# Error bounds for second order approximation

Where here we prove an approximation bound for twice continuously differentiable functions $f:\mathbb{R}^n \to \mathbb{R}$ with M-Lipschitzian Hessian, that is $\|\nabla^2 f(x) - \nabla^2 f(y)\| \leq M \|x-y\|$ for all $x,y$. In particular, we show that for all $x,y$

\begin{aligned} |f(y) - f(x) - \langle \nabla f(x), y-x\rangle - \frac{1}{2}\langle \nabla f^2(x)(y-x), y-x\rangle| \leq \frac{M}{6}\|x-y\|^3. \end{aligned}

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

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