# Gradient Descent vs Normal Equation for Regression Problems

# Gradient Descent vs Normal Equation for Regression Problems

### In this article, we will see the actual difference between gradient descent and the normal equation in a practical approach.

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*Gradient Descent v/s Normal Equation*

In this article, we will see the actual difference between gradient descent and the normal equation in a practical approach. Most of the newbie machine learning enthusiasts learn about gradient descent during the linear regression and move further without even knowing about the most underestimated Normal Equation that is far less complex and provides very good results for small to medium size datasets.

If you are new to machine learning, or not familiar with a normal equation or gradient descent, don’t worry I’ll try my best to explain these in layman’s terms. So, I will start by explaining a little about the regression problem.

## What Is a Linear Regression?

It is the entry-level **supervised** (both feature and target variables are given) machine learning algorithms. Suppose, we plot all these variables in the space, then the main task here is to **fit** the line in such a way that it **minimizes** the cost **function** or loss(don’t worry I’ll also explain this). There are various types of linear regression like a simple(one feature), multiple, and logistic(for classification). We have taken multiple linear regression in consideration for this article. The actual regression formula is:-

where **θ₀** and **θ₁** are the parameters that we have to find in such a way that they minimize the loss. In multiple regression, formula extended like θ₀+θ₁X₁+θ₂X₂. **Cost function** finds the error between the actual value and predicted value of our algorithm. It should be as **minimum** as possible. The formula for the same is:-

where m=number of examples or rows in the dataset, xᶦ=feature values of iᵗʰ example, yᶦ=actual outcome of iᵗʰ example.

## Gradient Descent

It is an **optimization** **technique** to find the best combination of parameters that **minimize** the cost function. In this, we start with random values of parameters(in most cases zero) and then keep changing the parameters to reduce **J(θ₀,θ₁)** or cost function until end up at a minimum. The formula for the same is:-

where **j** represents the no. of the parameter, **α** represents the learning rate. I will not discuss it in depth. You can find handwritten notes for these here.

## Normal Equation

It is an approach in which we can directly find the best values of **parameters** without using gradient descent. It is a very **effective** algorithm or ill say formula(as it consists of only **one** **line** ��)when you are working with smaller datasets.

The only problem with the normal equation is that finding the **inverse** of the **matrix** is computational very expensive in large datasets.

That’s a lot of theory, I know but that was required to understand the following code snippets. And I have only scratched the surface, so please google the above topics for in-depth knowledge.

## Prerequisites

I assume that you are familiar with **python** and already have installed the **python** **3** in your systems. I have used a **jupyter** **notebook** for this tutorial. You can use the **IDE** of your like. All required libraries come inbuilt in **anaconda** suite.

## Let’s Code

The **dataset** that I have used consists of **3** columns, two of them are considered as **features** and the other one as the **target** variable. Dataset available in GithHub.

**Firstly**, we need to import the **libraries** that we will use in this study. Here, `numpy`

is used to create NumPy arrays for training and testing data. `pandas`

for making the data frame of the dataset and retrieving values easily. `matplotlib.pyplot`

to plot the data like overall stock prices and predicted prices. `mpl_toolkits`

for plotting 3d data, `sklearn`

for splitting the dataset and calculating accuracy. We have also imported `time`

to calculate the time taken by each algorithm.

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`import pandas as pd`

`import numpy as np`

`from mpl_toolkits import mplot3d`

`import matplotlib.pyplot as plt`

`from sklearn.metrics import mean_squared_error`

`from sklearn.model_selection import train_test_split`

`import time`

We have loaded the dataset into the `pandas`

data frame and the shape of the dataset comes out to be **(1000,3),** then we simply print the first **5** rows with `head()`

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` df = pd.read_csv('student.csv')`

` print(df.shape)`

` df.head()`

*Preview of the dataset used*

Here, feature values namely **Math** and **Reading** are saved in variables **X1** and **X2** as NumPy arrays whereas the **Writing** column is considered as a target variable and its values saved in the **Y** variable. Then we have plotted this 3D data.

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` X1 = df['Math'].values`

` X2 = df['Reading'].values`

` Y = df['Writing'].valuesax = plt.axes(projection='3d')`

` ax.scatter(X1, X2, Y, c=Y, cmap='viridis', linewidth=0.5);`

*Visualization of data in 3D*

Now, **X****₀** is initialized as a `numpy`

array that consists of ones with the same dimensions as other features(It acts like **bias**). After that, we grouped all features in a single variable and transpose them to be in the right format. Then the data is **split** into **training** and **testing** with the help of t`rain_test_split`

and test size was **5%** that is **50** rows for testing. The shapes are given in the screenshot below.

