Just Interpolate: Kernel "Ridgeless" Regression Can Generalize
This addresses a theoretical puzzle in machine learning about generalization without explicit regularization, which is relevant for researchers in statistical learning theory.
The paper tackles the problem of understanding why kernel 'ridgeless' regression, which perfectly fits training data, can still generalize well on test data, and it derives an upper bound on out-of-sample error and provides experimental evidence on MNIST.
In the absence of explicit regularization, Kernel "Ridgeless" Regression with nonlinear kernels has the potential to fit the training data perfectly. It has been observed empirically, however, that such interpolated solutions can still generalize well on test data. We isolate a phenomenon of implicit regularization for minimum-norm interpolated solutions which is due to a combination of high dimensionality of the input data, curvature of the kernel function, and favorable geometric properties of the data such as an eigenvalue decay of the empirical covariance and kernel matrices. In addition to deriving a data-dependent upper bound on the out-of-sample error, we present experimental evidence suggesting that the phenomenon occurs in the MNIST dataset.