Redes Neurais - Fundamentos e derivação do algoritmo de retropropagação

Introduction

This tutorial will guide you through the fundamentals of neural networks and the backpropagation algorithm. Based on the lecture by Professor Marcos G. Quiles, you will learn how the backpropagation algorithm functions, including the derivation of weight update formulas for the output layer and hidden layers in a multi-layer perceptron (MLP). Understanding these concepts is essential for anyone interested in machine learning and artificial intelligence.

Step 1: Understanding Neural Networks

• Familiarize yourself with the basic structure of neural networks:

  • Input Layer: Receives initial data.
  • Hidden Layers: Processes data through neurons.
  • Output Layer: Produces the final output.

• Key concepts to grasp:

  • Neurons: Basic units that process information.
  • Weights: Parameters that adjust the input signal.
  • Activation Function: Determines the output of a neuron based on input.

Step 2: Introduction to Backpropagation

• Learn what backpropagation is:

  • An algorithm for training neural networks by minimizing error.
  • Works by calculating gradients and updating weights accordingly.

• Understand the significance:

  • Essential for learning in neural networks.
  • Enables the model to learn from errors and improve accuracy.

Step 3: Deriving Weight Update Formula for the Output Layer

• Follow these steps to derive the weight update formula:

  1. Calculate the error at the output layer:
     • Error = Target Output - Actual Output.
  2. Compute the gradient of the error with respect to weights:
     • Gradient = Error * Activation Function Derivative.
  3. Update the weights:
     • New Weight = Old Weight + Learning Rate * Gradient.

• Example formula:

  w_new = w_old + η * (y_true - y_pred) * f'(net_input)
  

  Where:

  • w_new: Updated weight
  • w_old: Previous weight
  • η: Learning rate
  • y_true: Target output
  • y_pred: Predicted output
  • f': Derivative of the activation function

Step 4: Deriving Weight Update Formula for Hidden Layers

• To derive the weight update for hidden layers, follow these steps:

  1. Calculate the error signal for hidden neurons:
     • Error = Sum of (Output Error * Weights of the next layer).
  2. Compute the gradient:
     • Gradient = Error * Activation Function Derivative.
  3. Update the weights similarly:
     • New Weight = Old Weight + Learning Rate * Gradient.

• Example formula:

  w_hidden_new = w_hidden_old + η * δ_hidden * f'(net_input)
  

  Where:

  • δ_hidden: Error term for hidden layer neurons.

Step 5: Implementing Backpropagation in Code

• Consider implementing the backpropagation algorithm in a programming language such as Python. Here’s a simple example outline:

def backpropagation(X, y, weights, learning_rate):
    # Forward pass
    output = forward_pass(X, weights)
    
    # Compute error
    error = y - output
    
    # Backward pass
    for layer in reversed(range(len(weights))):
        # Calculate gradients and update weights
        gradient = calculate_gradient(error, weights[layer])
        weights[layer] += learning_rate * gradient
    return weights

Conclusion

In this tutorial, you have learned the foundational concepts of neural networks and the backpropagation algorithm. You derived the weight update formulas for both the output and hidden layers of a multi-layer perceptron. These concepts are crucial for building and training effective neural network models. As a next step, consider implementing a complete neural network from scratch or using a library like TensorFlow or PyTorch to see backpropagation in action.