Kernel Alignment Risk Estimator: Risk Prediction from Training Data

TL;DR

This paper introduces the Signal Capture Threshold and Kernel Alignment Risk Estimator to predict and approximate the generalization risk of Kernel Ridge Regression directly from training data, enabling better kernel and hyperparameter selection.

Contribution

It proposes the KARE and SCT as novel tools for risk prediction in KRR, providing a data-dependent method for kernel and hyperparameter selection based on training data.

Findings

KARE accurately approximates KRR risk on real datasets.

The approach supports kernel and hyperparameter comparison directly from training data.

Numerical experiments validate the universality assumption and effectiveness of the method.

Abstract

We study the risk (i.e. generalization error) of Kernel Ridge Regression (KRR) for a kernel $K$ with ridge $\lambda>0$ and i.i.d. observations. For this, we introduce two objects: the Signal Capture Threshold (SCT) and the Kernel Alignment Risk Estimator (KARE). The SCT $\vartheta_{K,\lambda}$ is a function of the data distribution: it can be used to identify the components of the data that the KRR predictor captures, and to approximate the (expected) KRR risk. This then leads to a KRR risk approximation by the KARE $\rho_{K, \lambda}$, an explicit function of the training data, agnostic of the true data distribution. We phrase the regression problem in a functional setting. The key results then follow from a finite-size analysis of the Stieltjes transform of general Wishart random matrices. Under a natural universality assumption (that the KRR moments depend asymptotically on the first two moments of the observations) we capture the mean and variance of the KRR predictor. We numerically investigate our findings on the Higgs and MNIST datasets for various classical kernels: the KARE gives an excellent approximation of the risk, thus supporting our universality assumption. Using the KARE, one can compare choices of Kernels and hyperparameters directly from the training set. The KARE thus provides a promising data-dependent procedure to select Kernels that generalize well.