## 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.

## 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!

### And while I have you…

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I think the formuale below is wrong. You should apply the multivariable version of the chaine rule instead.

\frac{\partial E_{total}}{\partial w_{5}} = \frac{\partial E_{total}}{\partial out_{o1}} * \frac{\partial out_{o1}}{\partial net_{o1}} * \frac{\partial net_{o1}}{\partial w_{5}}

Hi, it seems your network visualization is giving an error. numInputs not defined. Please, let me know when it is working, I’d like to check whether I can reference or include it into my materials.

Best

Ingo Dahn

I’m assuming only one weight on the bias is just for simplicity?

Although this is a good post. I don’t think the writer understand about bias, that’s why there’s no concrete explanation about how to update bias in this post.

The author likely assigned the same bias for the hidden and output neurons for simplicity. The article does not show the step of updating the bias weights, presumably as it is the exact same process as is used to update other weights (it is actually simpler, since the bias node activation is always 1.)

Can give someone one example for updating one bias? I don’t know, how i can update one bias-

I like how you put this, will help for my exam tomorrow

Correct observation. The bias is not updated, therefore the total error in the first round and the second round goes like this:

Matt: E=0.298371109 -> E=0.291027924

If you correctly update the bias also, then it should be like this:

Barek: E=0.298371 -> E=0.280471

When I did not update the bias, just like Matt, then I got

Barek: E=0.298371 -> E=0.291028

This proves that Matt did not update the bias.

Anyway, this is a great post. Thank you, Matt!

- xmioviettel.net

Thanks to such the nice explanation!

I implement my first neural network by referring the post, thanks.

hello!

If there are multiple hidden layers, say two, three, or even more; How do you continue the back-propagation? Do you always take it two layers at a time?

If we are dealing with the leftmost hidden layer, do we need to track how changes to weights in this layer effect the following multiple hidden layers and then finally the output layer (working out error)

Tracing all routes that changes in a far left weight effect error seems to “blow up” very quickly so to speak.

Can anyone point me towards a worked example with at least 1 more hidden layer?

Hello Justin,

the back-propagation works also with multiple hidden layers.

You will find a good visualization for neural networks:

http://playground.tensorflow.org/

Sincerly

Albrecht

Machine Learning Roadmap 2022 - Codelivly

Thank you for your detailed explanation.

I wonder how you set the outputs to be 0.01 and 0.99 ?

It is arbitrary chosen? Does the amount of the inputs and their vaules impact of the decesion how to set the outputs?

This is multi-label classification, correct?

Otherwise I would be wondering why we don’t either use a softmax in the last layer? Also our outputs don’t sum to 1.

Thank you and kind regards

How to calculate the total error from an arbitrary hidden layer in a neural network back propogation? - PhotoLens

Hmm, in my case, after first round of backpropagation the total error is 0.291027774 instead 0.291027924. And after repeating 10,000 times, the total error is 0.0000351019 instead 0.0000351085. Тhe rest of the calculations matched perfectly

Same, I wonder if that is my own issue or some floating point magic. The network seems to be learning…

Comprendre le DeepLearning et les réseaux de neurones en 10 minutes ! – TUTO3D

机器学习整理（神经网络） - 算法网

I’m totally stumped, I’ve been over this article like 100 times and scrolled through as many comments and I cannot figure out how back propagation would work with multiple hidden layers. Using the example (I’m using d instead of the swirly thing because I’ve seen other comments do that and idk how to type the other character):

dEtotal / dW1 = dEtotal / dOutputH1 * stuff I get

dEtotal / dOutputH1 = dEo1/dNeto1 + same for O2

dEo1/dNeto1 = dEo1 / dOutputo1 * dOutputo1 / dNeto1

dEo1 / dOutputo1 = -(targeto1 – outputo1)

I understand that for the output layer, but if you have a second hidden layer in place of the output layer then for some hn

dEhn / dOutputhn = -(targethn – outputhn)

What the hell is the target value? I’ve been pulling my hair out for like 4 hours and I cannot figure it out.

