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Given the following text description, write Python code to implement the functionality described below step by step
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Description:
 <a href="http
Step1:  The class <code>plot_diagram</code> helps us to visualize the data space and the parameter space during training and has nothing to do with PyTorch.
Step2:  <!--Empty Space for separating topics-->
<h2 id="Makeup_Data">Make Some Data</h2>
Import PyTorch library
Step3:  Generate values from -3 to 3 that create a line with a slope of -3. This is the line you will estimate.
Step4:  Let us plot the line.
Step5:  Let us add some noise to the data in order to simulate the real data. Use <code>torch.randn(X.size())</code> to generate Gaussian noise that is the same size as <code>X</code> and has a standard deviation opf 0.1.
Step6:  Plot the <code>Y</code>
Step7:  <!--Empty Space for separating topics-->
<h2 id="Model_Cost">Create the Model and Cost Function (Total Loss)</h2>
In this section, let us create the model and the cost function (total loss) we are going to use to train the model and evaluate the result.
First, define the <code>forward</code> function $y=w*x$. (We will add the bias in the next lab.)
Step8:  Define the cost or criterion function using MSE (Mean Square Error)
Step9:  Define the learning rate <code>lr</code> and an empty list <code>LOSS</code> to record the loss for each iteration
Step10:  Now, we create a model parameter by setting the argument <code>requires_grad</code> to <code> True</code> because the system must learn it.
Step11:  Create a <code>plot_diagram</code> object to visualize the data space and the parameter space for each iteration during training
Step12:  <!--Empty Space for separating topics-->
<h2 id="Train">Train the Model</h2>
Let us define a function for training the model. The steps will be described in the comments.
Step13:  Let us try to run 4 iterations of gradient descent
Step14:  Plot the cost for each iteration
Step15:  <!--Empty Space for separating topics-->
<h3>Practice</h3>
Create a new learnable parameter <code>w</code> with an initial value of -15.0.
Step16:  Double-click <b>here</b> for the solution.
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Step17:  Double-click <b>here</b> for the solution.
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Step18:  Double-click <b>here</b> for the solution.
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Python Code:
# These are the libraries will be used for this lab.
import numpy as np
import matplotlib.pyplot as plt
Explanation: <a href="http://cocl.us/pytorch_link_top">
    <img src="https://cocl.us/Pytorch_top" width="750" alt="IBM 10TB Storage" />
</a> 
<img src="https://ibm.box.com/shared/static/ugcqz6ohbvff804xp84y4kqnvvk3bq1g.png" width="200" alt="cognitiveclass.ai logo" />
<h1>Linear Regression 1D: Training One Parameter</h1>
<h2>Table of Contents</h2>
<p>In this lab, you will train a model with PyTorch by using data that you created. The model only has one parameter: the slope.</p>
<ul>
    <li><a href="#Makeup_Data">Make Some Data</a></li>
    <li><a href="#Model_Cost">Create the Model and Cost Function (Total Loss)</a></li>
    <li><a href="#Train">Train the Model</a></li>
</ul>
<p>Estimated Time Needed: <strong>20 min</strong></p>
<hr>
<h2>Preparation</h2>
The following are the libraries we are going to use for this lab.
End of explanation
# The class for plotting
class plot_diagram():
    
    # Constructor
    def __init__(self, X, Y, w, stop, go = False):
        start = w.data
        self.error = []
        self.parameter = []
        self.X = X.numpy()
        self.Y = Y.numpy()
        self.parameter_values = torch.arange(start, stop)
        self.Loss_function = [criterion(forward(X), Y) for w.data in self.parameter_values] 
        w.data = start
        
    # Executor
    def __call__(self, Yhat, w, error, n):
        self.error.append(error)
        self.parameter.append(w.data)
        plt.subplot(212)
        plt.plot(self.X, Yhat.detach().numpy())
        plt.plot(self.X, self.Y,'ro')
        plt.xlabel("A")
        plt.ylim(-20, 20)
        plt.subplot(211)
        plt.title("Data Space (top) Estimated Line (bottom) Iteration " + str(n))
        plt.plot(self.parameter_values.numpy(), self.Loss_function)   
        plt.plot(self.parameter, self.error, 'ro')
        plt.xlabel("B")
        plt.figure()
    
