How Backpropagation Works in Neural Networks: A Beginner’s Explanation

1. Introduction

Neural networks are powerful machine learning models inspired by the brain, and backpropagation is the key algorithm that enables them to learn from data. If you’ve heard terms like “training a neural network” or “adjusting weights,” they mostly refer to backpropagation in action.

Despite its intimidating name, backpropagation is intuitive when broken down step by step. In this post, we’ll explain how it works without drowning in complex math. Using simple analogies and a hands-on example, you’ll see how a neural network learns from mistakes by sending error signals backward to improve accuracy.

By the end, you’ll have a clear understanding of what backpropagation does under the hood—and why it’s crucial for deep learning.

2. Neural Network Basics Refresher

Before diving into backpropagation, let’s set the stage with a simple neural network. Imagine a network with:

One input,
One neuron (with a weight and bias),
One output.

This structure is similar to a simple linear regression model:

\begin{matrix} o u t p u t = (w e i g h t \times i n p u t) + b i a s \end{matrix}

In more complex networks, neurons are stacked in layers, with each layer’s output becoming the next layer’s input. However, the learning principle remains the same—adjusting weights to minimize errors.

3. How Does the Network Learn?

Initially, weights are set randomly, so the output may be far from correct. To measure its performance, the network uses a loss function (also called a cost function). One common choice is Mean Squared Error (MSE), which calculates how far the predicted output is from the actual value:

\begin{matrix} L o s s = (p r e d i c t e d - t r u e)^{2} \end{matrix}

If the prediction is perfect, loss = 0.
The larger the error, the higher the loss (since the squared term amplifies bigger mistakes).

This loss guides the learning process—the goal is to adjust weights so the loss gets smaller over time. This is where backpropagation and gradient descent come into play, which we’ll explore next.

4. What is Backpropagation?

Backpropagation (short for backward propagation of errors) is the key algorithm that helps a neural network learn by updating its weights based on the error.

4.1 How Does It Work?

- The network makes a prediction.
- We calculate the loss (how far off the prediction is).
- The loss is sent backward through the network to determine how much each weight contributed to the error.
- Each weight is adjusted in the right direction to reduce the error for the next prediction.

4.2 A Simple Way to Think About It

Imagine the network makes a wrong prediction. Backpropagation acts like a detective, asking:

Which internal connections (weights) caused this mistake?
How should we adjust them to improve next time?

It’s like assigning blame for an error and tweaking the system to correct it.

4.3 The Role of Gradients

Backpropagation uses calculus (specifically, partial derivatives) to compute gradients, which measure how much each weight affects the loss.

If a weight greatly influences the loss, its gradient is large (meaning it needs a significant adjustment).
If a weight has little effect, its gradient is small (requiring a minor update).

These gradients help the network fine-tune its weights systematically, rather than adjusting them randomly. This process, combined with gradient descent, enables the network to gradually improve its accuracy over multiple learning cycles.

5. Gradient Descent and Weight Updates

Once we have the gradients (slopes that indicate how much each weight affects the loss), we need a way to update the weights to improve the network’s predictions. This is where gradient descent comes in.

5.1 How Gradient Descent Works

Gradient descent is an optimization algorithm that updates the weights by moving in the opposite direction of the gradient (downhill).

5.1.1 Why the opposite direction?

The gradient tells us the direction in which loss increases.
To minimize the loss, we move in the opposite direction—just like walking downhill instead of climbing up.

5.2 The Role of the Learning Rate (η)

The learning rate (η, eta) controls how big of a step we take during each update.

A large learning rate makes big changes (but risks overshooting the best values).
A small learning rate makes tiny adjustments (slower but more precise).

5.3 Weight Update Formula

For a given weight w, the update rule is:

\begin{matrix} w_{n e w} = w_{o l d} - η \times \frac{\partial w}{\partial L o s s} ​ \end{matrix}

Where:

$\frac{\partial Loss}{\partial w}$ is the gradient (slope of the loss function).
$η$ (learning rate) controls the step size.

6. Intuition: Adjusting Weights Based on Error

If a weight’s gradient is large and positive, it means increasing that weight increases the loss, so we decrease the weight.
If the gradient is negative, it means increasing the weight would have reduced the loss, so we increase the weight.

This adjustment process happens for all weights after each training pass.

6.1 Types of Gradient Descent

Depending on how often we update the weights, there are different types of gradient descent:

Stochastic Gradient Descent (SGD) – Updates weights after each individual data point (noisy but updates fast).
Batch Gradient Descent – Updates weights after computing the average gradient on the entire dataset (slower but stable).
Mini-Batch Gradient Descent – A balance between the two, updating weights after a small batch of data (commonly used in deep learning).

Using gradient descent with backpropagation, the network gradually refines its weights to make better predictions over time.

