Just Interpolate: Kernel "Ridgeless" Regression Can Generalize (1808.00387v2)

Abstract: In the absence of explicit regularization, Kernel "Ridgeless" Regression with nonlinear kernels has the potential to fit the training data perfectly. It has been observed empirically, however, that such interpolated solutions can still generalize well on test data. We isolate a phenomenon of implicit regularization for minimum-norm interpolated solutions which is due to a combination of high dimensionality of the input data, curvature of the kernel function, and favorable geometric properties of the data such as an eigenvalue decay of the empirical covariance and kernel matrices. In addition to deriving a data-dependent upper bound on the out-of-sample error, we present experimental evidence suggesting that the phenomenon occurs in the MNIST dataset.

• The paper derives a data-dependent upper bound on out-of-sample error for minimum-norm interpolated solutions, revealing implicit regularization effects.
• It highlights that high-dimensional data, kernel curvature, and favorable geometric properties balance bias and variance naturally.
• Empirical results on MNIST validate that interpolated solutions can outperform regularized models under specific classification tasks.

Overview of "Just Interpolate: Kernel 'Ridgeless' Regression Can Generalize"

In the paper titled "Just Interpolate: Kernel 'Ridgeless' Regression Can Generalize," authors Tengyuan Liang and Alexander Rakhlin investigate the counterintuitive phenomenon observed when kernel methods in machine learning, specifically Kernel 'Ridgeless' Regression, are used without explicit regularization. Despite the conventional wisdom suggesting the necessity of regularization to mitigate overfitting, this paper presents analytical and empirical evidence of implicit regularization mechanisms that allow interpolated solutions to generalize effectively to out-of-sample data.

Key Contributions

The primary contribution of the paper is the derivation of a data-dependent upper bound on the out-of-sample error for minimum-norm interpolated solutions obtained using Kernel 'Ridgeless' Regression. This bound is significant because it provides theoretical insight into when and why such interpolation methods work, linking performance to factors such as high dimensionality, kernel curvature, and the geometric properties of the data.

Methodology and Results

The authors formalize the problem by examining Kernel 'Ridgeless' Regression, where a non-linear kernel that typically requires regularization is set to interpolate perfectly through the training data. Through meticulous analysis, they find that several factors contribute to the implicit regularization effect:

1. High Dimensionality: The phenomenon is particularly pronounced in high-dimensional settings, where the number of features is comparable to or exceeds the number of samples.
2. Kernel Curvature: Curvature in the kernel function imposes a form of structural bias, which is crucial for achieving a balance between bias and variance without explicit regularization.
3. Geometric Properties: Favorable geometric properties such as spectral decay in the empirical covariance and kernel matrices play a substantial role.

Empirically, the authors validate their theoretical findings using the MNIST dataset. By systematically examining the effects of setting the regularization parameter to zero (i.e., employing interpolating solutions), they demonstrate that interpolating solutions frequently outperform those with regularization across various pairs of digit classification tasks within the dataset.

Implications for Machine Learning

The implications of this paper extend to both practical applications and theoretical understandings of machine learning models, particularly in regimes where the traditional bias-variance trade-off is expected. The findings suggest that in certain high-dimensional settings, interpolated models can provide robust performance, prompting a reevaluation of regularization strategies and potentially simplifying model deployment by reducing the parameter tuning overhead.

From a theoretical standpoint, the paper invigorates discussions around the inherent complexities of learning in high-dimensional spaces, urging further exploration into the implicit biases induced by data and model architecture.

Future Directions

The research opens several avenues for future work, including:

• Exploring the precise relationship between the eigenvalue decay of data matrices and the effectiveness of implicit regularization.
• Extending the analysis to other kernel types beyond the scope of the current investigation, such as RBF kernels.
• Investigating the impact of implicit regularization in neural networks, particularly in deep learning architectures where overparameterization is a common occurrence.

In summary, Liang and Rakhlin's work on Kernel 'Ridgeless' Regression contributes a nuanced understanding of when interpolation without regularization can succeed, challenging established notions and providing new insights into model generalization in machine learning.