Forem: Ben Kemp

Backpropagation Explained in Plain English (With a PyTorch Example)

Ben Kemp — Fri, 13 Mar 2026 10:06:31 +0000

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:

The network makes a prediction.
The prediction is compared to the correct answer.
The error is measured using a loss function.
Gradients are calculated.
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:

Computes gradients for each parameter.
Applies the chain rule from calculus.
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.

Understanding Representation Learning in Neural Networks (With PyTorch Example)

Ben Kemp — Thu, 12 Mar 2026 17:22:16 +0000

Deep learning systems are powerful because they learn representations of data automatically.

Instead of engineers manually designing features, neural networks discover patterns on their own during training. This capability is known as representation learning, and it is one of the core reasons why modern AI models outperform traditional machine learning approaches.

From image recognition to large language models, representation learning is the engine behind many breakthroughs in artificial intelligence.

What Is Representation Learning?

Representation learning refers to a model’s ability to transform raw input data into meaningful internal features that help solve a task.

Traditional machine learning often relied on manually engineered features.

For example:

Problem --- Traditional Features --- Learned Representations

Image classification --- edges, color histograms --- hierarchical visual features
Speech recognition --- handcrafted audio features --- learned phoneme patterns
NLP --- bag-of-words --- contextual embeddings

Deep neural networks learn these representations automatically through training.

Each layer transforms the input data into a more abstract representation.

How Representations Emerge in Deep Networks

Neural networks process information through multiple layers.

Each layer applies transformations that progressively refine the data representation.

For example in computer vision:

Layer progression might look like:

Edges
Textures
Object parts
Complete objects

The deeper the network, the more abstract the representation becomes.

This hierarchical structure is why deep neural networks are effective at modeling complex patterns.

Representation Learning in Modern AI

Representation learning plays a major role in several key AI technologies.

Computer Vision

Convolutional neural networks learn spatial features from raw pixel data.

Natural Language Processing

Transformer models learn contextual token representations.

Recommendation Systems

User behavior patterns are encoded into latent feature vectors.

Speech Recognition

Acoustic signals are transformed into linguistic representations.

These internal representations allow neural networks to generalize beyond the training data.

A Simple PyTorch Example

Below is a minimal neural network demonstrating how hidden layers transform input data into internal representations.

import torch
import torch.nn as nn

class SimpleRepresentationNet(nn.Module):

def __init__(self):
    super().__init__()
    self.layer1 = nn.Linear(10, 32)
    self.layer2 = nn.Linear(32, 16)
    self.output = nn.Linear(16, 2)

def forward(self, x):
    x = torch.relu(self.layer1(x))
    x = torch.relu(self.layer2(x))
    return self.output(x)

model = SimpleRepresentationNet()

Example input

x = torch.randn(1, 10)

Forward pass

prediction = model(x)

print(prediction)

What Happens Inside the Network?

The layers progressively transform the input:

Layer Transformation
Input Raw numeric features
Layer 1 First learned representation
Layer 2 Higher-level abstraction
Output Task prediction

During training, the network learns which representations best solve the task.

Why Representation Learning Matters

Representation learning solved one of the biggest problems in classical machine learning: feature engineering.

Previously, performance depended heavily on manually designed features.

Deep learning changed this paradigm.

Now models can:

discover patterns automatically
build hierarchical abstractions
adapt to complex data distributions

This is why deep learning works so well in areas like:

computer vision
speech recognition
natural language processing
generative AI

Representation Learning in Large Language Models

Large language models rely heavily on representation learning.

The process typically looks like this:

Tokens are converted into embeddings
Attention layers refine contextual relationships
Hidden states become rich semantic representations
Output layers convert these representations into predictions

This allows models to understand relationships like:

semantic similarity
syntax
context dependencies

All without explicit feature engineering.

Related Neural Network Concepts

Representation learning connects to several other important deep learning ideas:

Feature Learning
Embeddings
Latent Representations
Transformer Attention
Self-Supervised Learning

Together these form the foundation of modern AI architectures.

Final Thoughts

Representation learning is one of the key innovations that enabled modern deep learning.

By allowing models to discover meaningful features automatically, neural networks can scale to complex tasks and large datasets.

Whether you are building computer vision systems, training language models, or developing recommendation engines, understanding representation learning is essential.