The Downstream of A MLP from Scratch

In this blog, we will go step by step with how to create a mlp from scratch.

Some Utility Functions and The Dataset

# all of these libraries are used for plotting
import numpy as np
import matplotlib.pyplot as plt

# Plot the dataset
def plot_data(ax, X, Y):
    plt.axis('off')
    ax.scatter(X[:, 0], X[:, 1], s=1, c=Y, cmap='bone')

from sklearn.datasets import make_moons
X, Y = make_moons(n_samples=2000, noise=0.1)

MLP in NumPy and Define the Grid on Which The Classifier Will Be Evaluated

xx, yy = np.meshgrid(np.arange(x_min, x_max, .1),
                     np.arange(y_min, y_max, .1))

to_forward = np.array(list(zip(xx.ravel(), yy.ravel())))

# plot the decision boundary of our classifier


def plot_decision_boundary(ax, X, Y, classifier):
    # forward pass on the grid, then convert to numpy for plotting
    Z = classifier.forward(to_forward)
    Z = Z.reshape(xx.shape)
    
    # plot contour lines of the values of our classifier on the grid
    ax.contourf(xx, yy, Z>0.5, cmap='Blues')
    
    # then plot the dataset
    plot_data(ax, X,Y)

# Define the grid on which we will evaluate our classifier
xx, yy = np.meshgrid(np.arange(x_min, x_max, .1),
                     np.arange(y_min, y_max, .1))

to_forward = np.array(list(zip(xx.ravel(), yy.ravel())))

# plot the decision boundary of our classifier
def plot_decision_boundary(ax, X, Y, classifier):
    # forward pass on the grid, then convert to numpy for plotting
    Z = classifier.forward(to_forward)
    Z = Z.reshape(xx.shape)
    
    # plot contour lines of the values of our classifier on the grid
    ax.contourf(xx, yy, Z>0.5, cmap='Blues')
    
    # then plot the dataset
    plot_data(ax, X,Y)

Implementing the Linear Layer

class MyReLU(object):
    def forward(self, x):
        # the relu is y_i = max(0, x_i)
        # YOUR CODE HERE
        raise NotImplementedError()
        
    
    def backward(self, grad_output):
        # the gradient is 1 for the inputs that were above 0, 0 elsewhere
        # YOUR CODE HERE
        raise NotImplementedError()
    
    def step(self, learning_rate):
        # no need to do anything here, since ReLU has no parameters
        # YOUR CODE HERE
        raise NotImplementedError()

class MySigmoid(object):
    def forward(self, x):
        # the sigmoid is y_i = 1./(1+exp(-x_i))
        # YOUR CODE HERE
        raise NotImplementedError()
    
    def backward(self, grad_output):
        # the partial derivative is e^-x / (e^-x + 1)^2
        # YOUR CODE HERE
        raise NotImplementedError()
    
    def step(self, learning_rate):
        # no need to do anything here since Sigmoid has no parameters
        # YOUR CODE HERE
        raise NotImplementedError()

class MyLinear(object):
    def __init__(self, n_input, n_output):
        # initialize two random matrices for W and b (use np.random.randn)
        # YOUR CODE HERE
        raise NotImplementedError()

    def forward(self, x):
        # save a copy of x, you'll need it for the backward
        # return xW + b
        # YOUR CODE HERE
        raise NotImplementedError()

    def backward(self, grad_output):
        # y_i = \sum_j x_j W_{j,i}  + b_i
        # d y_i / d W_{j, i} = x_j
        # d loss / d y_i = grad_output[i]
        # so d loss / d W_{j,i} = x_j * grad_output[i]  (by the chain rule)
        # YOUR CODE HERE
        raise NotImplementedError()
        
        # d y_i / d b_i = 1
        # d loss / d y_i = grad_output[i]
        # YOUR CODE HERE
        raise NotImplementedError()
        
        # now we need to compute the gradient with respect to x to
        # continue the back propagation d y_i / d x_j = W_{j, i}
        # to compute the gradient of the loss, we have to sum over 
        # all possible y_i in the chain rule d loss / d x_j = \sum_i 
        # (d loss / d y_i) (d y_i / d x_j)
        # YOUR CODE HERE
        raise NotImplementedError()
    
    def step(self, learning_rate):
        # update self.W and self.b in the opposite direction of the 
        # stored gradients, for learning_rate
        # YOUR CODE HERE
        raise NotImplementedError()