    # Destructor
    def __del__(self):
        plt.close('all')
Explanation: The class <code>plot_diagram</code> helps us to visualize the data space and the parameter space during training and has nothing to do with PyTorch.
End of explanation
# Import the library PyTorch
import torch
Explanation: <!--Empty Space for separating topics-->
<h2 id="Makeup_Data">Make Some Data</h2>
Import PyTorch library:
End of explanation
# Create the f(X) with a slope of -3
X = torch.arange(-3, 3, 0.1).view(-1, 1)
f = -3 * X
Explanation: Generate values from -3 to 3 that create a line with a slope of -3. This is the line you will estimate.
End of explanation
# Plot the line with blue
plt.plot(X.numpy(), f.numpy(), label = 'f')
plt.xlabel('x')
plt.ylabel('y')
plt.legend()
plt.show()
Explanation: Let us plot the line.
End of explanation
# Add some noise to f(X) and save it in Y
Y = f + 0.1 * torch.randn(X.size())
Explanation: Let us add some noise to the data in order to simulate the real data. Use <code>torch.randn(X.size())</code> to generate Gaussian noise that is the same size as <code>X</code> and has a standard deviation opf 0.1.
End of explanation
# Plot the data points
plt.plot(X.numpy(), Y.numpy(), 'rx', label = 'Y')
plt.plot(X.numpy(), f.numpy(), label = 'f')
plt.xlabel('x')
plt.ylabel('y')
plt.legend()
plt.show()
Explanation: Plot the <code>Y</code>:
End of explanation
# Create forward function for prediction
def forward(x):
    return w * x
Explanation: <!--Empty Space for separating topics-->
<h2 id="Model_Cost">Create the Model and Cost Function (Total Loss)</h2>
In this section, let us create the model and the cost function (total loss) we are going to use to train the model and evaluate the result.
First, define the <code>forward</code> function $y=w*x$. (We will add the bias in the next lab.)
End of explanation
# Create the MSE function for evaluate the result.
def criterion(yhat, y):
    return torch.mean((yhat - y) ** 2)
Explanation: Define the cost or criterion function using MSE (Mean Square Error):
End of explanation
# Create Learning Rate and an empty list to record the loss for each iteration
lr = 0.1
LOSS = []
Explanation: Define the learning rate <code>lr</code> and an empty list <code>LOSS</code> to record the loss for each iteration:
End of explanation
w = torch.tensor(-10.0, requires_grad = True)
Explanation: Now, we create a model parameter by setting the argument <code>requires_grad</code> to <code> True</code> because the system must learn it.
End of explanation
gradient_plot = plot_diagram(X, Y, w, stop = 5)
Explanation: Create a <code>plot_diagram</code> object to visualize the data space and the parameter space for each iteration during training:
End of explanation
# Define a function for train the model
def train_model(iter):
    for epoch in range (iter):
        
        # make the prediction as we learned in the last lab
        Yhat = forward(X)
        
        # calculate the iteration
        loss = criterion(Yhat,Y)
        
        # plot the diagram for us to have a better idea
        gradient_plot(Yhat, w, loss.item(), epoch)
        
        # store the loss into list
        LOSS.append(loss)
        
        # backward pass: compute gradient of the loss with respect to all the learnable parameters
        loss.backward()
        
        # updata parameters
        w.data = w.data - lr * w.grad.data
        
        # zero the gradients before running the backward pass
        w.grad.data.zero_()
Explanation: <!--Empty Space for separating topics-->
<h2 id="Train">Train the Model</h2>
Let us define a function for training the model. The steps will be described in the comments.
End of explanation
# Give 4 iterations for training the model here.
train_model(4)
Explanation: Let us try to run 4 iterations of gradient descent:
End of explanation
# Plot the loss for each iteration
plt.plot(LOSS)
plt.tight_layout()
plt.xlabel("Epoch/Iterations")
plt.ylabel("Cost")
Explanation: Plot the cost for each iteration:
End of explanation
# Practice: Create w with the inital value of -15.0
# Type your code here
Explanation: <!--Empty Space for separating topics-->
<h3>Practice</h3>
Create a new learnable parameter <code>w</code> with an initial value of -15.0.
End of explanation
# Practice: Create LOSS2 list
# Type your code here
Explanation: Double-click <b>here</b> for the solution.
<!-- Your answer is below:
w = torch.tensor(-15.0, requires_grad=True)
-->
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Create an empty list <code>LOSS2</code>:
End of explanation
# Practice: Create your own my_train_model
gradient_plot1 = plot_diagram(X, Y, w, stop = 15)
Explanation: Double-click <b>here</b> for the solution.
<!-- Your answer is below:
LOSS2 = []
-->
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Write your own <code>my_train_model</code> function with loss list <code>LOSS2</code>. And run it with 4 iterations.
End of explanation
# Practice: Plot the list LOSS2 and LOSS
# Type your code here
Explanation: Double-click <b>here</b> for the solution.
<!-- Your answer is below:
def my_train_model(iter):
    for epoch in range (iter):
        Yhat = forward(X)
        loss = criterion(Yhat,Y)
        gradient_plot1(Yhat, w, loss.item(), epoch)
        LOSS2.append(loss)
        loss.backward()
        w.data = w.data - lr * w.grad.data
        w.grad.data.zero_()
my_train_model(4)
    -->
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Plot an overlay of the list <code>LOSS2</code> and <code>LOSS</code>.
End of explanation
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