All Random Features Representations are Equivalent

Random features are an important technique that make it possible to rewrite positive-definite kernels as infinite-dimensional dot products. Over time, increasingly elaborate random feature representations have been developed in pursuit of finite approximations with ever lower error. We resolve this arms race by deriving an optimal sampling policy, and show that under this policy all random features representations have the same approximation error. This establishes a lower bound that holds across all random feature representations, and shows that we are free to choose whatever representation we please, provided we sample optimally.