## 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.

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## 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, machine learning, and deep learning, I highly recommend checking out Adrian Rosebrock’s new book, Deep Learning for Computer Vision with Python. I really enjoyed the book and will have a full review up soon.

## 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!

Great tutorial!

But how can I update biases using back propagation ?

Same way. Just take the partial with respect to the bias instead of the weight.

It’s easy to generalize the method accroding to weight update. You just need use the similar way to calculate dEtotal/dbias to get the step for bias and update it.

very good

The best explanaition of backpropagation I’ve read! Really helped me, thank you!

A quick overview of back propagation | Panthmas

Thank you thank you thank you.

Just about to give up trying to understand back prop, before I saw this.

Thank you thank you thank you.

Just about to give up trying to understand back prop, before I saw this.

How to create a Neural Network in JavaScript in only 30 lines of code – Learning JavaScript

Very, nice.

Did not succeed at first, because i was using 0 and 1 in one of inputs.

I figured out that 0 does not adjust weight.

So i shifted inputs by 0.5 (-0.5 and 0.5 instead 0 and 1) and it worked.

Hi Matt,

One question. In this article you calculate the Error function at the end of the forward propagation process as

E_{total} = \sum \frac{1}{2}(target – output)^{2}

I understand that this way of calculating the Error function was mostly used in the past and now we should use cross entropy. However, getting back to the squared error function – because the difference between the target and output is power 2, the result is always positive (regardless whether target > output or vice versa). That means that regardless that the actual network output result (target – output) can be positive error or negative error, we always back propagate the positive E function and eventually use the fractions of it at any neuron to adjust its weight and bias.

So, the adjustment goes always in one direction. Since it was successfully used in the past, how that worked? Or getting back to your example, we can use different input number and come up with negative (target – output) but the Error function will still be positive, and so the weight and bios adjustments for each neuron.

Regards

Igor

The squared error is always positive. But for backpropagation you use the (partial) derivative of the error function, which is linear and hence can be positive or negative.

Man I’m studying for a final and this explained the algorithm better than the textbook. you’re actively the best

Amazing explanation!! can you please explain in the similar fashion about updating the bias. As i am confused bias is only for a layer how can we update for every neuron??

So what happens next? What do you mean by “after repeating this process 10,000 times, for example, the error plummets to 0.0000351085”? Should we keep using the same input record in all these 10000 iterations? I think I understood what has been explained by this text, but I wish you could ellaborate a bit more on the whole neural network learning process.

Also, can you provide a general idea on what is happening in your neural network visualization example? What were you feeding the network during all those many iterations?

Sorry for asking so many questions, it’s just I’m trying to get a deep understanding on this topic, but failing to find quality material that isn’t too difficult for beginner like me.

Thank you.

So what happens next? What do you mean by “after repeating this process 10,000 times, for example, the error plummets to 0.0000351085”? Should we keep using the same input record in all these 10000 iterations? I think I understood what has been explained by this text, but I wish you could ellaborate a bit more on the whole neural network learning process.

Also, can you provide a general idea on what is happening in your neural network visualization example? What were you feeding the network during all those many iterations?

Sorry for asking so many questions, it’s just I’m trying to get a deep understanding on this topic, but failing to find quality material that isn’t too difficult for beginner like me.

Thank you.

this network doesn’t work well, my outputs are exactly like the example above, but training 16000 inputs for xor problem, the error is still very big, with another net I got very small error with 2000 inputs and i didnt even touch eta

If I get your question correctly you’re asking, whether to keep using the same input for all training cycles, then the answer is no. The network learns by example. The more examples you show the more it will learn. If you only show one example, that’s the only case it will be able to work with. Depending on the complexity of the task you might need to teach tens of thousands of different samples throughout training. In some cases, it is appropriate to show the same sample multiple times. E.g. if you want to train XOR (you need at least one hidden layer for this) you have the possible samples (0,0 => 0) (1,0 => 1) (0,1 => 1) and (1,1 => 0). You run them in some random order until you have reached an error rate you’re comfortable with.

why should it be called backpropagation if you don’t update the weights after you calculed them? you can easily perform this operation from the input layer to the output layer and get the same result… are you sure about “we use the original weights, not the updated weights, when we continue the backpropagation algorithm below” ?

Updating and backpropagation are two separate steps. Backpropagation pushes the error back through the network to find out how much responsibility to assign to each weight. This responsibility is then used to update the weights. You can interleave backpropagation and update for each layer, but first, you have to calculate the error for the next layer before you can do the update.

Great explanation! Short and clear!

Great simpification a NN to a two layer single data structure. It’s really easy to learn for beginner. Thanks man.

Great tutorial just finished going through the math and managed to reproduce the calculation. Only a matter of time until I master it.

This was seriously helpful. Thanks for writing it!

why bias weight is not updated?

Simple and intuitive explanation !! This is what I was looking for. Thank you.

Really good tutorial! One of the most helpful ones I’ve come across.

Does the bias unit in each layer have only one weight or would there be a separate weight per connection with the nodes?

Man you explained it all, i have finally succeeded to implement hidden layer backpropagation now. THANKS!

Maybe I missed it, but I think you left out the update of the biases. It’s simple but still might not be obvious to everyone new to the topic.