# Derivation: Error Backpropagation & Gradient Descent for Neural Networks

The material in this post has been migraged with python implementations to my github pages website.

Posted on September 6, 2014, in Algorithms, Classification, Derivations, Gradient Descent, Machine Learning, Neural Networks, Optimization, Regression, Theory and tagged backprop derivation, backpropagation algorithm, backpropagation derivation, Derivation, Machine Learning, Neural Networks. Bookmark the permalink. 18 Comments.

Hi, this is the first write-up on backpropagation I actually understand. Thanks.

A few possible bugs:

1. Last part of Eq.8 should I think sum over a_i and not z_i.

2. Between Eq.3 and Eq.4 it should I think be z_k=b_k + … and not z_k=b_j …

3. Last section says Output layer bias while the derivation is for hidden layer bias. Also,

b_i seems to be used as the notation for hidden layer bias while it should be b_j.

All in all, a very helpful post.

Reblogged this on DaFeda's Blog and commented:

The easiest to follow derivation of backpropagation I’ve come across.

Probably the best derivation of BackProp I’ve ever seen on internet 🙂

Thanks. Nice clean explanation.

Thank you !

Second time benefited from your blog ..

Best introduction about back prop ever!

Thank you so much.

Really useful! Though there are a few typos, as daFeda has mentioned.

really helpful man,

I just have one small question Im hopeing somebody can answer..

I understand this algebraically, and I understand the iterative patterns created with the deltas when calculating the weights from different layers starting backwards,

but WHY does (a_k – t_k) * the derivative mean that the ERROR (which is equal to a_k – t_k) is being “BACK PROPAGATED”. What’s the intuition behind multiplying by the derivative which makes us saying this.

?

This really cleared up all the confusions that I had in backpropagation. Thanks a bunch !

Thank You ………… dustinstansbury. Finally, I understood back propagation.

25 years ago I had these formulae in my PhD, but I couldn’t retrieve a copy, luckily i found your blog (true story) and your very clear exposition refreshed my memory.

A very neat and simple derivation. Great job!!

Thanks, very interesting and helpful article. Great introduction to neural networks for beginners like me

Pingback: Derivation: Derivatives for Common Neural Network Activation Functions | The Clever Machine

Pingback: A Gentle Introduction to Artificial Neural Networks | The Clever Machine

Pingback: Some sites I found helpful in reviewing backprop – Into DL and Beyond

Pingback: Derivation: Error Backpropagation & Gradient Descent for Neural Networks – collection of dev articles

Pingback: Derivazione: backpropagazione di errori e discesa del gradiente per reti neurali - Sem Seo 4 You