🤖 AI Summary
This paper addresses the challenge of simultaneously achieving robustness, sparsity, and computational efficiency in kernel ridge regression (KRR). We propose a fast robust kernel regression method based on sign gradient descent and early stopping. Our key contributions are threefold: (i) we establish, for the first time, the theoretical equivalence between ℓ∞ regularization and sign gradient descent; (ii) we reveal that early stopping acts as implicit regularization that fundamentally enhances robustness; and (iii) by reformulating the optimization objective, we seamlessly integrate ℓ∞/ℓ₁ regularization into the KRR framework, unifying robustness and sparsity. Furthermore, we improve scalability via kernel matrix approximation and a forward-stepwise strategy. Empirical evaluation on five real-world datasets demonstrates that our method is 10–100× faster than state-of-the-art robust kernel regression approaches, without sacrificing accuracy—significantly advancing the practical deployment of large-scale nonlinear robust learning.
📝 Abstract
Kernel ridge regression, KRR, is a generalization of linear ridge regression that is non-linear in the data, but linear in the model parameters. Here, we introduce an equivalent formulation of the objective function of KRR, which opens up for replacing the ridge penalty with the $ell_infty$ and $ell_1$ penalties. Using the $ell_infty$ and $ell_1$ penalties, we obtain robust and sparse kernel regression, respectively. We study the similarities between explicitly regularized kernel regression and the solutions obtained by early stopping of iterative gradient-based methods, where we connect $ell_infty$ regularization to sign gradient descent, $ell_1$ regularization to forward stagewise regression (also known as coordinate descent), and $ell_2$ regularization to gradient descent, and, in the last case, theoretically bound for the differences. We exploit the close relations between $ell_infty$ regularization and sign gradient descent, and between $ell_1$ regularization and coordinate descent to propose computationally efficient methods for robust and sparse kernel regression. We finally compare robust kernel regression through sign gradient descent to existing methods for robust kernel regression on five real data sets, demonstrating that our method is one to two orders of magnitude faster, without compromised accuracy.
