# Hessian optimization (Newton method) using the direction given by the gradient to make the next iteration step of the parameters

Reading Deep Learning Book (page 86) I am having trouble understanding the reasons behind using the gradient ($$g$$) as the direction of the step of the parameters ($$x$$).

I understand that the Newton method consists on minimizing the second-order Taylor series approximation of the function ($$f(x_o + \delta x)$$) given by: $$f(x_o + \delta x) \approx f(x_o) + \delta x^T g +\frac{1}{2}\delta x^T \,H \,\,\delta x$$ Where $$g$$ is the gradient and $$H$$ is the hessian matrix. Thereby minimizing this expression w.r.t. $$\delta x$$ we obtain that the step direction should be $$\delta x= -H^{-1}\,g$$, so this is a direction different from the gradient.

But in the approach given in the text book, this step direction is given by a direction proportional to the gradient: $$\rightarrow \delta x = \alpha \,g$$ where $$\alpha$$ is the learning rate (scalar). Thereby minimizing $$f(x_o + \delta x)$$ with respect to $$\alpha$$ we can obtain that this learning rate should be the right term:

$$f(x_o + \delta x) \approx f(x_o)+ \alpha g^T g + \frac{1}{2} \alpha^2 g^T H g \,\,\,\,\,\,\,\,\,\,\rightarrow \,\,\,\,\,\,\,\,\,\,\alpha = \frac{g^Tg}{g^THg}$$

What I am having difficulties with is understanding if with this second approach we are able to make use of the curvature of the function, $$f(x)$$, in order to make the next step on the parameters ($$x$$). So my questions are:

1. Considering $$\delta x = \alpha g$$, are we able to take account of the curvature of the function in order to make the next iteration of $$x$$?
2. Which are the advantages of using $$\delta x = \alpha g$$ in comparison to $$\delta x= -H^{-1}\,g$$?

• I find it useful to realize that with the second approach of my question we don't need to invert the hessian $\rightarrow$ less computing cost. But I still don't understand if this second approach takes into acount the curvature. Any more help about that would be appreciated and I also appreciated your help Eric. – Javier TG Sep 5 '20 at 19:27
My bad, a few pages later the author explains that the second approach (using $$\delta x = \alpha g$$ with $$\alpha = g^Tg/(g^THg$$)) does not take account of the curvature.
In case it may help to someone and in order to visualize this, I have plotted the optimization paths for each method for the function $$4x^2 + y^2$$ and starting point $$(15,15)$$:
As expected we can see there that only the step made by the original Newton method ($$\delta x = -H^{-1}g$$) takes advantage of the curvature of the function.