A Step by Step Backpropagation Example – Matt Mazur

8/18/2019 A Step by Step Backpropagation Example – Matt Mazur

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M a t t M a z u r

A S t e p b y S t e p

B a c k p r o p a g a t i o n E x a m p l e

B a c k g r o u n d

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.

B a c k p r o p a g a t i o n i n P y t h o n

You can play around with a Python script tha t I wrote that implements the

backpropagation algorithm in this Github rep o.

B a c k p r o p a g a t i o n V i s u a l i z a t i o n

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

Neural Network visualization .

A d d i t i o n a l R e s o u r c e s

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 .

O v e r v i e w

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:

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In order to have some numbers to work with, here’s are the initial weights , thebiases , 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.

T h e F o r w a r d P a s s

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

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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.

Total net input is also referred to as just 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:

C a l c u l a t i n g t h e T o t a l E r r o r

We can now calculate the error for each output neuron using the squared error

function and sum them to get the total error:

Some sources refer to the target as the ideal and the output as the actual .

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The is included so that exponent is cancelled when we differentiate later

on. The result is eventually multiplied by a learning rate anyway so it doesn’t

matter that we introduce a constant here [ 1].

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:

T h e B a c k w a r d s P a s s

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.

O u t p u t L a y e r

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

aka .

is read as “the partial derivative of with respect to “. You can

also say “the gradient with respect to “.

By applying the chain rule we know that:

Visually, here’s what we’re doing:

http://en.wikipedia.org/wiki/Chain_rulehttp://en.wikipedia.org/wiki/Backpropagation#Derivation


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We need to figure out each piece in this equation.

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

is sometimes expressed as

When we take the partial derivative of the total error with respect to ,

the quantity becomes zero because does notaffect it which means we’re taking the derivative of a constant which is zero.

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:

http://en.wikipedia.org/wiki/Logistic_function#Derivative


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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):

Some sources use (alpha) to represent the learning rate, others use

(eta), and others even use (epsilon).

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).

Follow

http://void%280%29/http://web.cs.swarthmore.edu/~meeden/cs81/s10/BackPropDeriv.pdfhttps://www4.rgu.ac.uk/files/chapter3%20-%20bp.pdfhttp://aima.cs.berkeley.edu/http://en.wikipedia.org/wiki/Delta_rulehttp://en.wikipedia.org/wiki/Delta_rule


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H i d d e n L a y e r

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:

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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:


9/19



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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 0.000035085. 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!

S h a r e t h i s :

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Posted on March 17, 2015 by Mazur. This entry was posted in Machine Learning and tagged ai,

backpropagation, machine learning, neural networks. Bookmark the permalink .

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← Older Comments

1 1 5 t h o u g h t s o n “ A S t e p b y S t e p B a c k p r o p a g a t i o n E x a m p l e ”

Mostafa Razavi— December 7, 2015 at 1:09 pm

That was heaven, thanks a million.Reply

Sonal Shrivastava— December 8, 2015 at 11:40 am

That was awesome. Thank a ton.

Reply

Nayantara— December 9, 2015 at 7:29 am

Hi Matt, Can you also please provide a similar example for a convolutional neural

network which uses at least 1 convolutional layer and 1 pooling layer ? Surprisingly, I

haven’t been able to find ANY similar example for backpropagation, on the internet,

for Conv. Neural Network.

TIA.

Reply

Mazur — December 9, 2015 at 8:36 am

I haven’t learnt that yet. If you find a good tutorial please let me know.

Reply

payamrastogi— December 11, 2015 at 4:24 am

All hail to “The” Mazur

Reply

A Step by Step Backpropagation Example | Matt Mazur | tensorflowgraphs P i n g !

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11/19



Louis Hong— December 11, 2015 at 4:41 pm

Thank you so much for your most comprehensive tutorial ever on the internet.

Reply

ad— December 17, 2015 at 1:49 am

why is bias not updated ?

Reply

Mazur — December 17, 2015 at 9:23 am

Hey, in the tutorials I went through they didn’t update the bias which is why I

didn’t include it here.

