Backpropagation Explained in Plain English (With a PyTorch Example)

If neural networks are powerful learning systems, backpropagation is the engine that trains them.

Without backpropagation, deep learning would not exist.

It is the algorithm that allows neural networks to learn from mistakes, adjusting millions (or even billions) of parameters so the model gradually improves during training.

In this article, we’ll explain what backpropagation is, how it works conceptually, and show a small PyTorch example.

What Is Backpropagation?

Backpropagation (short for backward propagation of errors) is the process used to compute how much each weight in a neural network contributed to the model’s error.

The goal is simple:

Determine how every parameter should change to reduce prediction error.

Backpropagation works together with an optimization algorithm like gradient descent.

The process looks like this:

1. The network makes a prediction.
2. The prediction is compared to the correct answer.
3. The error is measured using a loss function.
4. Gradients are calculated.
5. Model weights are updated to reduce the loss.

This cycle repeats thousands or millions of times during training.

The Training Loop of Neural Networks

A typical neural network training process follows these steps:

1. Forward Pass

Input data flows through the network to produce a prediction.

Input → Hidden Layers → Output

2. Loss Calculation

The prediction is compared to the true label.

Example loss functions:

• Mean Squared Error (MSE)
• Cross Entropy Loss
• Hinge Loss

The result is a numerical measure of error.

3. Backward Pass (Backpropagation)

The loss is propagated backward through the network.

Gradients are computed for every weight.

These gradients tell us:

How much each parameter influenced the final error.

4. Weight Update

An optimizer updates the model parameters.

Example update rule (simplified):

weight = weight - learning_rate * gradient

Over time, these updates improve model performance.

Why Backpropagation Is So Important

Before backpropagation was widely used, training multi-layer neural networks was extremely difficult.

Backpropagation enabled:

• deep neural networks
• convolutional networks
• transformer models
• large language models

Without it, modern AI systems like GPT-style models would not be possible.

A Minimal PyTorch Example

Let’s train a tiny neural network using backpropagation.

import torch
import torch.nn as nn
import torch.optim as optim

Simple neural network

model = nn.Sequential(
nn.Linear(2, 8),
nn.ReLU(),
nn.Linear(8, 1)
)

Example dataset

X = torch.tensor([[0.,0.],
[0.,1.],
[1.,0.],
[1.,1.]])

y = torch.tensor([[0.],
[1.],
[1.],
[0.]])

Loss function

criterion = nn.MSELoss()

Optimizer

optimizer = optim.Adam(model.parameters(), lr=0.01)

Training loop

for epoch in range(1000):

predictions = model(X)

loss = criterion(predictions, y)

optimizer.zero_grad()

loss.backward()

optimizer.step()

print("Final loss:", loss.item())

What Happens When loss.backward() Runs?

This single line triggers the entire backpropagation process.

PyTorch automatically:

1. Computes gradients for each parameter.
2. Applies the chain rule from calculus.
3. Propagates gradients backward through all layers.

These gradients are then used by the optimizer to update model weights.

The Chain Rule Behind Backpropagation

Backpropagation relies on the chain rule from calculus.

If a function depends on intermediate variables, the chain rule lets us compute the gradient step by step.

Example conceptually:

Loss → Output → Hidden Layer → Input

Gradients flow backward through the network, adjusting weights based on their contribution to the final error.

Backpropagation in Large AI Models

• Even the largest modern AI systems still rely on this same principle.
• Training models like large language models involves:
• trillions of gradient updates

massive datasets

distributed GPU training

But at the core, the algorithm is still backpropagation combined with gradient descent.

Related Neural Network Concepts

Backpropagation is closely connected to several other key ideas:

• Gradient Descent
• Loss Functions
• Optimization Algorithms
• Vanishing Gradients
• Training Stability

Understanding these concepts helps explain how modern deep learning systems are trained.

Final Thoughts

Backpropagation is one of the most important algorithms in machine learning.

It allows neural networks to learn from data by gradually improving their internal parameters.

Every modern deep learning system—from image recognition models to large language models—depends on this simple but powerful idea.

If you understand backpropagation, you understand the core mechanism that trains neural networks.

This article is part of the Neural Network Lexicon project, a growing resource explaining the most important concepts behind modern AI systems.