Asymptotics of Random Feature Regression in High-Dimensional Polynomial Scaling

Główne pojęcia

Random feature ridge regression exhibits a trade-off between approximation and generalization power in high-dimensional polynomial scaling.

Streszczenie

Recent advances in machine learning challenge traditional notions of model complexity and generalization capabilities. The study investigates random feature ridge regression (RFRR) in the high-dimensional polynomial scaling regime, revealing insights into the impact of parametrization on model performance. RFRR shows a trade-off between approximation and generalization, with distinct behaviors based on the relationship between the number of parameters and sample size. The analysis provides a comprehensive understanding of RFRR's test error behavior under different parametrization scenarios.

Statystyki

p ≳ √n random features are sufficient to match kernel ridge regression performance.
For optimal test error, overparametrization is needed until p/n → ∞.
Taking λ → 0+ achieves optimal test error in critical regimes κ1 = κ2.

Cytaty

Kluczowe wnioski z

Asymptotics of Random Feature Regression Beyond the Linear Scaling Regime

by Hong Hu,Yue ... o arxiv.org 03-14-2024

https://arxiv.org/pdf/2403.08160.pdf

Głębsze pytania

How does the trade-off between approximation and generalization change with different scalings

In the context of random feature models, the trade-off between approximation and generalization changes with different scalings by considering the impact of the number of parameters and sample size relative to model complexity and generalization capabilities.

Overparametrized Regime: When there are significantly more parameters than samples (p ≫ n), the model's performance is limited by the number of samples, leading to a scenario where RFRR behaves as if it had an infinite number of random features. In this case, achieving optimal test error is akin to Kernel Ridge Regression (KRR) with infinitely many parameters.

Underparametrized Regime: Conversely, in situations where there are far fewer parameters than samples (n ≫ p), the statistical error dominates, making the sample size a limiting factor for learning. Here, RFRR matches its best approximation error over the random feature class as if it had an infinite number of training samples.

Critical Parametrization Regime: At critical parametrizations where p is comparable to n or when both scale similarly in relation to d (κ1 = κ2), RFRR test error interpolates between KRR test error and approximation error. This regime showcases peaks at specific points like interpolation thresholds due to factors such as matrix conditioning numbers.

What implications do these findings have for practical applications of random feature models

The implications of these findings for practical applications of random feature models are significant:

Optimal Model Complexity Selection: The results provide insights into choosing an optimal number of features relative to sample size for efficient learning without sacrificing performance.

Computational Efficiency: Understanding how different scaling affects model behavior can guide computational strategies in selecting appropriate parameter sizes for improved efficiency without compromising accuracy.

Generalization Performance: By identifying trade-offs between approximation and generalization errors under various scaling scenarios, practitioners can make informed decisions on model architecture selection based on their dataset characteristics.

Benign Overfitting Phenomenon: Insights from these findings shed light on phenomena like benign overfitting observed in deep learning models trained with excessive parameters compared to data points.

How can these results be extended to other types of loss functions and regularization beyond the linear regime

Extending these results beyond linear regression settings involves exploring other types of loss functions and regularization methods outside linear regimes within random feature models:

Loss Function Variations:

Investigating how different loss functions impact model behavior under varying scalings could provide insights into optimizing neural network architectures using non-linear activation functions.

Regularization Techniques:

Extending analyses to include diverse regularization techniques such as Lasso or Elastic Net can offer a broader understanding of how regularization impacts model performance across different scaling regimes.

Non-linear Activation Functions:

Exploring effects on approximation-generalization trade-offs when employing non-linear activation functions like ReLU or Sigmoid can enhance our understanding beyond traditional linear settings.

By delving into these aspects, researchers can deepen their comprehension of complex machine learning systems' behaviors under diverse conditions beyond simple linear regression setups found in standard statistical analyses involving random feature models."