Multi-Layer Perceptron Learning in Tensorflow

Multi-Layer Perceptron (MLP) consists of fully connected dense layers that transform input data from one dimension to another. It is called multi-layer because it contains an input layer, one or more hidden layers and an output layer. The purpose of an MLP is to model complex relationships between inputs and outputs.

Components of Multi-Layer Perceptron (MLP)

• Input Layer: Each neuron or node in this layer corresponds to an input feature. For instance, if you have three input features the input layer will have three neurons.
• Hidden Layers: MLP can have any number of hidden layers with each layer containing any number of nodes. These layers process the information received from the input layer.
• Output Layer: The output layer generates the final prediction or result. If there are multiple outputs, the output layer will have a corresponding number of neurons.

Every connection in the diagram is a representation of the fully connected nature of an MLP. This means that every node in one layer connects to every node in the next layer. As the data moves through the network each layer transforms it until the final output is generated in the output layer.

Working of Multi-Layer Perceptron

Let's see working of the multi-layer perceptron. The key mechanisms such as forward propagation, loss function, backpropagation and optimization.

1. Forward Propagation

In forward propagation the data flows from the input layer to the output layer, passing through any hidden layers. Each neuron in the hidden layers processes the input as follows:

1. Weighted Sum: The neuron computes the weighted sum of the inputs:

z = \sum_{i} w_i x_i + b

Where:

• x_i​ is the input feature.
• w_i​ is the corresponding weight.
• b is the bias term.

2. Activation Function: The weighted sum z is passed through an activation function to introduce non-linearity. Common activation functions include:

• Sigmoid: \sigma(z) = \frac{1}{1 + e^{-z}}
• ReLU (Rectified Linear Unit): f(z) = \max(0, z)
• Tanh (Hyperbolic Tangent): \tanh(z) = \frac{2}{1 + e^{-2z}} - 1

2. Loss Function

Once the network generates an output the next step is to calculate the loss using a loss function. In supervised learning this compares the predicted output to the actual label.

For a classification problem the commonly used binary cross-entropy loss function is:

L = -\frac{1}{N} \sum_{i=1}^{N} \left[ y_i \log(\hat{y}_i) + (1 - y_i) \log(1 - \hat{y}_i) \right]

Where:

• y_i is the actual label.
• \hat{y}_i is the predicted label.
• N is the number of samples.

For regression problems the mean squared error (MSE) is often used:

MSE = \frac{1}{N} \sum_{i=1}^{N} (y_i - \hat{y}_i)^2

3. Backpropagation

The goal of training an MLP is to minimize the loss function by adjusting the network's weights and biases. This is achieved through backpropagation:

1. Gradient Calculation: The gradients of the loss function with respect to each weight and bias are calculated using the chain rule of calculus.
2. Error Propagation: The error is propagated back through the network, layer by layer.
3. Gradient Descent: The network updates the weights and biases by moving in the opposite direction of the gradient to reduce the loss: w = w - \eta \cdot \frac{\partial L}{\partial w}

Where:

• w is the weight.
• \eta is the learning rate.
• \frac{\partial L}{\partial w}​ is the gradient of the loss function with respect to the weight.

4. Optimization

MLPs rely on optimization algorithms to iteratively refine the weights and biases during training. Popular optimization methods include:

• Stochastic Gradient Descent (SGD): Updates the weights based on a single sample or a small batch of data: w = w - \eta \cdot \frac{\partial L}{\partial w}
• Adam Optimizer: An extension of SGD that incorporates momentum and adaptive learning rates for more efficient training:
  • m_t = \beta_1 m_{t-1} + (1 - \beta_1) \cdot g_t
  • v_t = \beta_2 v_{t-1} + (1 - \beta_2) \cdot g_t^2
• Here g_t​ represents the gradient at time t and \beta_1, \beta_2 are decay rates.

Now that we are done with the theory part of multi-layer perception, let's go ahead and implement code in python using the TensorFlow library.

Implementing Multi Layer Perceptron

In this section, we will guide through building a neural network using TensorFlow.

