Notes on Backward Algorithm and Gradient Descent Algorithm in Neural Networks

Forward Algorithm Review

• The forward algorithm calculates the output of a neural network from the input layer to the output layer.

  • In a multilayer perceptron, each layer is fully connected to the next layer.

  • Each neuron calculates its output based on the inputs it receives, weights, and bias applied with an activation function.

Backward Algorithm

• The backward algorithm allows for the calculation of gradients of the loss function concerning the weights by propagating the error back through the network.

  • It utilizes the derivatives calculated during the forward pass to update weights.

  • It follows the principle that knowing the output layer error enables the calculation of hidden layer errors recursively.

Purpose of Backward Algorithm

• To minimize the error by adjusting the weights in the neural network, facilitating the learning process.

• Allows for weight updates via the chain rule of derivatives, taking the error from the output layer back through to the input layer.

Gradient Descent Algorithm

• Gradient descent is the optimization method used to update the weights of the neural network after training on input/output pairs.

  • The primary aim is to minimize the loss function, typically by adjusting weights in the opposite direction of the gradient.

• The update rule for weights is given by: $\Delta w = - \eta \frac{\partial E}{\partial w}$

  • Where:

  • ( Delta w ): Change in weight

  • ( eta ): Learning rate

  • ( frac{partial E}/{partial w} ): Derivative of the error with respect to the weight

Importance of Learning Rate ((eta))

• The learning rate controls how much to change the weights during training.

  • A large learning rate may cause overshoot of the minimum error.

  • A small learning rate converges slowly and may get stuck in local minima.

• It is essential to tune the learning rate for optimal performance.

Update Equations for Weights

• For weights in the output layer:
  w{kj} = w{kj} + Delta w_{kj}

• For weights in the hidden layer:
  w{ji} = w{ji} + Delta w_{ji}

• Where the changes ( Delta w ) are calculated using the errors propagated back from the output layer.

Error Backpropagation

• The error for each neuron in the output layer is calculated as: $E<em>k = d</em>k - y_k$

  • Where ( dk ) is the desired output and ( yk ) is the output after applying the activation function.

• To propagate the error back through the network:

  • Calculate the error term for each layer, using:
    $\Delta<em>k = E</em>k \cdot \phi'(v_k)$

  • Where ( \phi'(vk) ) is the derivative of the activation function at the output layer and ( Ek ) is the error at that neuron.

Updating Weights Using Errors

• After calculating ( \Delta ), update the weights:

  • For output layer:
    $\Delta w<em>{kj} = \eta \Delta</em>k \cdot y_j$

  • For hidden layers:
    $\Delta w<em>{ji} = \eta \Delta</em>j \cdot x_i$

  • Where ( x_i ) is the input to the hidden layer neuron.

Conclusion

• The backward algorithm and gradient descent work together in neural networks to ensure that the model learns correctly from the errors to produce accurate predictions.

• The iterative process of forward propagation followed by backward propagation allows for efficient learning of weights in a neural network.

Note: The mathematical expressions provided in this note should be verified for mathematical accuracy and formatted correctly as per notation standards in mathematical texts, particularly when resolved in a LaTeX-compatible format for publications or presentations.