Using the BCE loss

class Sequential(object):
    def __init__(self, layers):
        # YOUR CODE HERE
        raise NotImplementedError()
        
    def forward(self, x):
        # YOUR CODE HERE
        raise NotImplementedError()
    
    def compute_loss(self, out, label):
        # use the BCE loss
        # -(label * log(output) + (1-label) * log(1-output))
        # save the gradient, and return the loss      
        # beware of dividing by zero in the gradient.
        # split the computation in two cases, one where the label is 
        # 0 and another one where the label is 1
        # add a small value (1e-10) to the denominator
        # YOUR CODE HERE
        raise NotImplementedError()

    def backward(self):
        # apply backprop sequentially, starting from the gradient of the loss
        # YOUR CODE HERE
        raise NotImplementedError()
    
    def step(self, learning_rate):
        # take a gradient step for each layers
        # YOUR CODE HERE
        raise NotImplementedError()

h=50

# define your network with your Sequential
# it should be a linear layer with 2 inputs and h outputs, followed by a ReLU
# then a linear layer with h inputs and 1 outputs, followed by a sigmoid
# feel free to try other architectures

# YOUR CODE HERE
raise NotImplementedError()

# unfortunately animation is not working on colab
# you should comment the following line if on colab
%matplotlib notebook
fig, ax = plt.subplots(1, 1, facecolor='#4B6EA9')
ax.set_xlim(x_min, x_max)
ax.set_ylim(y_min, y_max)
losses = []
learning_rate = 1e-2
for it in range(10000):
    # pick a random example id
    j = np.random.randint(1, len(X))

    # select the corresponding example and label
    example = X[j:j+1]
    label = Y[j]

    # do a forward pass on the example
    # YOUR CODE HERE
    raise NotImplementedError()

    # compute the loss according to your output and the label
    # YOUR CODE HERE
    raise NotImplementedError()
    
    # backward pass
    # YOUR CODE HERE
    raise NotImplementedError()
    
    # gradient step
    # YOUR CODE HERE
    raise NotImplementedError()

    # draw the current decision boundary every 250 examples seen
    if it % 250 == 0 : 
        plot_decision_boundary(ax, X,Y, net)
        fig.canvas.draw()
plot_decision_boundary(ax, X,Y, net)
fig.canvas.draw()

%matplotlib inline
plt.plot(losses)

Using A Pytorch Module

import torch
import torch.nn as nn

# y = xw + b
class MyLinear_mod(nn.Module):
    def __init__(self, n_input, n_output):
        super(MyLinear_mod, self).__init__()
        # define self.A and self.b the weights and biases
        # initialize them with a normal distribution
        # use nn.Parameters
        # YOUR CODE HERE
        raise NotImplementedError()

    def forward(self, x):
        # YOUR CODE HERE
        raise NotImplementedError()
        
class MyReLU_mod(nn.Module):
    def __init__(self):
        super(MyReLU_mod, self).__init__()
        
    def forward(self, x):
        # YOUR CODE HERE
        raise NotImplementedError()

Subsequent section defines the network using MyLinearmod, MyReLUmod and nn.Sigmoid

from torch import optim
optimizer = optim.SGD(net.parameters(), lr=1e-2)

X_torch = torch.from_numpy(X).float()
Y_torch = torch.from_numpy(Y).float()

# you should comment the following line if on colab
%matplotlib notebook
fig, ax = plt.subplots(1, 1, facecolor='#4B6EA9')
ax.set_xlim(x_min, x_max)
ax.set_ylim(y_min, y_max)

losses = []
criterion = nn.BCELoss()
for it in range(10000):
    # pick a random example id 
    j = np.random.randint(1, len(X))

    # select the corresponding example and label
    example = X_torch[j:j+1]
    label = Y_torch[j:j+1].unsqueeze(1)

    # do a forward pass on the example
    # YOUR CODE HERE
    raise NotImplementedError()

    # compute the loss according to your output and the label
    # YOUR CODE HERE
    raise NotImplementedError()

    # zero the gradients
    # YOUR CODE HERE
    raise NotImplementedError()

    # backward pass
    # YOUR CODE HERE
    raise NotImplementedError()

    # gradient step
    # YOUR CODE HERE
    raise NotImplementedError()

    # draw the current decision boundary every 250 examples seen
    if it % 250 == 0 : 
        plot_decision_boundary(ax, X,Y, net)
        fig.canvas.draw()
plot_decision_boundary(ax, X,Y, net)
fig.canvas.draw()
%matplotlib inline
plt.plot(losses)

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