Lecture 4: Backpropagation

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Backpropagation

COMP 4630 | Winter 2026 Charlotte Curtis

Overview

• A brief review of the history of neural networks
• Neurons, perceptrons, and multilayer perceptrons
• Backpropagation
• References and suggested reading:
  • Scikit-learn book: Chapter 10, introduction to artificial neural networks
  • Deep Learning Book: Chapter 6, deep feedforward networks

The rise and fall of neural networks

In between each era of excitement and advancement there was an “AI winter”

Model of a neuron

Threshold Linear Units (TLUs)

Training a perceptron

Multilayer perceptrons (MLPs)

• If a perceptron can’t even solve XOR, how can it do higher order logic?

• Consider that XOR can be rewritten as:

  A xor B = (A and !B) or (!A and B)
  

• A perceptron can solve and and or and not… so what if the input to the or perceptron is the output of two and perceptrons?

A solution to XOR

Backpropagation

• I just gave you the weights to solve XOR, but how do we actually find them?
• Applying the perceptron learning rule no longer works, need to know how much to adjust each weight relative to the overall output error
• Solution presented in 1986 by Rumelhart, Hinton, and Williams
• Key insight: Good old chain rule! Plus some recursive efficiencies

Training MLPs with backpropagation

1. Initialize the weights, through some random-ish strategy

2. Perform a forward pass to compute the output of each neuron

3. Compute the loss of the output layer (e.g. MSE)

4. Calculate the gradient of the loss with respect to each weight

5. Update the weights using gradient descent (minibatch, stochastic, etc)

6. Repeat steps 2-5 until stopping criteria met

   Step 4 is the “backpropagation” part

Example: forward pass

Example: calculate error and gradient

• We never picked a loss function! Let’s assume we’re using MSE
• For a single sample: $$\mathcal{L}({\mathbf{w}}^{(1)},{\mathbf{w}}^{(2)})=\frac{1}{2}(\hat{y}-y{)}^2=\frac{1}{2}{\left(\sum _{j=1}^2{w}_j^{(2)}\sum _{i=1}^2{x}_i{w}_{ij}^{(1)}-y\right)}^2$$ with the $1/2$ added for convenience
• The goal is to update each weight by a small amount to minimize the loss
• Fortunately, we know how to find a small change in a function with respect to one of the variables: the partial derivative!

Recursively applying the chain rule

• Weights in the second layer (connecting hidden and output): $$\frac{\partial \mathcal{L}}{\partial w_j^{(2)}}=\frac{\partial \mathcal{L}}{\partial \hat{y}}\frac{\partial \hat{y}}{\partial w_j^{(2)}}=(\hat{y}-y)\frac{\partial \hat{y}}{\partial w_j^{(2)}}=(\hat{y}-y)\sum _i{x}_i{w}_{ij}^{(1)}$$

• For the first layer (connecting inputs to hidden): $$\frac{\partial \mathcal{L}}{\partial w_{ij}^{(1)}}=\frac{\partial \mathcal{L}}{\partial \hat{y}}\frac{\partial \hat{y}}{h_j}\frac{\partial h_j}{\partial w_{ij}^{(1)}}=(\hat{y}-y)w_j^{(2)}{x}_i$$ where $h_j=x_i{w}_{ij}^{(1)}$ is the output of the hidden layer

Bias terms

• The toy example did not include bias terms, but these are very important (as seen in the perceptron examples)
• With a single layer we can add a column of 1s to $\mathbf{X}$, but with multiple layers we need to add bias at every layer
• The forward pass becomes: $$\hat{\mathbf{y}}=(\mathbf{X}{\mathbf{W}}^{(1)}+{\mathbf{b}}^{(1)}){\mathbf{W}}^{(2)}+{\mathbf{b}}^{(2)}$$
• Or in summation form: $$\hat{y}=\sum _{j=1}^2{w}_j^{(2)}\left(\sum _{i=1}^2{x}_i{w}_{ij}^{(1)}+b^{(1)}\right)+b^{(2)}$$

