A Step by Step Backpropagation Example

Matt Mazur


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…

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Alan Turing’s Patterns in Nature, and Beyond from Wired Magazine


I found this ‘Interesting’ article on wired.com and was filled with some sort of an inexplicable joy. The buzzkill, however was that I had to click on each photograph to read about it. Hence this post…

This originally appeared in http://www.wired.com on February 22, 2011 at 7:00 am and is written by Brandon Keim.

Thanks Brandon Keim…

Alan Turing’s Biology PaperImage

Near the end of his life, the great mathematician Alan Turing wrote his first and last paper on biology and chemistry, about how a certain type of chemical reaction ought to produce many patterns seen in nature.

Called “The Chemical Basis of Morphogenesis,” it was an entirely theoretical work. But in following decades, long after Turing tragically took his own life in 1954, scientists found his speculations to be reality.

First found in chemicals in dishes, then in the stripes and spirals and whorls of animals…

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Camels in the Cambrian? A Geology Mnemonic

Life in Pen and Ink

Sitting camel

How did you learn the geological timescale?

Geology is not a standard subject in the UK Curriculum, so those few students who arrive at university having done it at GCSE or A-Level, have usually been taught it by non-tradtitional means.  They are more exposed to the whim and wit of their teacher than they would be in any other subject.

In fact, it was partly the charisma and enthusiasm of my A-Level Geology teacher that prompted me to apply to Geology at university, and…well, the rest is history.  Initially planning to take science subjects and apply for biochemistry, I chose Geology at A-Level on a bit of a whim – having always enjoyed physical geography.  Imperceptibly, as the weeks passed, all thoughts of biochemistry slipped away, and I realised I’d been a geologist all along.  Starting it at university was a bit of a shock to the system, and it…

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Siamese Cats and the Optic Chiasm

the sensitive motor

Anatomy constrains function | Function drives anatomy

As a general rule of thumb, information about the right side of the body is represented in the left side of the brain, and information about the left side is represented on the right.  This is called lateralization.  While that’s probably only mildly interesting to most people, it can help neurologists determine where a nervous system injury (eg, stroke) has occurred.

But to me, even though I study the motor and somatosensory systems where this holds (mostly) true, I think a cooler example of decussation (crossing from one side to the other) is the optic chiasm.   More than that, it’s a great example of what neuroanatomy can tell us about the body and how it functions in its environment.  Similarly, the anatomy and behavior of an animal tell us about its brain.

Before I get in to the details, I should cover a few terms: visual fields, monocular/binocular…

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In what way can philosophy, or philosophical thinking contribute to the physical sciences?

The MOOC's Essays

86px-richard_feynman_nobel Richard P. Feynman

Physicist hero and Nobel laureate, Richard Feynman, was known for not being particularily fond of philosophy. In his Auckland lecture on Quantum Mechanics, he addresses philosophy with the polemic challenge that “if you don’t like the universe as it is, go somewhere else, to another universe where the rules are simpler” [1]. As much as this statement reflects a clear-cut scientific realism, criticizing what he disdained as wishful thinking, this essay takes a more differentiated approach. It is trying to investigate the question how much philosophy, from which physics had emanated, can make contributions to the physical sciences. In trying to argue that science without philosophy runs the risk of being disoriented, it investigates the following question: How could philosophical thinking help avoid physical sciences drifting off into the wrong direction?

Albeit it is the obvious objection that science has to be free to investigate in whatever…

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