## 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 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 0.000035085. 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!

thank you very much

it was so good :D

Really great tutorial. How would the pattern of back propagation carry on with multiple hidden layers instead of just the one? What would the overall algorithm look like where the hidden layer count, weights, etc, are all variable?

OMG you saved my life!

Thank you an amazing article! Everywhere else the description of the algorithm is extremely theoretically, so I am glad that you got your hands dirty by really describing the actual implementation with numbers and not just by loads of symbols, which particularly in the case of backpropagation was very confusing for me. Cheers from Edinburgh.

Having a read now – seems really helpful. Thank you!

Hello Caspar, did you get information about how to implement it for multiple hidden layer ? I am also looking for this. Thank you.

Thanks for great article!

Question: how do we tune the weights of biases?

Yes how do we find the updated biases??

Hi there,

I think those are same with same layer’s weight but the output value won’t update.( in example suppose 1 for biases weight )

Sorry Matt please update my post with this:

Hi there,

I think those are same with same layer’s weight but the value of biases won’t update.( in example supposed 1 for biases weight )

Nicely explained ! :) Thank you.

Exploring Neural Networks… – Cluster Chord

great

Thanks alot for such a wonderful explanation.

Awesome!

This was a pleasure to read with beautiful diagrams and numbers to make it real. Thank you.

Best explanation so far, thanks so much!

This is probably the best explanation I’ve seen so far. Thank you so much!

Thank you for your great explanation! but i have some questions about it.

in your post, at [The Backwards Pass // Output Layer] part,

(d E_total/ d out_o1) = 0.74136507.

and at [The Backwards Pass // Hidden Layer] part, you said

‘We can calculate (d E_o1/d out_o1) using values we calculated earlier.’

(d E_o1/d out_o1) = 0.74136507.

(d E_o1/d out_o1) = (d E_total/ d out_o1) ?

is this two partial deviation values are same? please reply me!

I wonder this too, is there anybody?

Yeah, I wonder this too, anybody?

I guess it is because, E_O2 is independent of Out_O1 and is treated as a constant and when you take the derivative it is zero. Could be wrong.

maybe:

E_total = E_o1 + E_o2

d(E_total) = d(E_o1 + E_o2) = d(E_o1) + d(E_o2)

so, with respect to d_out1, the second part will be zero.

As @Shree Ranga Raju said.

I think, Yes

those are same:

E_total = E_output1 + E_output2 => E_total = E_output1 + 0 for E_o1 and E_total = 0 + E_output2 for E_o2.

for calculate E_o1, E_output2 can’t effect and is equal 0 then they have same amount.

Reblogged this on irusin.

A very helpful explanation thank you.

amazing article, those diagrams are very clear. thanks.

i have same question, it was asked before. how do we update biases weights?

Thanks! Best explanation I have ever read.

Question: Does it make any difference if I choose bias to be -1?

Hi, thanks for this explanation. What I don’t understand is why the weight adjustment is dEtot/dw instead of (dw/dEtot)*(Etot)*(eta).

Thanks, Rich

Nice explanation. Very helpful article.

Best explanation I’ve seen so far on backpropagation. Great job, many thanks!

Thanks much…

Learning Machine Learning | ebc

Why is the weight for the bias the same for a layer? For instance, for the input layer, the bias going into the hidden layer nodes h1, h1 has weight b2.

amazing! Thanks!

7 Steps to Understanding Deep Learning – What does a Ph.D need?

How can I implement it for multiple hidden layers (more than one) ?

I think, You can do it for any number of hidden layer just you need to repeat process between output layer and hidden layer(Section Hidden Layer) for extra hidden layer you have.

Nice tutorial thanks. But how can we make this to work for multiple hidden layers(not a single hidden layer) ?

Nice Explanation. Thank you.

Awesome!

Thanks.

Very good article, however:

neth1 = 0.15*0.5 + 0.1*0.2 + 1*0.35 is not equal 0.3775 but 0.445

I suggest you examining all the calculations in order to improve your credibility

Sorry, I was wrong. You have 0.05 not 0.5. I made a mistake by wrong copying data.

You are explaining that dE/dO1 = 1/2 * 2 * (target1 – output1) * (-1) = 0.7414

But dE/dO2 = 1/2 * 2 * (target2 – output2) * (-1) = -0.217

Above you have considered it as +0.217

By taking it as negative, makes w7+ = 0.48871 and not 0.51130 as you have calculated.

Could you verify it?

Sorry my bad

Thank you very much, this was a great explanation and I could check the results of my neural network output by confronting them with your calculations. I also managed to improve the error reduction after 10000 cycles by updating the biases of neurons during backpropagation.

How I learn Neural Network (and Deep Learning) | Malioboro;

Thank you so much! That was brilliant!

Excellent! Thanks for writing this up.

Thank you. This is easy to follow. This helps me a lot.

As far as I understand, all these calculations are made to get two-dimensional output vector (target) as a result of two-dimensional input vector. I think, one linear system of two equations can give the same result. Am I right?

It was an amazing tutorial. Succint, yet comprehensive. Thanks a million.

Nicely done! Thanks for doing this – helped me a lot!

Backpropagation | Headbirths

Very nice explanation with sound examples.

Thank you,

Very clear and easy to understand.

It gave me an end-to-end vision.

Other articles only explain math calculus, but not a systematic diagrama.