Solving Kernel Ridge Regression with Gradient-Based Optimization Methods

Kernel ridge regression, KRR, is a non-linear generalization of linear ridge regression. Here, we introduce an equivalent formulation of the objective function of KRR, opening up both for using other penalties than the ridge penalty and for studying kernel ridge regression from the perspective of gradient descent. Using a continuous-time perspective, we derive a closed-form solution, kernel gradient flow, KGF, with regularization through early stopping, which allows us to theoretically bound the differences between KGF and KRR. We generalize KRR by replacing the ridge penalty with the $\ell_1$ and $\ell_\infty$ penalties and utilize the fact that analogously to the similarities between KGF and KRR, the solutions obtained when using these penalties are very similar to those obtained from forward stagewise regression (also known as coordinate descent) and sign gradient descent in combination with early stopping. Thus the need for computationally heavy proximal gradient descent algorithms can be alleviated. We show theoretically and empirically how these penalties, and corresponding gradient-based optimization algorithms, produce signal-driven and robust regression solutions, respectively. We also investigate kernel gradient descent where the kernel is allowed to change during training, and theoretically address the effects this has on generalization. Based on our findings, we propose an update scheme for the bandwidth of translational-invariant kernels, where we let the bandwidth decrease to zero during training, thus circumventing the need for hyper-parameter selection. We demonstrate on real and synthetic data how decreasing the bandwidth during training outperforms using a constant bandwidth, selected by cross-validation and marginal likelihood maximization. We also show that using a decreasing bandwidth, we are able to achieve both zero training error and a double descent behavior.