Gradient with respect to the bias terms

• For layer 2 (the output layer):

$$\frac{\partial \mathcal{L}}{\partial b_j^{(2)}}=\frac{\partial \mathcal{L}}{\partial \hat{y}}\frac{\partial \hat{y}}{\partial b^{(2)}}=(\hat{y}-y)(1)$$

• For layer 1: $$\frac{\partial \mathcal{L}}{\partial b^{(1)}}=\frac{\partial \mathcal{L}}{\partial \hat{y}}\frac{\partial \hat{y}}{h}\frac{\partial h}{\partial b^{(1)}}=(\hat{y}-y)\sum _i{w}_{ij}^{(2)}$$

  where $h={\sum }_i{x}_i{w}_{ij}^{(1)}+b^{(1)}$ is the input to the hidden layer

Summary in matrix form

Computational considerations

• Many of the terms computed in the forward pass are reused in the backward pass (such as the inputs to each layer)

• Similarly, gradients computed in layer $l+1$ are reused in layer $l$

• Typically each intermediate value is stored, but modern networks are big

  Model	Parameters
  Our example	6
  AlexNet (2012)	60 million
  GPT-3 (2020)	175 billion

Choices in neural network design

Activation functions

• The simple example used a linear activation function (identity)

• To include other activation functions, the forward pass becomes: $$\hat{\mathbf{y}}={\mathbf{f}}_2({\mathbf{f}}_1(\mathbf{X}{\mathbf{W}}^{(1)}+{\mathbf{b}}^{(1)}){\mathbf{W}}^{(2)}+{\mathbf{b}}^{(2)})$$

• The gradient in the output layer becomes: $$\frac{\partial \mathcal{L}}{\partial {\mathbf{W}}^{(2)}}=\frac{\partial \mathcal{L}}{\partial {\mathbf{f}}_2}\frac{\partial {\mathbf{f}}_2}{\partial {\mathbf{z}}^{(2)}}\frac{\partial {\mathbf{z}}^{(2)}}{\partial {\mathbf{W}}^{(2)}}$$ where ${\mathbf{z}}^{(2)}={\mathbf{f}}_1(\mathbf{X}{\mathbf{W}}^{(1)}+{\mathbf{b}}^{(1)}){\mathbf{W}}^{(2)}+{\mathbf{b}}^{(2)}$, or the summation the second layer before applying the activation function

• Problem! That step function in the original perceptron is not differentiable

Activation functions

• A common early choice was the sigmoid function: $$\sigma (z)=\frac{1}{1+e^{-z}},\frac{d\sigma }{dz}=\sigma (z)(1-\sigma (z))$$
• A more computationally efficient choice common today is the “ReLU” (Rectified Linear Unit) function: $$\text{ReLU}(z)=\max (0,z),\frac{d\text{ReLU}}{dz}=\{\begin{array}{ll}0& z<0\\ 1& z>0\end{array}$$

Activation functions in hidden layers

The design of hidden units is an extremely active area of research and does not yet have many definitive guiding theoretical principles. – Deep Learning Book, Section 6.3

• Activation functions in hidden layers serve to introduce nonlinearity
• Common for multiple hidden layers to use the same activation function
• Sigmoid, ReLU, and tanh (hyperbolic tangent) are common choices
• Also “leaky” ReLU, Parameterized ReLU, absolute value, etc
• Can be considered a hyperparameter of the network

Loss functions

• The choice of loss function is very important!
• Depends on the task at hand, e.g.:
  • Regression: MSE, MAE, etc
  • Classification: Usually some kind of cross-entropy (log likelihood)
• May or may not include regularization terms
• Must be differentiable, just like the activation functions

Activation functions in the output layer

• Activation functions in the output layer should be chosen based on the loss function (and thus the task)
  • Regression: linear
  • Binary classification: sigmoid
  • Multiclass classification: softmax (generalization of sigmoid)
• Again, must be differentiable

A complete fully connected network

Next up: Classification loss functions and metrics