Applying Backpropagation: Step-by-step Calculations for Deep Network Training

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Backpropagation is a fundamental algorithm used to train deep neural networks. It involves calculating gradients of the loss function with respect to each weight in the network, allowing for weight updates that minimize errors. This article provides a step-by-step overview of the backpropagation process, focusing on the calculations involved in training a deep network.

Forward Pass

The process begins with a forward pass, where input data is propagated through the network. Each neuron computes a weighted sum of its inputs, adds a bias, and applies an activation function. The output of each layer serves as the input for the next layer until the final prediction is obtained.

Calculating the Loss

Once the network produces an output, the loss function measures the difference between the predicted output and the true label. Common loss functions include Mean Squared Error and Cross-Entropy. The goal is to minimize this loss through weight adjustments.

Backward Pass: Gradient Computation

The core of backpropagation involves computing the gradients of the loss with respect to each weight. Starting from the output layer, the error term is calculated and propagated backward through the network. This involves applying the chain rule to compute derivatives at each layer.

For each neuron, the error term is determined by multiplying the derivative of the activation function with the weighted sum of errors from the subsequent layer. These error terms are then used to compute gradients for each weight and bias.

Updating Weights

Using the computed gradients, weights are updated typically via gradient descent. The update rule subtracts a fraction of the gradient from the current weight, controlled by the learning rate. This process reduces the loss over successive iterations.

Summary of Key Steps

  • Perform a forward pass to compute predictions.
  • Calculate the loss between predictions and true labels.
  • Compute error terms starting from the output layer backward.
  • Calculate gradients for each weight and bias.
  • Update weights using gradient descent.