7. Step-by-Step Example (A Tiny Network)

Step 1: Define the Network

We’ll use a basic one-input, one-output neural network, ignoring bias for simplicity:

Equation – Output

output = w \cdot x

Given:

Input: $x = 2$
True output: $y_{true} = 4$
Initial weight: $w = 1.0$ (randomly chosen)

Step 2: Forward Pass (Prediction)

The network makes a prediction:

y = w \times x = 1.0 \times 2 = 2.0

Step 3: Compute Loss

Using Mean Squared Error (MSE):

Loss = {(y_{pred} - y_{true})}^{2} = {(2.0 - 4.0)}^{2} = 4.0

Since the loss is high, the model needs to adjust its weight to improve predictions.

Step 4: Backpropagation (Compute Gradient)

To update $w$ , we compute the gradient of the loss with respect to $w$ .

\frac{\partial Loss}{\partial w} = 2 \times (w \times 2 - 4) \times 2

Substituting $w = 1.0$ :

= 2 \times (1.0 \times 2 - 4) \times 2 = 2 \times (2 - 4) \times 2 = 2 \times (- 2) \times 2 = - 8

A negative gradient means increasing $w$ would reduce the loss.

Step 5: Update Weight Using Gradient Descent

Using a learning rate of η = 0.1, we update the weight:

$w_{new} = w_{old} - η \times \frac{\partial Loss}{\partial w}$

$w_{new} = 1.0 - (0.1 \times - 8) = 1.0 + 0.8 = 1.8$

The weight increases from 1.0 to 1.8, bringing the prediction closer to the true value.

Step 6: Repeat the Process

New Forward Pass:

Equation 1

y = 1.8 \times 2 = 3.6

New Loss Calculation:

Equation 2

Loss = {(3.6 - 4)}^{2} = 0.16

New Gradient Calculation:

Equation 3

\frac{\partial Loss}{\partial w} = 2 \times (1.8 \times 2 - 4) \times 2 = - 1.6

New Weight Update:

Equation 4

w_{new} = 1.8 - (0.1 \times - 1.6) = 1.96

The model continues adjusting the weight with each iteration, gradually reducing the loss until it converges to the optimal weight 𝑤 = 2.0

In real-world neural networks, this process happens simultaneously for thousands or millions of weights, using vectorized operations for efficiency (e.g., in TensorFlow or PyTorch)

8. Key Intuition and Analogy: Backpropagation as a Teacher Correcting Homework

Imagine a student taking a math test and getting some answers wrong. Instead of just giving a final score, the teacher provides feedback on which mistakes the student made and how to fix them.

Student completes the test → (Neural network makes a prediction).
Teacher checks errors and marks mistakes → (Calculate the loss).
Teacher gives feedback on each mistake → (Backpropagation assigns blame to specific weights).
Student corrects mistakes and tries again → (Weights get updated, improving predictions).
Repeating the process → (After many rounds, the student gets better and learns the right answers).

Just like a teacher guiding a student to improve, backpropagation guides a neural network by adjusting its weights based on mistakes. Over time, the network learns to reduce errors and make more accurate predictions—just like a student improving with practice!

9. Why Backpropagation Matters

Backpropagation was a game-changer in machine learning because it made training multi-layer neural networks possible.

9.1 The Breakthrough

Before backpropagation (1970s–early 1980s), researchers struggled to train deep neural networks efficiently. The discovery (or re-discovery) of backpropagation in the mid-1980s showed that:
✔ Gradients for all weights in a neural network can be computed systematically.
✔ Multi-layer networks (deep learning) can be trained effectively.
✔ The algorithm scales well, making it feasible to adjust millions of parameters.

Today, modern deep learning frameworks like TensorFlow and PyTorch automate this process. For example, in PyTorch, calling .backward() on a loss automatically performs backpropagation, computing gradients for all network parameters.

9.2 Backpropagation Alone Isn’t Enough

While backpropagation is essential, it doesn’t guarantee a perfect model. Successful training also depends on:

Quality of data – Garbage in, garbage out!
Model architecture – A poor design won’t learn well.
Hyperparameters – Learning rate, batch size, and regularization affect training stability.

Without proper tuning, networks can:
❌ Get stuck in local minima (suboptimal solutions).
❌ Diverge if the learning rate is too high.

9.3 Why Differentiability Matters

Backpropagation requires that all functions in the network be differentiable (i.e., you can compute gradients for them).

This is why activation functions like ReLU and sigmoid are widely used.
Non-differentiable components break the gradient flow, preventing backpropagation from working.

10. Conclusion

In summary, backpropagation is the engine that allows a neural network to learn from its mistakes. By computing how much each weight contributes to the error, it adjusts them step by step to improve predictions.

While the underlying math involves calculus, the core idea is simple:

The network makes a guess.
Backpropagation figures out what went wrong.
The weights are tweaked slightly to reduce future errors.

Next time you hear “the network is training,” picture numbers flowing forward to make predictions and error signals flowing backward to fine-tune the system—like a feedback loop that happens millions of times.

Thanks to backpropagation (and lots of data!), neural networks can recognize images, understand speech, and solve complex problems—all through iterative improvement.