Reply

justaguy— December 24, 2015 at 8:54 pm

Typically, bias error is equal to the sum of the errors of the neurons

that the bias connects to. For example, in regards to your example,

b1_error = h1_error + h2_error. Updating the bias’ weight would be

adding the product of the summed errors and the learning rate to the

bias, ex. b1_weight = b1_error * learning_rate. Although many

problems can be learned by a neural network without adjusting

biases and there may be better ways to adust bias weights. Also,

updating bias weights may cause problems with learning as opposed

to keeping them static. As usual with neural networks, through

experimentation you may discover more optimal designs.

Reply

patriczhao— January 13, 2016 at 1:30 am

nice explanations, thanks.

Reply

Ahad Khan— December 20, 2015 at 2:26 am

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12/19



This is perfect. I am able to visualize back propagation algo better after reading this

article. Thanks once again!

Reply

sunlyt— December 21, 2015 at 12:57 am

Brilliant. Thank-you!

Reply

garky— December 24, 2015 at 8:25 am

If we have more than one sample in our dataset how we can train it by considering all

samples, not just one sample?

Reply

Daniel Zukowski— December 24, 2015 at 2:32 pm

Invaluable resource you’ve produced. Thank you for this clear, comprehensive, visual

explanation. The inner mechanics of backpropagation are no longer a mystery to me.

Reply

Long Pham— December 26, 2015 at 10:58 am

precisely, intuitively, very easy to understand, great work, thank you.

Reply

Dionisius AN— December 27, 2015 at 1:16 pm

Thank you very much ,it’s help me well, u really give detail direction to allow me

imagine how it works. I really appreciate it. May God repay your kindness thousand

time than u do.

Reply

singhrocks91

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— December 28, 2015 at 1:35 am

Thank You. I have a better insight now

Reply

DGelling

— January 1, 2016 at 6:48 pm

Shouldn’t the derivative of out_o1 wrt net_o1 be net_o1*(1-net_o1)?

Reply

NaanTadow— February 24, 2016 at 1:10 am

No the one stated above is correct, see here for the steps on the gradient of

the activation function with respect to its input value (net):

https://theclevermachine.wordpress.com/2014/09/08/derivation-derivatives-

for-common-neural-network-activation-functions/

Oh and thanks for this Matt – was able to work through your breakdown of the

partial derivatives for the Andrew Ng ML Course on coursera :D

Reply

Aro— January 10, 2016 at 6:23 pm

thanks so much, I haven’t see tutorial before like this.

Reply

Derive Me— January 12, 2016 at 1:22 am

Hello. I don’t understand, below the phrase “First, how much does the total error

change with respect to the output?”, why there is a (*-1) in the second equation, that

eventually changes the result to -(target – output) instead of just (target – output). Can

you help me understand?

Thank you!

Reply

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angie1pecht— January 17, 2016 at 8:52 pm

This helped me a lot. Thank you so much!

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LEarning AI again— January 18, 2016 at 4:26 pm

This was awesome. Thanks so much!

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Ashish— January 19, 2016 at 7:21 am

Thanks a lot Matt… Appreciated the effort, Kudos

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Tariq— January 20, 2016 at 12:03 pm

If the error is “squared” but simply E = sum (target – output) , you can still do the

calculus to work out the error gradient .. and then update the weights. Where did I go

wrong with this logic?

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Elliot— January 28, 2016 at 9:03 am

Good afternoon, dear Matt Mazur!

Thank you very much for writing so complete and comprehensive tutorial, everything

is understandable and written in accessible way! If is it posdible may I ask following

question if I need to compute Jacobian Matrix elements in formula for computing Error

Gradient with respect to weight dEtotal/dwi I should just percieve Etotal not as the full

error from all outputs but as an error from some certain single output, could you

please say is this correct? Could you please say are you not planning to make a

simillar tutorial but for computing second order derivatives (backpropagation with

partial derivatives of second order)? I have searching internet for tutorial of calculating

second order derivatives in backpropagation but did not found anything. Maybe you

know some good tutorials for it? I have know that second order partial derivatives

Learning How To Code Neural Networks | ipythonblog P i n g !

