Corollary 4.14 (Convergence rates in probability). Suppose the regularisation strategy g α has qualification q ν + s. Suppose that Y ∈ L ψ 2 (P; Y), X ∈ L ψ 2 (P; X ), θ ⋆ ∈ Ω(ν, R), and 0 < α < 1. Let δ ∈ (0, 1 e ] and s ∈ [0, 1 2 ]. If the regularisation parameter α = α n is chosen to depend on the number n of **data** points via
α n := 1 √ n 1 ν+1
, then, for n n 0 := max X 4 L ψ 2 (P;X ) , 1152e 2 X 4 L ψ 2 (P;X ) log(1/δ)
1 ν 1+ν ,
with P ⊗n -probability at least 1 − 2δ,
θ ⋆ − θ αn C s XX S 2 (X ,Y) 3κ log(1/δ) 1 √ n s+ν 1+ν
, whereκ := max{κ ν,C XXγ R, 64κ D,B eB ψ 2 } < ∞ is obtained from the constants appearing in Propositions 4.12 and 4.13.