An Agnostic View on the Cost of Overfitting in Kernel Ridge Regression

Core Concepts

Understanding the cost of overfitting in kernel ridge regression from an agnostic perspective.

Abstract

The content delves into the cost of overfitting in noisy kernel ridge regression, analyzing benign, tempered, and catastrophic overfitting. It discusses the impact of overfitting on generalization performance and provides insights into different regimes of overfitting. The analysis is based on Gaussian universality ansatz and closed-form risk estimates. Various examples are provided to illustrate different scenarios of benign, tempered, and catastrophic overfitting. Introduction: Large neural networks' ability to generalize challenged understanding of overfitting. Progress in understanding overfitting in kernel methods. Problem Formulation: Bi-criterion optimization in kernel ridge regression. Mercer's Decomposition: Understanding kernel decomposition using Mercer's theorem. Closed-Form Risk Estimate: Theoretical works providing closed-form equations for estimating test risk. Cost of Overfitting: Defining and analyzing the cost of overfitting under different scenarios. Benign Overfitting: Conditions for benign overfitting based on effective rank analysis. Tempered Overfitting: Analysis and estimation of tempered overfitting based on effective rank ratios. Catastrophic Overfitting: Discussion and conditions leading to catastrophic overfitting. Application: Inner-Product Kernels: Application to inner-product kernels in the polynomial regime with detailed analysis and results. For detailed proofs and further insights, refer to the full content above.

Stats

We study the cost of overfitting in noisy kernel ridge regression (KRR). The test error can approach Bayes optimality even when models interpolate noisy training data. Effective ranks play a crucial role in characterizing the cost of overfitting across different settings.

Quotes

"The ability of large neural networks to generalize has significantly challenged our understanding." "We take an 'agnostic' view by considering costs as a function of sample size for any target function." "Our analysis provides a refined characterization of benign, tempered, and catastrophic overfittings."

Key Insights Distilled From

An Agnostic View on the Cost of Overfitting in (Kernel) Ridge Regression

by Lijia Zhou,J... at arxiv.org 03-25-2024

https://arxiv.org/pdf/2306.13185.pdf

Deeper Inquiries

How does this agnostic view change traditional perspectives on model fitting

The agnostic view presented in the context challenges traditional perspectives on model fitting by focusing on the direct effect of overfitting rather than asymptotic behavior and consistency. This approach considers the cost of overfitting as a function of sample size for any target function, even if the sample size is not large enough for consistency or if the target is outside the Reproducing Kernel Hilbert Space (RKHS). By doing so, it separates the effects of overfitting from other factors like model difficulty and appropriateness, providing a more refined understanding.

What implications do these findings have for real-world applications beyond theoretical frameworks

These findings have significant implications for real-world applications beyond theoretical frameworks. Understanding benign, tempered, and catastrophic overfitting can help practitioners make better decisions when developing machine learning models. For example: Model Selection: Knowing when overfitting is likely to be benign can guide model selection processes. Generalization Performance: Insights into different types of overfitting can improve generalization performance in various applications. Risk Management: Identifying catastrophic overfitting scenarios can help mitigate risks associated with unreliable predictions.

How might these insights be applied to improve current machine learning practices

These insights could be applied to improve current machine learning practices in several ways: Regularization Techniques: Tailoring regularization techniques based on whether overfitting is benign or catastrophic. Model Evaluation: Developing new evaluation metrics that consider different types of overfitting. Hyperparameter Tuning: Optimizing hyperparameters based on an understanding of how different types of overfitting affect model performance.