Where here we prove an approximation bound for twice continuously differentiable functions with *M*-Lipschitzian Hessian, that is for all . In particular, we show that for all

This is stated as Lemma 1.2.4 in: Y. Nesterov, *Introductory Lectures on Convex Optimization – A basic course*, Kluwer Ac. Publishers, 2004.

Since is twice continuously differentiable, we may use the fact that

for some .

We may write this equivalently as (and this is maybe more convenient)

Since is -Lipschitz,

where . We may now easily plug (3) into (2) and prove the original inequality. In particular, because of (1):

where the first integral in (4) is

and the second integral’s absolute value is

[…] derived – for real-valued functions – when second order information is available. Read this post for details. In that case, the right-hand side involves a cubic term […]

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