Valid generalisation from approximate interpolation

Let and be sets of functions from domain X to ℝ. We say that validly generalises from approximate interpolation if and only if for each η > 0 and ∈, δ ∈ (0,1) there is m0(η, ∈, δ) such that for any function t ∈ and any probability distribution on X, if m > m0 then with m-probability at least 1 – δ, a sample X = (x1, X2,…,xm) ∈ Xm satisfiesWe find conditions that are necessary and sufficient for to validly generalise from approximate interpolation, and we obtain bounds on the sample length m0{η,∈,δ) in terms of various parameters describing the expressive power of .