# Newton's method optimization for Deep Learning

I'm reading this paper "Deep learning via Hessian-free optimization" by J. Martens, I am having difficulty figure out the following statement:

In the standard Newton's method, $$q_{\theta}(p)$$ is optimized by computing the $$N\times N$$ matrix $$B$$ and then solving the system $$Bp = −\nabla f(\theta)$$.

(section 3 of the paper)

Is there any theorem, or statement anywhere regarding why the above system needs to be solved to optimize the local approximation? I came across another paper that has a reference to J. Martens and has used the same statement.

The central idea motivating Newton’s method is that $$f$$ can be locally approximated around each $$\theta$$, up to 2nd-order, by the quadratic: $$f(\theta + p) \approx q_\theta(p) \equiv f(\theta) + \nabla f(\theta)^Tp + \frac{1}{2} p^TBp \, \, (1)$$ where $$B = H(\theta)$$ is the Hessian matrix of $$f$$ at $$\theta$$. Finding a good search direction then reduces to minimizing this quadratic with respect to $$p$$.
To minimize, you need to take the derivative of (1) with respect to $$p$$ and set it to zero:
$$\Rightarrow \nabla f(\theta) + Bp = 0$$
which is equivalent to $$Bp = -\nabla f(\theta)$$.