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ADALINE with Stochastic Gradient Descent (Minibatch) | Implementing Stochastic Gradient Descent



In this notebook, we are implementing ADALINE "by hand" without using PyTorch's autograd capabilities.



Import Necessary Packages

import pandas as pd
import matplotlib.pyplot as plt
import torch
%matplotlib inline

Load & Prepare a Toy Dataset

df = pd.read_csv('./datasets/iris.data', index_col=None, header=None)
df.columns = ['x1', 'x2', 'x3', 'x4', 'y']
df = df.iloc[50:150]
df['y'] = df['y'].apply(lambda x: 0 if x == 'Iris-versicolor' else 1)
df.tail()

output:











# Assign features and target

X = torch.tensor(df[['x2', 'x3']].values, dtype=torch.float)
y = torch.tensor(df['y'].values, dtype=torch.int)

# Shuffling & train/test split

torch.manual_seed(123)
shuffle_idx = torch.randperm(y.size(0), dtype=torch.long)

X, y = X[shuffle_idx], y[shuffle_idx]

percent70 = int(shuffle_idx.size(0)*0.7)

X_train, X_test = X[shuffle_idx[:percent70]], X[shuffle_idx[percent70:]]
y_train, y_test = y[shuffle_idx[:percent70]], y[shuffle_idx[percent70:]]

# Normalize (mean zero, unit variance)

mu, sigma = X_train.mean(dim=0), X_train.std(dim=0)
X_train = (X_train - mu) / sigma
X_test = (X_test - mu) / sigma

plt.scatter(X_train[y_train == 0, 0], X_train[y_train == 0, 1], label='class 0')
plt.scatter(X_train[y_train == 1, 0], X_train[y_train == 1, 1], label='class 1')
plt.legend()
plt.show()

output:













plt.scatter(X_test[y_test == 0, 0], X_test[y_test == 0, 1], label='class 0')
plt.scatter(X_test[y_test == 1, 0], X_test[y_test == 1, 1], label='class 1')
plt.legend()
plt.show()

output:













Implement ADALINE Model

class Adaline1():
    def __init__(self, num_features):
        self.num_features = num_features
        self.weights = torch.zeros(num_features, 1, 
                                   dtype=torch.float)
        self.bias = torch.zeros(1, dtype=torch.float)

    def forward(self, x):
        netinputs = torch.add(torch.mm(x, self.weights), self.bias)
        activations = netinputs
        return activations.view(-1)
        
    def backward(self, x, yhat, y):  
        
        grad_loss_yhat = 2*(yhat - y)
        
        grad_yhat_weights = x
        grad_yhat_bias = 1.
        
        # Chain rule: inner times outer
        grad_loss_weights = torch.mm(grad_yhat_weights.t(),
                                         grad_loss_yhat.view(-1, 1)) / y.size(0)

        grad_loss_bias = torch.sum(grad_yhat_bias*grad_loss_yhat) / y.size(0)
        
        # return negative gradient
        return (-1)*grad_loss_weights, (-1)*grad_loss_bias


Define Training and Evaluation Functions

####################################################
##### Training and evaluation wrappers
###################################################

def loss(yhat, y):
    return torch.mean((yhat - y)**2)


def train(model, x, y, num_epochs,
          learning_rate=0.01, seed=123, minibatch_size=10):
    cost = []
    
    torch.manual_seed(seed)
    for e in range(num_epochs):
        
        #### Shuffle epoch
        shuffle_idx = torch.randperm(y.size(0), dtype=torch.long)
        minibatches = torch.split(shuffle_idx, minibatch_size)
        
        for minibatch_idx in minibatches:

            #### Compute outputs ####
            yhat = model.forward(x[minibatch_idx])

            #### Compute gradients ####
            negative_grad_w, negative_grad_b = \
                model.backward(x[minibatch_idx], yhat, y[minibatch_idx])

            #### Update weights ####
            model.weights += learning_rate * negative_grad_w
            model.bias += learning_rate * negative_grad_b
            
            #### Logging ####
            minibatch_loss = loss(yhat, y[minibatch_idx])
            print('    Minibatch MSE: %.3f' % minibatch_loss)

