Generalization Properties of Learning with Random Features

We study the generalization properties of ridge regression with random features in the statistical learning framework. We show for the first time that O(1/ √ n) learning bounds can be achieved with only O( √ n log n) random features rather than O(n) as suggested by previous results. Further, we prove faster learning rates and show that they might require more random features, unless they are sampled according to a possibly problem dependent distribution. Our results shed light on the statistical computational trade-offs in large scale kernelized learning, showing the potential effectiveness of random features in reducing the computational complexity while keeping optimal generalization properties