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 satisfies
![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS096354830000198X/resource/name/S096354830000198XeqnU1.gif?pub-status=live)
We 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
.