Kernel Truncated Randomized Ridge Regression: Optimal Rates and Low Noise Acceleration
This work addresses a foundational problem in machine learning for regression tasks, offering improved theoretical guarantees.
The paper tackles nonparametric least square regression in RKHS by proposing a randomized algorithm that achieves optimal generalization error bounds, closing a gap between upper and lower bounds, and shows faster rates when Bayes risk is small.
In this paper, we consider the nonparametric least square regression in a Reproducing Kernel Hilbert Space (RKHS). We propose a new randomized algorithm that has optimal generalization error bounds with respect to the square loss, closing a long-standing gap between upper and lower bounds. Moreover, we show that our algorithm has faster finite-time and asymptotic rates on problems where the Bayes risk with respect to the square loss is small. We state our results using standard tools from the theory of least square regression in RKHSs, namely, the decay of the eigenvalues of the associated integral operator and the complexity of the optimal predictor measured through the integral operator.