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Backpropagation in Neural Networks

Backpropagation is an algorithm for training neural networks by iteratively correcting prediction errors. It calculates the gradient of the loss function with respect to each model parameter , enabling updates via Gradient Descent to minimize the loss.

Mathematically, backpropagation employs the chain rule from calculus to propagate errors backward through the network. Each layer computes a partial derivative, which is used to adjust its weights. This iterative process continues until a convergence criterion is met, typically when the loss change falls below a threshold.

Backpropagation is especially critical in Supervised Learning, where models learn from labeled data to recognize patterns.

Important Notes

  • Gradient descent minimizes the loss by updating parameters in the direction of the negative gradient:
    .
  • Backpropagation adjusts weights based on computed error gradients, enabling effective deep model optimization.
  • Computational Cost: Backpropagation can be expensive for deep networks, requiring computation of gradients across all layers.
  • Vanishing/Exploding Gradients: In deep networks, gradients can become too small or too large, hindering effective training.

Example Application

A Feed Forward Neural Network trained for image classification uses backpropagation to minimize cross-entropy loss. The gradient of the loss is calculated layer-by-layer and weights are updated using an optimizer like Adam.

Backpropagation Step-by-Step (Manual Calculation)

When dealing with many parameters, like in neural networks, it can be helpful to think in terms of computation graphs.

Basic backpropagation procedure through a computation graph:

  1. Work right to left: starting from the output layer back towards the input.
  2. For each node:
    • Calculate the local derivative(s) (i.e., the derivative of the node’s output with respect to its input).
    • Combine this with the derivative of the loss with respect to the node using the chain rule.

Definition:
The “local derivative(s)” are the derivatives of the current node’s output with respect to each of its inputs or parameters.

Computation Graph Example (using Sympy)

To manually calculate gradients through a computation graph, symbolic differentiation can be used.

from sympy import symbols, diff
 
# Define symbols
x, w, b = symbols('x w b')
y = symbols('y')  # true label
 
# Define a simple model: prediction = wx + b
prediction = w*x + b
 
# Define a loss function: squared error
loss = (prediction - y)2
 
# Compute gradients
grad_w = diff(loss, w)
grad_b = diff(loss, b)
 
print(grad_w)
print(grad_b)

This symbolic approach helps verify gradient calculations during small-scale experiments.

Follow-Up Questions

  • How does backpropagation compare to optimization methods like Newton’s Method or evolutionary strategies?
  • What role does Regularisation play in addressing overfitting when training deep networks with backpropagation?

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