Kernel alignment risk estimator: Risk prediction from training data

Arthur Jacot, Berfin Simsek, Francesco Spadaro, Clément Hongler, Franck Gabriel

Research output: Contribution to journalConference articlepeer-review


We study the risk (i.e. generalization error) of Kernel Ridge Regression (KRR) for a kernel K with ridge ? > 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 ?K,? 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 ?K,?, 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.

Original languageEnglish (US)
JournalAdvances in Neural Information Processing Systems
StatePublished - 2020
Event34th Conference on Neural Information Processing Systems, NeurIPS 2020 - Virtual, Online
Duration: Dec 6 2020Dec 12 2020

ASJC Scopus subject areas

  • Computer Networks and Communications
  • Information Systems
  • Signal Processing


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