Hessian矩阵的梯度下降牛顿法

Question

I am implementing gradient descent for regression using newtons method as explained in the 8.3 section of the Machine Learning A Probabilistic Perspective (Murphy) book. I am working with two dimensional data in this implementation. I am using following notations.
x = input data points m*2
y = labelled outputs(m) corresponding to input data
H = Hessian matrix is defined as

gradient descent update

where loss function is defined as

In my case is array and H is

Here is my python implementation. However this is not working as cost is increasing in each iteration.

def loss(x,y,theta):
   m,n = np.shape(x)
   cost_list = []
   for i in xrange(0,n): 
     x_0 = x[:,i].reshape((m,1)) 
     predicted = np.dot(x_0, theta[i])    
     error = predicted - y
     cost = np.sum(error ** 2) /  m
     cost_list.append(cost)

   cost_list = np.array(cost_list).reshape((2,1))
   return cost_list


def NewtonMethod(x,y,theta,maxIterations):
   m,n = np.shape(x)
   xTrans  = x.transpose()
   H       = 2 * np.dot(xTrans,x) / m
   Hinv    = np.linalg.inv(H)
   thetaPrev = np.zeros_like(theta)
   best_iter = maxIterations
   for i in range(0,maxIterations):
     cost = loss(x,y,theta)
     theta  = theta - np.dot(Hinv,cost))
     if(np.allclose(theta,thetaPrev,rtol=0.001,atol=0.001)):
        break;
     else:
       thetaPrev = theta
       best_iter = i

   return theta

Here are the sample values I used

import numpy as np

x = np.array([[-1.7, -1.5],[-1.0 , -0.3],[ 1.7 ,  1.5],[-1.2, -0.7 ][  0.6,  0.1]])  
y = np.array([ 0.3 ,  0.07, -0.2,  0.07,  0.03 ])
theta = np.zeros(2)
NewtonMethod(x,y,theta,100)

Need help / suggestions to fix this problem.
Thanks

Answer 1

You are effectively using a step size of 1. Try reducing the step size and see if that helps. That is, Instead of

Do this:

with a smaller value than 1.