TITLE: Newton's method intuition
QUESTION [10 upvotes]: In optimisation the Newton step is $-\nabla^2f(x)^{-1}\nabla f(x)$. Could someone offer an intuitive explanation of why the Newton direction is a good search direction? For example I can think of steepest gradient decent as a ball rolling down a hill (in three dimensions), I'm struggling to think of a similar analogy for the Newton direction.

REPLY [5 votes]: In Newton's method we pick $\Delta x$ so that $\nabla f(x_k + \Delta x) \approx 0$.  Note that $\nabla f(x_k + \Delta x) \approx \nabla f(x_k) + \nabla^2 f(x_k) \Delta x$.  Setting this equal to $0$ and solving for $\Delta x$ gives us the Newton step.
Another viewpoint is to start with 
\begin{equation*}
f(x_k + \Delta x) \approx f(x_k) + \langle \nabla f(x_k), \Delta x \rangle + \frac12 \langle \Delta x, \nabla^2 f(x_k) \Delta x \rangle.
\end{equation*}
Minimizing the right hand side with respect to $\Delta x$ gives us the Newton step.