https://ipythonblog.wordpress.com/2016/01/20/learning-how-to-code-neural-networks/http://mattmazur.com/2015/03/17/a-step-by-step-backpropagation-example/comment-page-2/#comment-19745http://mattmazur.com/2015/03/17/a-step-by-step-backpropagation-example/?replytocom=19603#respondhttp://mattmazur.com/2015/03/17/a-step-by-step-backpropagation-example/comment-page-2/#comment-19603http://makeyourownneuralnetwork.blogspot.com/http://mattmazur.com/2015/03/17/a-step-by-step-backpropagation-example/?replytocom=19583#respondhttp://mattmazur.com/2015/03/17/a-step-by-step-backpropagation-example/comment-page-2/#comment-19583http://mattmazur.com/2015/03/17/a-step-by-step-backpropagation-example/?replytocom=19574#respondhttp://mattmazur.com/2015/03/17/a-step-by-step-backpropagation-example/comment-page-2/#comment-19574http://mattmazur.com/2015/03/17/a-step-by-step-backpropagation-example/?replytocom=19559#respondhttp://mattmazur.com/2015/03/17/a-step-by-step-backpropagation-example/comment-page-2/#comment-19559http://angiepecht.wordpress.com/


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(elements of Hessian Matrix) can be approximated by multiplaying Jacobians but

wanted to find it’s exact non approximated calculation. Thank you in advance for your

reply!

Sincerely

Reply

Pulley— February 1, 2016 at 9:52 pm

hello Matt, Can you please tell me that after updating all weights in first iteration I

should update the values of all ‘h’ at-last in first iteration or not.

Reply

Behroz Ahmad Ali— February 6, 2016 at 8:01 am

Thank you for such a comprehensive explanation of backpropagation. I have been

trying to understand backpropagation for months but today I finally understood it after

reading your this post.

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Tariq— February 8, 2016 at 10:57 am

i am writing a gentle intro to neural networks – aimed at being accessible to

someone at school approx age 15… here is a draft which includes a very very

gentle intro to backprop

https://goo.gl/7uxHlm

i’d appreciate feedback to @myoneuralnet

Reply

Rebeka Sultana— February 16, 2016 at 12:59 am

Thank you so much.

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Ron— February 21, 2016 at 1:10 pm

http://mattmazur.com/2015/03/17/a-step-by-step-backpropagation-example/comment-page-2/#comment-20143http://mattmazur.com/2015/03/17/a-step-by-step-backpropagation-example/?replytocom=20069#respondhttp://mattmazur.com/2015/03/17/a-step-by-step-backpropagation-example/comment-page-2/#comment-20069http://mattmazur.com/2015/03/17/a-step-by-step-backpropagation-example/?replytocom=19956#respondhttps://goo.gl/7uxHlmhttp://mattmazur.com/2015/03/17/a-step-by-step-backpropagation-example/comment-page-2/#comment-19956http://makeyourownneuralnetwork.blogspot.co.uk/http://mattmazur.com/2015/03/17/a-step-by-step-backpropagation-example/?replytocom=19916#respondhttp://mattmazur.com/2015/03/17/a-step-by-step-backpropagation-example/comment-page-2/#comment-19916https://www.facebook.com/app_scoped_user_id/10205797312227101/http://mattmazur.com/2015/03/17/a-step-by-step-backpropagation-example/?replytocom=19835#respondhttp://mattmazur.com/2015/03/17/a-step-by-step-backpropagation-example/comment-page-2/#comment-19835http://mattmazur.com/2015/03/17/a-step-by-step-backpropagation-example/?replytocom=19745#respond


16/19



Firstly, thank you VERY much for a great walkthrough of all the steps involved with

real values. I managed to create a quick implementation of the methods used, and

was able to train successfully.

I was looking to use this setup (but with 4 inputs / 3 outputs) for the famous iris data

(http://archive.ics.uci.edu/ml/datasets/Iris ). The 3 outputs would be 0.0-1.0 for each

classification, as there would be an output weight towards each type.

Unfortunately it doesn’t seem to be able to resolve to an always low error value, and

fluctuates drastically as it trains. Is this an indication that a second layer is needed for

this type of data?

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Werner — February 22, 2016 at 5:44 am

The first explanation I read that actually makes sense to me. Most just seem to start

shovelling maths in your face in the name of “not making it simpler that they should”.

Now let’s hope my AI will finally be able to play a game of draughts.