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` X0 = np.ones(len(X1))`

` X = np.array([X0,X1,X2]).T`

` x_train,x_test,y_train,y_test = train_test_split(X,Y,test_size=0.05)`

` print("X_train shape:",x_train.shape,"\nY_train shape:",y_train.shape)`

` print("X_test shape:",x_test.shape,"\nY_test shape:",y_test.shape)`

*Shapes of training, testing sets*

### Gradient Descent

Here, **Q** denotes the list of parameters, which in our case are 3 (X₀, X₁, X₂), they are initialized as (0,0,0). **n** is just an integer with value equals to the number of training examples. Then we have defined our **cost** **function** that will be used in the gradient descent function to calculate the **cost** for every combination of **parameters**.

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` Q = np.zeros(3)`

` n = len(X1)`

` def cost_function(X,Y,Q):`

` return np.sum(((X.dot(Q)-Y)**2)/(2*n))`

This is the **gradient descent** function which takes **features**, **target** **variable**, **parameters**, **epochs**(number of iterations), and **alpha**(learning rate) as arguments. In the function, `cost_history`

is initialized to append the cost of every combination of parameters. After that, we started a loop to repeat the process of finding the parameters. Then we calculated the **loss** and the **gradient** term and updated the set of parameters. Here, you don't see the partial derivative term because the formula here is used after computing partial derivate(for reference, see the square term in the formula described above is canceled out with 2 in the denominator). At last, we call cost function to calculate the cost and append it in the `cost_history `

.

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` def gradient_descent(X,Y,Q,epochs,alpha):`

` cost_history = np.zeros(epochs)`

` for i in range(epochs):`

` pred = X.dot(Q)`

` loss = pred-Y`

` gradient = X.T.dot(loss)/n`

` Q = Q-gradient*alpha`

` cost_history[i] = cost_function(X,Y,Q)`

` return cost_history,Q`

Here, we have started the `timer`

before calling the above function and set epochs =**1000** and alpha =**0.0001**(should be low as possible) and the timer stops just after the execution of the function. So our gradient descent takes around **82** milliseconds for execution(1000 epochs).

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` start = time.time()`

` cost_his,parameters = gradient_descent(x_train,y_train.flatten(),Q,1000,0.0001)`

` end = time.time()`

` print(end - start)`

Here, we plotted the graph for `cost_history`

. As we can see the graph converges at around **400** epochs, so I run the gradient descent function with epochs=**400**, and this time it takes around **25.3** milliseconds. You can test this yourself by using the notebook from my GitHub.

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` x = [i for i in range(1,1001)]`

` plt.plot(x,cost_his)`

*Iteration V/S Cost*

It is time for **testing**. The `mean squared error`

comes out to be around **3.86** which is very acceptable.

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` y_pred = x_test.dot(parameters)`

` np.sqrt(mean_squared_error(y_pred,y_test))`

*Accuracy using Gradient Descent*

## Normal Equation

Normal Equation is very **simple** and you can see this in the code itself(only **single line**). As above, we have measured the time taken by the formula to calculate parameters. Don’t worry about the non-invertible matrix, `NumPy`

covers all of them here. It takes around **3** milliseconds(average). Isn’t it great!

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` start = time.time()`

` Q1 = np.linalg.inv(x_train.T.dot(x_train)).dot(x_train.T).dot(y_train)`

` end = time.time()`

` print(end - start)`

Finally, we have calculated the `mean squared error`

of **Normal** **Equation** which comes out to be **3.68**

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` pred_y = x_test.dot(Q1)`

` np.sqrt(mean_squared_error(pred_y,y_test))`

*Accuracy using Normal Equation*

*Consider Normal Equation before Gradient Descent*

## Conclusion

It turns out that **Normal** **Equation** takes **less** **time** to compute the **parameters** and gives almost similar results in terms of accuracy, also it is quite easy to use. According to me, the **Normal** **Equation** is better than **Gradient** **Descent** if the dataset size is not too large(**~20,000**). Due to the good computing capacity of today’s modern systems, the Normal Equation is the first algorithm to consider in cases of **regression**.

The source code is available on GitHub. Please feel free to make improvements.

Thank you for your precious time. And I hope you like this tutorial.

Opinions expressed by DZone contributors are their own.

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