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(cross-posted from data-science StackExchange)(someone recommended that this community is more appropriate for my problem)

I am trying to implement a simple multivariate linear regression model without using any inbuilt machine libraries. So far, I have been able to get a root mean squared error for training about $2.93$ and the model from the normal (closed-form) equation is able to produce a training RMSE of $~2.3$. I am looking for ways in which I can improve my implementation of the gradient descent algorithm. Below is my implementation:

My gradient descent method looks like this: $\theta = \theta - [(\alpha/N) * X (X\theta - Y)]$ where $\theta$ is the model parameter, $N$ is the number of training elements, $X$ is the input and $Y$ are the target elements. $\alpha$ is the step size.

def gradientDescent(self):
    for i in range(self.iters):
        # T = T - (\alpha/N) * X*(XT - Y)
        self.theta = self.theta - (self.alpha/len(self.X)) * np.sum(self.X * (self.X @ self.theta.T - self.Y), axis=0)
    return errors

I had set the $\alpha$ as $0.1$ and number of iterations as 1000. The gradient descent reaches convergence at around 700-800 iterations (checked).

My error function is like:

def error_function(self):
        # Error function: (1/2N) * (XT - Y)^2 where T is theta
        error_values = np.power(((self.X @ self.theta.T) - self.Y), 2)
        return np.sum(error_values)/(2 * len(self.X))

I was expecting the training error from the gradient descent and the normal equations would turn out to be similar, but they have a bit of a huge difference. So, I wanted to know whether I am doing anything wrong or not.

PS I have not normalized the data, yet. Normalizing leads to a much lower RMSE (~$0.22$)

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    $\begingroup$ Gradient descent is pretty much the worst way to estimate linear regression parameters. To actually get it to work, you'll have to spend a lot of computational time and perhaps implement tricks like learning rate scheduling and momentum. Gradient descent is only useful to know about because of its utility in neural network optimization; in nearly all other contexts, better methods are available. stats.stackexchange.com/questions/160179/… Unless your sole purpose is to learn about g.d., use another method. $\endgroup$ – Sycorax Sep 18 '19 at 14:19
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    $\begingroup$ Also, in math you wrote $\frac{\alpha}{2N}$ but in code you wrote $\frac{\alpha}{N}$ for gradientDescent. The difference just means you have a different learning rate between the two expressions, but I think you would prefer to be consistent. $\endgroup$ – Sycorax Sep 18 '19 at 14:21
  • $\begingroup$ Well, I am just getting started into the field and was assigned to do the same, just to test my understanding. Although, thanks for the alternative methods. I will surely look up to them, sooner. Also, $\alpha/2N$ looks like a newbish mistake. I will patch that up. $\endgroup$ – MaJoR21 Sep 18 '19 at 14:25
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My approach without using any built-in libraries.

Libraries used: pandas,numpy,argparse

import numpy as np
import pandas as pd
import argparse

eta,t = 0.01, 0.00001  #optimal hyperparameters

#********Matrix Calculations***************
def mat(dataset):
    ds = np.genfromtxt(dataset,delimiter=',')

    x = ds[:,0:-1].reshape(-1, ds.shape[1]-1)
    ones = np.ones([x.shape[0],1])
    x = np.concatenate([ones, x],1)

    y = ds[:,-1].reshape(-1,1)

    w = np.zeros([1, x.shape[1]])

    return x, y, w

#********Calculating SSE********************
def sqrerror(x, y, w):    
    e = np.power(((x@w.T)-y),2)
    return np.sum(e)

#********Implementing Batch Linear Regression*************
def regressor(dataset,eta,t):
    x, y, w = mat(dataset)
    costmat,res=[],[]
    key=0
    while True:
        currres=[]
        currres.append(key)
        key += 1
        cost = sqrerror(x,y,w)
        for ol in w:
            for il in ol:
                currres.append(il)
        currres.append(cost)
        costmat.append(cost)
        res.append(currres)
        w = w-(eta)*np.sum((x@w.T-y)*x,axis=0) #Gradient Calculation 

        if key>1:
            if (costmat[-2]-costmat[-1]) <= t:
                break
    return res

#************Driver code*************************
if __name__ == "__main__":

    parser = argparse.ArgumentParser()
    parser.add_argument('--data',
                       help='your_dataset.csv')

    parser.add_argument('--eta',
                       help='learning rate = 0.0001')

    parser.add_argument('--threshold',
                       help='threshold value = 0.0001')

    args = parser.parse_args()

    dataset = args.data
    learning_rate = float(args.eta)
    threshold = float(args.threshold)

    op=regressor(dataset,learning_rate,threshold)
    print(pd.DataFrame(op))

Works even with masked data.

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