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 dimensionality N (λ), particularly its dependence on the parameters of the family of distributions b and β. We discuss the mistake and provide a correct bound in Section 1. Its dependence on the regularization parameter λ, however, was correct, so the proof of Theorem 1 carries through with the exact same strategy. The proof was written in such a way, though, that it is not immediately obvious that it still holds for the corrected bound; we thus provide a more detailed explication of the proof, showing it is still valid. This note will make little sense without a copy of the original paper at hand. Numbered theorems and equation references always refer to those of Caponnetto and de Vito (2007); equations in this document are labeled alphabetically. A trivial correction First, we note a tiny mistake: Theorem 4 needs Cη = 96 log 2 6 η , rather than 32 log 2 6 η , because the last line of its proof dropped the constant 3 in front of S1(λ, z) and S2(λ, z) in (36). 1 Bound on the effective dimensionality Part of Proposition 3 is the claim that for p ∈ P(b, c), with c ∈ [1, 2] and b ∈ (1,∞), N (λ) ≤ βb b− 1 λ− 1 b . (a) The argument starts like this: N (λ) = Tr [ (T + λI)−1T ]