Fixing an error in Caponnetto and de Vito (2007)

The seminal paper of Caponnetto and de Vito (2007) provides minimax-optimal rates for kernel ridge regression in a very general setting. Its proof, however, contains an error in its bound on the effective dimensionality. In this note, we explain the mistake, provide a correct bound, and show that the main theorem remains true. The mistake lies in Proposition 3's bound on the effective dimensionalityN (λ), particularly its dependence on the parameters of the family of distributionsb and β . We discuss the mistake and provide a correct bound in Section 1.