        #### Logging ####
        yhat = model.forward(x)
        curr_loss = loss(yhat, y)
        print('Epoch: %03d' % (e+1), end="")
        print(' | MSE: %.5f' % curr_loss)
        cost.append(curr_loss)

    return cost

Train Model

model = Adaline1(num_features=X_train.size(1))
cost = train(model, 
             X_train, y_train.float(),
             num_epochs=20,
             learning_rate=0.1,
             seed=123,
             minibatch_size=10)

Output:

Minibatch MSE: 0.500 Minibatch MSE: 0.341 Minibatch MSE: 0.220 Minibatch MSE: 0.245 Minibatch MSE: 0.157 Minibatch MSE: 0.133 Minibatch MSE: 0.144 Epoch: 001 | MSE: 0.12142 Minibatch MSE: 0.107 Minibatch MSE: 0.147 Minibatch MSE: 0.064 Minibatch MSE: 0.079 Minibatch MSE: 0.185 Minibatch MSE: 0.063 Minibatch MSE: 0.135 Epoch: 002 | MSE: 0.09932 Minibatch MSE: 0.093 Minibatch MSE: 0.064 ... ...



Evaluate ADALINE Model


Plot Loss (MSE)

plt.plot(range(len(cost)), cost)
plt.ylabel('Mean Squared Error')
plt.xlabel('Epoch')
plt.show()

output:











Compare with analytical solution


print('Weights', model.weights)
print('Bias', model.bias)

Output:

Weights tensor([[-0.0763],

[ 0.4181]])

Bias tensor([0.4888])



def analytical_solution(x, y):
    Xb = torch.cat( (torch.ones((x.size(0), 1)), x), dim=1)
    w = torch.zeros(x.size(1))
    z = torch.inverse(torch.matmul(Xb.t(), Xb))
    params = torch.matmul(z, torch.matmul(Xb.t(), y))
    b, w = torch.tensor([params[0]]), params[1:].view(x.size(1), 1)
    return w, b

w, b = analytical_solution(X_train, y_train.float())
print('Analytical weights', w)
print('Analytical bias', b)

Output:

Analytical weights tensor([[-0.0703],

[ 0.4219]]) Analytical bias tensor([0.4857])



Evaluate on Evaluation Metric (Prediction Accuracy)

ones = torch.ones(y_train.size())
zeros = torch.zeros(y_train.size())
train_pred = model.forward(X_train)
train_acc = torch.mean(
    (torch.where(train_pred > 0.5, 
                 ones, 
                 zeros).int() == y_train).float())

ones = torch.ones(y_test.size())
zeros = torch.zeros(y_test.size())
test_pred = model.forward(X_test)
test_acc = torch.mean(
    (torch.where(test_pred > 0.5, 
                 ones, 
                 zeros).int() == y_test).float())

print('Training Accuracy: %.2f' % (train_acc*100))
print('Test Accuracy: %.2f' % (test_acc*100))

Output:

Training Accuracy: 90.00

Test Accuracy: 96.67



Decision Boundary

##########################
### 2D Decision Boundary
##########################

w, b = model.weights, model.bias - 0.5

x_min = -3
y_min = ( (-(w[0] * x_min) - b[0]) 
          / w[1] )

x_max = 3
y_max = ( (-(w[0] * x_max) - b[0]) 
          / w[1] )


fig, ax = plt.subplots(1, 2, sharex=True, figsize=(7, 3))

ax[0].plot([x_min, x_max], [y_min, y_max])
ax[1].plot([x_min, x_max], [y_min, y_max])

ax[0].scatter(X_train[y_train==0, 0], X_train[y_train==0, 1], label='class 0', marker='o')
ax[0].scatter(X_train[y_train==1, 0], X_train[y_train==1, 1], label='class 1', marker='s')

ax[1].scatter(X_test[y_test==0, 0], X_test[y_test==0, 1], label='class 0', marker='o')
ax[1].scatter(X_test[y_test==1, 0], X_test[y_test==1, 1], label='class 1', marker='s')

ax[1].legend(loc='upper left')
plt.show()

output:


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