You need to work backwards through the layers and use the ‘delta’ from the next layer’s neurons. This delta should be stored whilst you are working backwards. The delta is the derivative of the current neurons output multiplied by the sum of all output neurons deltas * the weight between that neuron and the current one.

So what you see in the blue box “You might also see this written as” works but replace input i, with the current neurons output passed through the derivative of the activation function

Same. Have been searching for 2 days till now…

I also found a website with clear formulas and definitions to compute multiple hidden layer weights. Hope this help!

https://brilliant.org/wiki/backpropagation

Hi, i noticed that the value of out_h2 is incorrect. since its a logistic function, the sum of the two output can never exceed one but should approximate one in the nearest whole number. i go the value as 0.34468163

OMG I’m a postgrad student and have learnt about CV,NLP which all covers NN but I’m still confusing with the math behind the learning. Thanks for your detailed formulas and calculations!!

Cheers,

Jiayu

This is such a good, detailed exaplanation. Thank you very much, it helped a lot.

#2 BACKPROPAGATION algorithm. How a neural network learn ? A step by step demonstration. - DAOPOSTS

I have written my own implementation and can verify I get the same results. Great! Now I am trying to add the bias.

Firstly, My understanding was that the bias b1 would have 2 distinct weights being fed in to h1 and h2. And similarly b2 would have 2 distinct weights for o1 and o2. The article suggest that it is a single common weight for b1 -> h1,h2 and a single weight for b2 -> o1,o2. Which is correct?

Secondly, when using the derivative of the activation function for the bias, is this just f'(1) always since the bias ‘output’ is always 1?

1.

You really should have distinct bias values for h1 and h2, but both can start from the given 0.35. These values will change during backpropagation. The same is true for o1 and o2. So after one backpropagation step:

hidden_layer->b[0]=0.345614 hidden_layer->b[1]=0.345023

output_layer->b[0]=0.530751 output_layer->b[1]=0.619049

2.

When using the derivative of the activation function, you start from out values and get back to net values. This is the “activation backward” step, and bias has no effect here. It will be updated in the next step, the “layer backward” step, which is matrix calculation.

Answer Key 2022 Capf

[Solved] My neural network backpropagates and converges at a total error of around 0.5 after many iterations. What have I done wrong? – Ten-tools.com

Problema preciso con la función de activación sigmoide para Tensorflow/Keras 2.3.1 - neural-network en Español

Critical Learning Periods in Deep Networks – Ramsey Elbasheer | History & ML

Critical Learning Periods in Deep Networks – Commonwealth of Research & Analysis

brilliant!!! thank you a lot!! you helped me very much!!!

Machine Learning Roadmap 2022 – Codelivly

Is there a mistake for w7+ ? I find 0.431 instead of 0.511. The other w+ seem fine.

Great detailed explanation. Thanks a million, it helped a lot.

Backpropagation from Perspektif Matematika dan Statistika - Techdigipro

Can you please send me BPNN python code using keras library

Very nicely explained. Can you please explain if there are more than one hidden layer what is done? I mean the repetitive portion which is iterated backward any worked example will be a great help.

In 2023 this blog still benefits newbie in learning neural networks. What an attempt! Thank you.

Classifying fruits with a Convolutional Neural Network in Keras | Daniel Pradilla

Thank you very mach Matt.

I develop my own Neural Net using Delphi, and the values are exactly the same!

I adding hidden layers:

1 Hidden layer -> 0.0000351020 (the same of the article)

2 Hidden layer -> 0.0000061176

4 Hidden layer -> 0.0000049050

I put 0.5 to all bias(s), and change the learning rate (with 1 hidden layer):

Learning rate 0.50 -> 0.0000277005

Learning rate 0.75 -> 0.0000116993

Learning rate 0.85 -> 0.0000086999

Best Configuration (4 hidden layers, bias = 0.5, learning rate = 0.85)

0.0000008945