What is wrong in this Python code for Regularized Linear Regression?

Question

I wrote code with numpy(theta, X is numpy array):

def CostRegFunction(X, y, theta, lambda_):
    m = len(X)
    # add bias unit
    X = np.concatenate((np.ones((m,1)),X),1)

    H = np.dot(X,theta)

    J = (1 / (2 * m)) * (np.sum([(H[i] - y[i][0])**2 for i in range(len(H))])) + (lambda_ / (2 * m)) * np.sum(theta[1:]**2)

    grad_ = list()

    grad_.append((1 / m) * np.sum([(H[j] - y[j][0]) for j in range(len(H))]))
    for i in range(len(theta)-1):
        grad_.append((1 / m) * np.sum([(H[j] - y[j]) * X[j][i+1] for j in range(len(H))]) + (lambda_ / m) * theta[i+1])

    return J, grad_



def TrainLinearReg(X, y, theta, lambda_, alpha, iter):

    JHistory = list()
    for i in range(iter):
        J, grad = CostRegFunction(X, y, theta, Lambda_)
        JHistory.append(J)
        for j in range(len(theta)):
            theta[j] = theta[j] - alpha * grad[j]

    return theta, JHistory

Theta, JH = TrainLinearReg(X, y, th, Lambda_, 0.01, 50)

But when I try learn theta this code gives me a realy huge grow of theta and value of J. For example first iteration grad = [-15.12452, 598.435436] - it is correct. J is 303.3255 2nd iteration - grad = [10.23566,-3646.2345] J = 7924 and so on J grows faster and faster but on idea of LR it must be lower.

But if I use Normal Linear Equation in gives me a good Theta.

What is wrong in that code?

Answer 1

numpy version

. edited on Oct 17th

I rewrote parts of the code using numpy libraries.

All vectors are now columns numpy arrays.

import numpy as np
from copy import deepcopy as dc
from matplotlib import pyplot as plt

_norm = np.linalg.norm    

def CostRegFunction(X, y, theta, lambda_):
    m = len(X)
    H = np.dot(X,theta)
    J = (1 / (2 * m)) * _norm(H-y)**2 + (lambda_ / (2 * m)) * _normal(theta[1:])**2
    grad_ = np.array(sum(H-y)/m,ndmin=2).T
    for i in range(theta.shape[0]-1):
        grad_=np.concatenate((grad_,np.array(sum((H-y)*np.array(X[:,1],ndmin=2).T)/m + (lambda_/m) * theta[i+1],ndmin=2).T),0)
    return J, grad_

def TrainLinearReg(X, y, theta, lambda_, alpha, iter):
    JHistory = list()
    # add bias unit -> it's better to do it here, before entering the loop
    X = np.concatenate((np.ones((X.shape[0],1)),X),1)
    for i in range(iter):
        J, grad = CostRegFunction(X, y, theta, lambda_)
        JHistory.append(J)
        theta = theta -  alpha*grad
    return theta, JHistory

Then I generated a simple set of xy polynomial data with white noise and fitted the polynom-equation using the TrainLinearReg function.

x = np.concatenate((np.array(np.linspace(0,10,100) + np.random.normal(0,0.01,100),ndmin=2).T,\
np.array(np.linspace(0,10,100)**2 + np.random.normal(0,0.01,100),ndmin=2).T),1)
y = 2 + -3*np.array(x[:,0],ndmin=2).T + np.random.normal(0,3,[100,1]) - 2*np.array(x[:,1],ndmin=2).T
th = np.array([1,2,3],ndmin=2).T
alpha = 0.001
lambda_ = 0.1
Theta, JH = TrainLinearReg(x, y, dc(th), lambda_, alpha, 10000)

What I get is the following.

plt.plot(x[:,0],y,'o',label='Original Data',alpha = 0.5)
x2 = np.linspace(0,10,10)
plt.plot(x2,Theta[0]+x2*Theta[1]+x2**2*Theta[2],'-',label='Fitted     Curve',lw=1.5,alpha=0.8,color='black')
plt.gca().set_xlabel('x')
plt.gca().set_ylabel('y')
plt.legend()

Output >> Theta = array([[ 1.29259285],
                         [-2.97763304],
                         [-1.98758321]])

I hope I have been of some help.

Best regards, Gabriel