe that in many cases one can sparsify the solution by up to two orders of magnitude without any considerable loss of performance, even though the GMM has no underlying sparsity structure. For $\ell_\infty$ regularization we show that, for large enough regularization strength, the optimal weights concentrate around two values of opposite sign. We observe that in many cases the corresponding "compression" of each weight to a single bit leads to very little loss in performance. These latter observations can have significant practical ramifications.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Regularized linear regression is a promising approach for binary classification problems in which the training set has noisy labels since the regularization term can help to avoid interpolating the mislabeled data points. | background 1.00 | 0.10 | 0.10 | hedged possibility 0.66 |
| s1 | In this paper we provide a systematic study of the effects of the regularization strength on the performance of linear classifiers that are trained to solve binary classification problems by minimizing a regularized least-squares objective. | contribution 1.00 | 0.86 | 0.06 | asserted 0.80 |
| s2 | We consider the over-parametrized regime and assume that the classes are generated from a Gaussian Mixture Model (GMM) where a fraction $c<\frac{1}{2}$ of the training data is mislabeled. | method 0.97 | 0.44 | 0.03 | asserted 0.93 |
| s3 | Under these assumptions, we rigorously analyze the classification errors resulting from the application of ridge, $\ell_1$, and $\ell_\infty$ regression. | method 0.61 | 0.92 | 0.05 | asserted 0.90 |
| s4 | In particular, we demonstrate that ridge regression invariably improves the classification error. | result 1.00 | 0.92 | 0.32 | asserted 1.00 |
| s5 | We prove that $\ell_1$ regularization induces sparsity and observe that in many cases one can sparsify the solution by up to two orders of magnitude without any considerable loss of performance, even though the GMM has no underlying sparsity structure. | result 1.00 | 0.95 | 0.12 | asserted 0.90 |
| s6 | For $\ell_\infty$ regularization we show that, for large enough regularization strength, the optimal weights concentrate around two values of opposite sign. | result 0.99 | 0.94 | 0.07 | asserted 1.00 |
| s7 | We observe that in many cases the corresponding "compression" of each weight to a single bit leads to very little loss in performance. | result 0.89 | 0.93 | 0.09 | asserted 0.70 |
| s8 | These latter observations can have significant practical ramifications. | interpretation 1.00 | 0.39 | 0.06 | hedged possibility 0.73 |
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The paper
Original abstract
Regularized linear regression is a promising approach for binary classification problems in which the training set has noisy labels since the regularization term can help to avoid interpolating the mislabeled data points. In this paper we provide a systematic study of the effects of the regularization strength on the performance of linear classifiers that are trained to solve binary classification problems by minimizing a regularized least-squares objective. We consider the over-parametrized regime and assume that the classes are generated from a Gaussian Mixture Model (GMM) where a fraction $c<\frac{1}{2}$ of the training data is mislabeled. Under these assumptions, we rigorously analyze the classification errors resulting from the application of ridge, $\ell_1$, and $\ell_\infty$ regression. In particular, we demonstrate that ridge regression invariably improves the classification error. We prove that $\ell_1$ regularization induces sparsity and observe that in many cases one can sparsify the solution by up to two orders of magnitude without any considerable loss of performance, even though the GMM has no underlying sparsity structure. For $\ell_\infty$ regularization we show that, for large enough regularization strength, the optimal weights concentrate around two values of opposite sign. We observe that in many cases the corresponding "compression" of each weight to a single bit leads to very little loss in performance. These latter observations can have significant practical ramifications.
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How the literature uses it
2 arXiv papers cite this in 3 sentences
Papers citing it · 1 sentence without a confident label
Cites it as background 2 papers
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In the case of binary classification, there has been a recent surge of results that provide a sharp analysis for a variety of methods tailored to different models (see, e.g., [ref] and the references therein).
One-Bit Quantization and Sparsification for Multiclass Linear Classification with Strong Regularization · Related works · 0.94
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The performances of various classification algorithms have been analyzed in depth using the standard CGMT for the case of scalar $Σ_1$ and $Σ_2$ (see, for example, [ref] ).
A Novel Gaussian Min-Max Theorem and its Applications · Binary classification for general GMMs · 0.96
What it cites
Cites as background 8 papers
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These works typically pose the over-parameterized binary classification problem as an optimization problem and employ either the Convex Gaussian Min-max Theorem (CGMT) [ref] or the Approximate Message Passing (AMP) (see, e.g, [ref] ) approach to obtain formulas for the generalization error that involve solutions to a system of non-linear equations (in a small number of variables) that often do not admit closed-form expressions.
A framework to characterize performance of LASSO algorithms · RELATED WORKS AND OUR CONTRIBUTION · 0.85
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The CGMT framework has been developed to analyze the properties of the solutions to non-smooth regularized convex optimization problems and has been successfully applied to characterize the precise performance in numerous applications such as $M$-estimators, generalized lasso, massive MIMO, phase retrieval, regularized logistic regression, adversarial training, max-margin classifiers, distributionally robust regression, and others (see [ref] .
The generalization error of max-margin linear classifiers: Benign overfitting and high dimensional asymptotics in the overparametrized regime · Convex Gaussian Min-Max Theorem · 0.88
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The CGMT framework has been developed to analyze the properties of the solutions to non-smooth regularized convex optimization problems and has been successfully applied to characterize the precise performance in numerous applications such as $M$-estimators, generalized lasso, massive MIMO, phase retrieval, regularized logistic regression, adversarial training, max-margin classifiers, distributionally robust regression, and others (see [ref] .
Wasserstein Distributionally Robust Estimation in High Dimensions: Performance Analysis and Optimal Hyperparameter Tuning · Convex Gaussian Min-Max Theorem · 0.88
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