![]() dot ( A, w ) - b ) # record weight and cost weight_history. Newton’s method (also known as the Newton-Raphson method) is a centuries-old algorithm that is popular due to its speed in solving various optimization problems. In the example, you can cast the problem as looking for a root of the function f ( x) x ² - y. ![]() ![]() size ( hess_eval )) ** ( 0.5 ))) # solve second order system system for weight update A = hess_eval + epsilon * np. For those who simply want the code, skip right to the Coding section. It finds roots of an equation, i.e., values of x for which f ( x) 0. ![]() # using an automatic differentiator - like the one imported via the statement below - makes coding up gradient descent a breeze from autograd import grad from autograd import hessian # newtons method function - inputs: g (input function), max_its (maximum number of iterations), w (initialization) def newtons_method ( g, max_its, w, ** kwargs ): # compute gradient module using autograd gradient = grad ( g ) hess = hessian ( g ) # set numericxal stability parameter / regularization parameter epsilon = 10 ** ( - 7 ) if 'epsilon' in kwargs : beta = kwargs # run the newtons method loop weight_history = # container for weight history cost_history = # container for corresponding cost function history for k in range ( max_its ): # evaluate the gradient and hessian grad_eval = gradient ( w ) hess_eval = hess ( w ) # reshape hessian to square matrix for numpy linalg functionality hess_eval. For the single input case of the second order Taylor series approximation centered at a point $v$ we can solve for the stationary point of the quadratic approximation by setting the first derivative of $h(w)$ to zero and solving. ![]()
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |