## Background

Backpropagation is a common method for training a neural network. There is no shortage of papers online that attempt to explain how backpropagation works, but few that include an example with actual numbers. This post is my attempt to explain how it works with a concrete example that folks can compare their own calculations to in order to ensure they understand backpropagation correctly.

If this kind of thing interests you, you should sign up for my newsletter where I post about AI-related projects that I’m working on.

## Backpropagation in Python

You can play around with a Python script that I wrote that implements the backpropagation algorithm in this Github repo.

## Backpropagation Visualization

For an interactive visualization showing a neural network as it learns, check out my Neural Network visualization.

## Additional Resources

If you find this tutorial useful and want to continue learning about neural networks and their applications, I highly recommend checking out Adrian Rosebrock’s excellent tutorial on Getting Started with Deep Learning and Python.

## Overview

For this tutorial, we’re going to use a neural network with two inputs, two hidden neurons, two output neurons. Additionally, the hidden and output neurons will include a bias.

Here’s the basic structure:

In order to have some numbers to work with, here are the initial weights, the biases, and training inputs/outputs:

The goal of backpropagation is to optimize the weights so that the neural network can learn how to correctly map arbitrary inputs to outputs.

For the rest of this tutorial we’re going to work with a single training set: given inputs 0.05 and 0.10, we want the neural network to output 0.01 and 0.99.

## The Forward Pass

To begin, lets see what the neural network currently predicts given the weights and biases above and inputs of 0.05 and 0.10. To do this we’ll feed those inputs forward though the network.

We figure out the *total net input* to each hidden layer neuron, *squash* the total net input using an *activation function* (here we use the *logistic function*), then repeat the process with the output layer neurons.

*net input*by some sources.

Here’s how we calculate the total net input for :

We then squash it using the logistic function to get the output of :

Carrying out the same process for we get:

We repeat this process for the output layer neurons, using the output from the hidden layer neurons as inputs.

Here’s the output for :

And carrying out the same process for we get:

### Calculating the Total Error

We can now calculate the error for each output neuron using the squared error function and sum them to get the total error:

For example, the target output for is 0.01 but the neural network output 0.75136507, therefore its error is:

Repeating this process for (remembering that the target is 0.99) we get:

The total error for the neural network is the sum of these errors:

## The Backwards Pass

Our goal with backpropagation is to update each of the weights in the network so that they cause the actual output to be closer the target output, thereby minimizing the error for each output neuron and the network as a whole.

### Output Layer

Consider . We want to know how much a change in affects the total error, aka .

By applying the chain rule we know that:

Visually, here’s what we’re doing:

We need to figure out each piece in this equation.

First, how much does the total error change with respect to the output?

Next, how much does the output of change with respect to its total net input?

The partial derivative of the logistic function is the output multiplied by 1 minus the output:

Finally, how much does the total net input of change with respect to ?

Putting it all together:

You’ll often see this calculation combined in the form of the delta rule:

Alternatively, we have and which can be written as , aka (the Greek letter delta) aka the *node delta*. We can use this to rewrite the calculation above:

Therefore:

Some sources extract the negative sign from so it would be written as:

To decrease the error, we then subtract this value from the current weight (optionally multiplied by some learning rate, eta, which we’ll set to 0.5):

We can repeat this process to get the new weights , , and :

We perform the actual updates in the neural network *after* we have the new weights leading into the hidden layer neurons (ie, we use the original weights, not the updated weights, when we continue the backpropagation algorithm below).

### Hidden Layer

Next, we’ll continue the backwards pass by calculating new values for , , , and .

Big picture, here’s what we need to figure out:

Visually:

We’re going to use a similar process as we did for the output layer, but slightly different to account for the fact that the output of each hidden layer neuron contributes to the output (and therefore error) of multiple output neurons. We know that affects both and therefore the needs to take into consideration its effect on the both output neurons:

Starting with :

We can calculate using values we calculated earlier:

And is equal to :

Plugging them in:

Following the same process for , we get:

Therefore:

Now that we have , we need to figure out and then for each weight:

We calculate the partial derivative of the total net input to with respect to the same as we did for the output neuron:

Putting it all together:

You might also see this written as:

We can now update :

Repeating this for , , and

Finally, we’ve updated all of our weights! When we fed forward the 0.05 and 0.1 inputs originally, the error on the network was 0.298371109. After this first round of backpropagation, the total error is now down to 0.291027924. It might not seem like much, but after repeating this process 10,000 times, for example, the error plummets to 0.0000351085. At this point, when we feed forward 0.05 and 0.1, the two outputs neurons generate 0.015912196 (vs 0.01 target) and 0.984065734 (vs 0.99 target).

If you’ve made it this far and found any errors in any of the above or can think of any ways to make it clearer for future readers, don’t hesitate to drop me a note. Thanks!

My First Neural Net – Overthink

BP神经网络 | Codeba

Thanks for such a clear explanation of backprop

Wonderful tutorial. I have a question with the initial calculation in the forward propagation. The calculation performed above states that the input for h1 is (0.15 * 0.05) + (0.2 * 0.1)+bias (weights 1 & 2). However, the graph, I think, shows (0.15 * 0.05) + (0.25 * 0.1)+bias (weights 1 & 3).

Have I looked at the graph incorrectly?

Thanks!

If you look at the first graph, the one without the weight values, you can see that w1 and w2 is weights for h1 and that w3 and w4 are weights for h2. The graph with the values may make us mix up w2 and w3, because of the labels disposition.

Great tutorial, btw :)

I have found here an explanation on why the gradient for the output layer was simply “output – target”. Now, I got it. 1000 thanks for taking time to share :)

Very good explanation with nice example. Thank you very much!!!

A Deep Learning primer for all » Prithiviraj Damodaran

Thnaks a ton for such a crystal clear explanation! Made my day :)

this is the best explanation about backpropagation that I’ve ever found. thanks!

Actually why is that? I don’t see the explanations here… to me the formula here is (output – target) * g'(z). And I still don’t know why I read in some other materials that the error term of the output layer is just (output – target).

Very nice explanation !

I hope you could clarify one thing. In your diagram, the same bias is applied to all perceptrons in a given layer (b1 is applied to h1 and h2 , b2 is applied to o1 and o2).

Is that the normal approach? Shouldn’t we have a bias per perceptron?

Thanks a lot !

It appears there are several ways of doing it. One way is a single bias setting, no weights, to all nodes.

Another is, as above, using one setting for each layer. Equal to that would be ONE bias (of say 1) but different weights to each layer.

Yet another is one setting, but unique weights to each node.

Since the only goal is to help the logistic or sigmoid function not pass through the origin, they all work – but I think the reason to have them be different is to avoid shifting the entire function to the right or left simultaneously at all nodes, forcing adjustments to be very miniscule. With different weights, it offers the chance to adjust different nodes in different ways.

This example doesn’t include it, but some people also update the weights on the bias settings.

Awesome tutorial!

best resource on understanding back propagation I have came across.

Thank u for such a needful explanation

This valuable explanation saved a lot of time…Thanks a lot for the very clear explanation of BP function.

Thank you for the great tutorial!

When you do:

d(E_total) / d(out_o1) = 2 * 1 / 2 * (target_o1 – out_o1)^(2-1) * -1 + 0

Where does the part >> * -1<< come from ???

@henry

(Fog)’ = g’ * (f’og)

With g being a-x g’ is -1

F being (a-x)^2

F’og is 2 (a-x)

Very clear tutorial. Thank you very much