1. Importing Modules and Loading Dataset

First we import necessary libraries such as TensorFlow, NumPy and Matplotlib for visualizing the data. We also load the MNIST dataset.

import tensorflow as tf
import numpy as np
import matplotlib.pyplot as plt
from tensorflow.keras.models import Sequential
from tensorflow.keras.layers import Flatten, Dense

(x_train, y_train), (x_test, y_test) = tf.keras.datasets.mnist.load_data()

2. Loading and Normalizing Image Data

Next we normalize the image data by dividing by 255 (since pixel values range from 0 to 255) which helps in faster convergence during training.

gray_scale = 255

x_train = x_train.astype('float32') / gray_scale
x_test = x_test.astype('float32') / gray_scale

print("Feature matrix (x_train):", x_train.shape)
print("Target matrix (y_train):", y_train.shape)
print("Feature matrix (x_test):", x_test.shape)
print("Target matrix (y_test):", y_test.shape)

Output:

3. Visualizing Data

To understand the data better we plot the first 100 training samples each representing a digit.

fig, ax = plt.subplots(10, 10)
k = 0
for i in range(10):
    for j in range(10):
        ax[i][j].imshow(x_train[k].reshape(28, 28), aspect='auto')
        k += 1
plt.show()

Output:

4. Building the Neural Network Model

Here we build a Sequential neural network model. The model consists of:

• Flatten Layer: Reshapes 2D input (28x28 pixels) into a 1D array of 784 elements.
• Dense Layers: Fully connected layers with 256 and 128 neurons, both using the relu activation function.
• Output Layer: The final layer with 10 neurons representing the 10 classes of digits (0-9) with sigmoid activation.

model = Sequential([
    Flatten(input_shape=(28, 28)),
    Dense(256, activation='sigmoid'),  
    Dense(128, activation='sigmoid'), 
    Dense(10, activation='softmax'),  
])

5. Compiling the Model

Once the model is defined we compile it by specifying:

• Optimizer: Adam for efficient weight updates.
• Loss Function: Sparse categorical cross entropy, which is suitable for multi-class classification.
• Metrics: Accuracy to evaluate model performance.

model.compile(optimizer='adam',
              loss='sparse_categorical_crossentropy',
              metrics=['accuracy'])

6. Training the Model

We train the model on the training data using 10 epochs and a batch size of 2000. We also use 20% of the training data for validation to monitor the model’s performance on unseen data during training.

mod = model.fit(x_train, y_train, epochs=10, 
          batch_size=2000, 
          validation_split=0.2)
          
print(mod)

Output:

7. Evaluating the Model

After training we evaluate the model on the test dataset to determine its performance.

results = model.evaluate(x_test, y_test, verbose=0)
print('Test loss, Test accuracy:', results)

Output:

Test loss, Test accuracy: [0.2682029604911804, 0.9257000088691711]

We got the accuracy of our model 92% by using model.evaluate() on the test samples.

8. Visualizing Training and Validation Loss VS Accuracy

plt.figure(figsize=(12, 5))

plt.subplot(1, 2, 1)
plt.plot(mod.history['accuracy'], label='Training Accuracy', color='blue')
plt.plot(mod.history['val_accuracy'], label='Validation Accuracy', color='orange')
plt.title('Training and Validation Accuracy', fontsize=14)
plt.xlabel('Epochs', fontsize=12)
plt.ylabel('Accuracy', fontsize=12)
plt.legend()
plt.grid(True)

plt.subplot(1, 2, 2)
plt.plot(mod.history['loss'], label='Training Loss', color='blue')
plt.plot(mod.history['val_loss'], label='Validation Loss', color='orange')
plt.title('Training and Validation Loss', fontsize=14)
plt.xlabel('Epochs', fontsize=12)
plt.ylabel('Loss', fontsize=12)
plt.legend()
plt.grid(True)

plt.suptitle("Model Training Performance", fontsize=16)
plt.tight_layout()
plt.show()

Output:

The model is learning effectively on the training set, but the validation accuracy and loss levels off which might indicate that the model is starting to overfit.

Advantages of Multi Layer Perceptron

• Versatility: MLPs can be applied to a variety of problems, both classification and regression.
• Non-linearity: Using activation functions MLPs can model complex, non-linear relationships in data.
• Parallel Computation: With the help of GPUs, MLPs can be trained quickly by taking advantage of parallel computing.

Disadvantages of Multi Layer Perceptron

• Computationally Expensive: MLPs can be slow to train especially on large datasets with many layers.
• Prone to Overfitting: Without proper regularization techniques they can overfit the training data, leading to poor generalization.
• Sensitivity to Data Scaling: They require properly normalized or scaled data for optimal performance.

In short Multilayer Perceptron has the ability to learn complex patterns from data makes it a valuable tool in machine learning.