4.1 Principles of backpropagation and automatic differentiation

Backpropagation is the backbone of neural network training. It efficiently computes gradients, allowing weights to be updated through gradient descent. The process involves a forward pass to compute outputs and a backward pass to propagate error gradients.

The chain rule is crucial for gradient computation in neural networks. It allows for the calculation of gradients through multiple layers, forming the basis of backpropagation equations. Automatic differentiation techniques and computational efficiency strategies further enhance the training process.

Fundamentals of Backpropagation

Principles of backpropagation

• Backpropagation algorithm efficiently computes gradients in neural networks enabling weight updates during training (Gradient descent)
• Gradient descent optimization iteratively minimizes loss function adjusting weights in steepest descent direction (Stochastic gradient descent)
• Forward pass computes network outputs given input data applying activation functions at each layer (ReLU, Sigmoid)
• Backward pass propagates error gradients from output to input layers computing partial derivatives for weights and biases
• Error function measures difference between predicted and actual outputs (Mean Squared Error, Cross-Entropy)

Chain rule for gradient computation

• Chain rule of calculus $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$ computes derivatives of composite functions
• Gradient computation in neural networks applies chain rule to calculate gradients through multiple layers
• Backpropagation equations:
  1. Error gradient for output layer: $\frac{\partial E}{\partial y_j} = \frac{\partial E}{\partial a_j} \cdot f'(z_j)$
  2. Error gradient for hidden layer: $\frac{\partial E}{\partial h_i} = \sum_j \frac{\partial E}{\partial y_j} \cdot w_{ji}$
  3. Weight update rule: $\Delta w_{ji} = -\eta \frac{\partial E}{\partial w_{ji}}$
• Jacobian matrix contains all first-order partial derivatives of vector-valued function

Automatic Differentiation and Computational Efficiency

Automatic differentiation techniques

• Computational graphs represent mathematical expressions as directed acyclic graphs nodes for operations edges for data flow
• Forward-mode automatic differentiation computes derivatives alongside forward computation efficient for few inputs many outputs
• Reverse-mode automatic differentiation computes derivatives in reverse order of operations efficient for many inputs few outputs
• Dual numbers represent both function value and its derivative used in some automatic differentiation implementations
• Operator overloading implements automatic differentiation in object-oriented programming

Complexity of backpropagation algorithms

• Time complexity of backpropagation $O(W)$ where W is number of weights in network linear in connections between neurons
• Space complexity stores activations and gradients during forward and backward passes
• Optimization strategies:
  • Mini-batch gradient descent reduces memory requirements increases training speed
  • Parallel computation utilizes GPUs or distributed systems for faster processing
  • Sparse updates focus on updating only most significant weights
• Gradient checkpointing trades computation time for reduced memory usage selectively stores intermediate activations
• Pruning and quantization reduce model size and computational requirements by removing unnecessary connections or reducing weight precision