P1: Backward Algorithm and Gradient Descent Insights

Discussion of Backward Algorithm and Gradient Descent

• The focus today includes understanding the Backward Algorithm and Gradient Descent Algorithm.
• Important context: Previous class covered the Feed Forward Algorithm and conditions necessary for it.

Recap of Feed Forward Algorithm

• In a neural network, we discussed the Forward Pass method, going from the Input Layer to the Output Layer.
• Key formula discussed: $y<em>l = f(z</em>l)$ Where:
  • $zl = Wl imes y{l-1} + bl$
  • $W_l$ is the weight matrix,
  • $y_{l-1}$ is the output from the previous layer,
  • $b_l$ is the bias.

Introduction to Backward Algorithm

• Need for Backward Algorithm:
  • To minimize the error obtained during the output pass.
  • Important in the context of learning from outputs.

Single Layer Perceptron (SLP)

• Begin with a simplified model with only the output layer and no hidden layers:
  • Formula for weights:
    $z<em>1 = W</em>1 imes x + b_1$
  • Followed by activation to compute outputs:
    $y<em>1 = f(z</em>1)$

Computing Errors

• For outputs $d1$ and $d2$ (desired outputs):

  • Calculate error:
    $e<em>1 = d</em>1 - y<em>1$$e</em>2 = d<em>2 - y</em>2$

• Use these errors to update weights:
  $ext{Update Rule: } \Delta W<em>1 = ext{learning rate} imes e</em>1 imes x<em>1$$ext{Update Rule: } \Delta W</em>2 = ext{learning rate} imes e<em>2 imes x</em>2$

  • Each weight adjustment is based on the corresponding input and error value.

Gradient Descent Algorithm

• Explanation: A method to minimize the loss function by iteratively adjusting weights.
• Fundamental principle:
  • Learning rate ($ ext{eta}$) is a crucial parameter determining weight updates.

Multilayer Network (Deep Learning)

• Multilayer perceptron involves hidden layers:
  • Forward pass for a deeper network follows similar principles with more complexity:
  • Need to adjust weights for connections not directly involving the output.
• New challenge: Calculate errors for neurons in hidden layers.
  • Errors must be propagated backward through the network using the previously calculated weights and activations.

Weight Updates in Multilayer Networks

• Incorporate outputs from earlier layers to determine subsequent errors.
• Each layer has weights that need to be adjusted based on cumulative errors processed through layers:
  • The formula for weight updates can become complex:
    $W<em>{ij}^{new} = W</em>{ij}^{old} + ext{learning rate} imes e<em>j imes a</em>i$
• Here, $ej$ is the error from the output computed backwards, and $ai$ is the activation from the previous layer.

Conclusion

• Understanding both Backward Algorithm and Gradient Descent is essential for effective learning in neural networks.
• Advanced concepts can integrate various types of activation functions and deeper network structures leading to effective learning and minimizing error.
• Further mathematical details can be explored in the provided slides for a deeper understanding of these algorithms and their implementations.