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admin— February 22, 2016 at 9:20 am

It helps me a lot. thanks for the work!!!

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Name(required)— February 24, 2016 at 9:04 pm

Great tutorial. By any chance do you know how do backpropagate 2 hidden layers?

Reply

Mazur — February 25, 2016 at 8:22 am

I do not, sorry.

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Kiran— February 25, 2016 at 12:29 am

Thank you so much! The explanation was so intuitive.

http://mattmazur.com/2015/03/17/a-step-by-step-backpropagation-example/comment-page-2/#comment-20188http://mattmazur.com/2015/03/17/a-step-by-step-backpropagation-example/?replytocom=20193#respondhttp://mattmazur.com/2015/03/17/a-step-by-step-backpropagation-example/comment-page-2/#comment-20193http://www.mattmazur.com/http://mattmazur.com/2015/03/17/a-step-by-step-backpropagation-example/?replytocom=20183#respondhttp://mattmazur.com/2015/03/17/a-step-by-step-backpropagation-example/comment-page-2/#comment-20183http://mattmazur.com/2015/03/17/a-step-by-step-backpropagation-example/?replytocom=20148#respondhttp://mattmazur.com/2015/03/17/a-step-by-step-backpropagation-example/comment-page-2/#comment-20148http://writeconomy.com/http://mattmazur.com/2015/03/17/a-step-by-step-backpropagation-example/?replytocom=20147#respondhttp://mattmazur.com/2015/03/17/a-step-by-step-backpropagation-example/comment-page-2/#comment-20147http://mattmazur.com/2015/03/17/a-step-by-step-backpropagation-example/?replytocom=20143#respondhttp://archive.ics.uci.edu/ml/datasets/Iris


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Anon— February 25, 2016 at 11:18 pm

Thank you! The way you explain this is very intuitive.

Reply

tariq— February 26, 2016 at 9:38 am

I’d love your feedback on my attempt to explain the maths and ideas underlying

neuralnetworks and backrpop.

Here’s an early draft online. The aim for me is to reach as many people as possible

inck teenagers with school maths.

http://makeyourownneuralnetwork.blogspot.co.uk/2016/02/early-draft-feedback-

wanted.html

Reply

Garett Ridge AndThenSomeMoreWords— March 1, 2016 at 5:45 pm

I have a presentation tomorrow on neural networks in a grad class that I’m

drowning in. This book is going to save my life

Reply

falcatrua— February 29, 2016 at 2:23 pm

It’s a great tutorial but I think I found an error:

at forward pass values should be:

neth1 = 0.15 * 0.05 + 0.25 * 0.1 + 0.35 * 1 = 0.3825

outh1 = 1/(1 + e^-0.3825) = 0,594475931neth2 = 0.20 * 0.05 + 0.30 * 0.1 + 0.35 * 1 = 0.39

outh2 = 1/(1 + e^-0.39) = 0.596282699

Reply

Garett Ridge AndThenSomeMoreWords— March 1, 2016 at 9:37 pm

The labels go the other way in his drawing, where the label that says w_2

goes with the line it’s next to (on the right of it) and the value of w_2 gets

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written to the left; look at the previous drawing without the values to see what I

mean

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Bill— March 2, 2016 at 3:09 am

Good stuff ! Professors should learn from you. Most professors make complex things

complex. A real good teacher should make complex things simple.

Reply

b— March 2, 2016 at 3:11 am

Also , recommend this link if you want to find a even simpler example than this one.

http://www.cs.toronto.edu/~tijmen/csc321/inclass/140123.pdf Reply

Priti— March 2, 2016 at 4:27 am

Can you give an example for backpropagation in optical networks

Reply

Moboluwarin— March 2, 2016 at 2:13 pm

Hey there very helpful indeed, in the line for net01 = w5*outh1 + ‘w6’*outh2+b2*1, is it

not meant to be ‘w7’ ??

Cheers

Reply

Dara— March 4, 2016 at 9:17 am

Can anyway help me explaining manual calculation for testing outputs with trained

weights and bias? Seems it does not give the correct answer when I directly substitute

my inputs to the equations. Answers are different than I get from MATLAB NN toolbox.

Reply

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A Step by Step Backpropagation Example – Matt Mazur