Question

I understand what Gradient Descent does. Basically it tries to move towards the local optimal solution by slowly moving down the curve. I am trying to understand what is the actual difference between the plain gradient descent and the Newton’s method?

From Wikipedia, I read this short line "Newton’s method uses curvature information to take a more direct route." What does this intuitively mean?

Answer 1

At a local minimum (or maximum) x, the derivative of the target function f vanishes: f'(x) = 0 (assuming sufficient smoothness of f).

Gradient descent tries to find such a minimum x by using information from the first derivative of f: It simply follows the steepest descent from the current point. This is like rolling a ball down the graph of f until it comes to rest (while neglecting inertia).

Newton’s method tries to find a point x satisfying f'(x) = 0 by approximating f' with a linear function g and then solving for the root of that function explicitely (this is called Newton’s root-finding method). The root of g is not necessarily the root of f', but it is under many circumstances a good guess (the Wikipedia article on Newton’s method for root finding has more information on convergence criteria). While approximating f', Newton’s method makes use of f'' (the curvature of f). This means it has higher requirements on the smoothness of f, but it also means that (by using more information) it often converges faster.