(Solved): 5. Newton's Method in Univariate Case ( 10 points \( ) \) : Consider minimizing a convex smooth fu ...

5. Newton's Method in Univariate Case ( 10 points \( ) \) : Consider minimizing a convex smooth function of one scalar variable, \( f(\beta) \). We gress an initial point \( \beta^{(0)} \), and take a second order Taylor expansion around \( \beta^{(0)} \) : \[ f(\beta) \approx f\left(\beta^{(0)}\right)+\left.\left(\beta-\beta^{(0)}\right) \frac{d f}{d w}\right|_{\beta=\beta^{(0)}}+\left.\frac{1}{2}\left(\beta-\beta^{(0)}\right)^{2} \frac{d^{2} f}{d w^{2}}\right|_{\beta=\beta^{(0)}} \] Show that to minimize the right-hand side, the next value of \( \beta \) to guess is: \[ \beta^{(1)}=\beta^{(0)}-\frac{f^{\prime}\left(\beta^{(0)}\right)}{f^{\prime \prime}\left(\beta^{(0)}\right)} \] where \( \left.\frac{d f}{d w}\right|_{\beta=\beta^{(0)}}=f^{\prime}\left(\beta^{(0)}\right) \) and \( \left.\frac{d^{2} f}{d w^{2}}\right|_{\beta=\beta^{(0)}}=f^{\prime \prime}\left(\beta^{(0)}\right) \). (hint: take derivative w.r.t. \( \beta \) ) In general, Newton's Method in multivariate case can be found at Slides #44 - 50 on https://www.cs.cmu.edu/ mgormley/courses/10701-f16/slides/lecture5.pdf We strongly recommend you to self-study these slides, as you will need Newton's Method for the next hands-on project anyway.