Backpropagation is a fundamental algorithm used in neural networks for training. It's a method to calculate the gradient of the loss function with respect to the weights in the network, which is then used to update the weights to minimize the loss.

Key Concepts

  • Neural Network: A series of algorithms that attempt to recognize underlying relationships in a set of data through a process that mimics the way the human brain operates.
  • Loss Function: A function that maps an output variable onto a real number that can be used to quantify a model's performance.

Steps of Backpropagation

  1. Forward Propagation: The input data is propagated forward through the network to generate an output.
  2. Compute Loss: The loss is calculated using the loss function.
  3. Backward Propagation: The gradient of the loss function is calculated with respect to the weights in the network.
  4. Update Weights: The weights are updated using the gradient.

Example

Let's say we have a simple neural network with one input, one hidden layer with one neuron, and one output. The input is ( x ), the hidden layer neuron is ( h ), and the output is ( y ).

  • The activation function for the hidden layer is ( \sigma(h) = \frac{1}{1 + e^{-h}} )
  • The output layer uses a linear activation function: ( y = wx + b )

The loss function is the mean squared error: ( L = \frac{1}{2}(y - t)^2 )

Where ( t ) is the target value.

Forward Propagation

  1. Compute the hidden layer output: ( h = wx + b )
  2. Compute the output: ( y = \sigma(h) )

Compute Loss

( L = \frac{1}{2}(y - t)^2 )

Backward Propagation

  1. Compute the gradient of the loss with respect to the output: ( \frac{\partial L}{\partial y} = y - t )
  2. Compute the gradient of the output with respect to the hidden layer output: ( \frac{\partial y}{\partial h} = \sigma'(h) = \sigma(h)(1 - \sigma(h)) )
  3. Compute the gradient of the hidden layer output with respect to the weights: ( \frac{\partial h}{\partial w} = x )

Update Weights

( w := w - \eta \frac{\partial L}{\partial w} )

Where ( \eta ) is the learning rate.

More Resources

For a deeper understanding of backpropagation, check out our Deep Learning Tutorial.

Neural Network

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