使用C＃进行线性回归梯度下降

Question

我现在正在参加Coursera机器学习课程而且我无法通过渐变下降线性回归函数来最小化。我使用：一个因变量，一个截距，以及四个x和y值，因此方程式非常简单。 Gradient Decent方程的最终值根据alpha和beta的初始值而变化很大，我无法弄清楚原因。 我只编写了大约两周的时间，所以我的知识至少可以说，如果你花时间帮忙，请记住这一点。

using System;
namespace LinearRegression
{
class Program
{
    static void Main(string[] args)
    {
        Random rnd = new Random();
        const int N = 4;

        //We randomize the inital values of alpha and beta
        double theta1 = rnd.Next(0, 100);
        double theta2 = rnd.Next(0, 100);

        //Values of x, i.e the independent variable
        double[] x = new double[N] { 1, 2, 3, 4 };
        //VAlues of y, i.e the dependent variable
        double[] y = new double[N] { 5, 7, 9, 12 };
        double sumOfSquares1;
        double sumOfSquares2;
        double temp1;
        double temp2;
        double sum;
        double learningRate = 0.001;
        int count = 0;

        do
        {
            //We reset the Generalized cost function, called sum of squares 
            //since I originally used SS to 
            //determine if the function was minimized
            sumOfSquares1 = 0;
            sumOfSquares2 = 0;
            //Adding 1 to counter for each iteration to keep track of how 
            //many iterations are completed thus far
            count += 1;

            //First we calculate the Generalized cost function, which is
            //to be minimized
            sum = 0;
            for (int i = 0; i < (N - 1); i++)
            {
                sum += Math.Pow((theta1 + theta2 * x[i] - y[i]), 2);
            }
            //Since we have 4 values of x and y we have 1/(2*N) = 1 /8 = 0.125
            sumOfSquares1 = 0.125 * sum;

            //Then we calcualte the new alpha value, using the derivative of 
            //the cost function. 
            sum = 0;
            for (int i = 0; i < (N - 1); i++)
            {
                sum += theta1 + theta2 * x[i] - y[i];
            }
            //Since we have 4 values of x and y we have 1/(N) = 1 /4 = 0.25
            temp1 = theta1 - learningRate * 0.25 * sum;

            //Same for the beta value, it has a different derivative
            sum = 0;
            for (int i = 0; i < (N - 1); i++)
            {
                sum += (theta1 + theta2 * x[i]) * x[i] - y[i];
            }
            temp2 = theta2 - learningRate * 0.25 * sum;

            //WE change the values of alpha an beta at the same time, otherwise the 
            //function wont work                
            theta1 = temp1;
            theta2 = temp2;

            //We then calculate the cost function again, with new alpha and beta values 
            sum = 0;
            for (int i = 0; i < (N - 1); i++)
            {
                sum += Math.Pow((theta1 + theta2 * x[i] - y[i]), 2);
            }
            sumOfSquares2 = 0.125 * sum;

            Console.WriteLine("Alpha: {0:N}", theta1);
            Console.WriteLine("Beta: {0:N}", theta2);
            Console.WriteLine("GCF Before: {0:N}", sumOfSquares1);
            Console.WriteLine("GCF After: {0:N}", sumOfSquares2);
            Console.WriteLine("Iterations: {0}", count);
            Console.WriteLine(" ");

        } while (sumOfSquares2 <= sumOfSquares1 && count < 5000);
        //we end the iteration cycle once the generalized cost function
        //cannot be reduced any further or after 5000 iterations            
        Console.ReadLine();
    }
}
}