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.