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Backpropagation calculus | Chapter 4, Deep learning

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3Blue1Brown

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This one is a bit more symbol-heavy, and that’s actually the point. The goal here is to represent in somewhat more formal terms the intuition for how backpropagation works in part 3 of the series, hopefully providing some connection between that video and other texts/code that you come across later.

For more on backpropagation:
http://neuralnetworksanddeeplearning.com/chap2.html
https://github.com/mnielsen/neural-networks-and-deep-learning
http://colah.github.io/posts/2015-08-Backprop/

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https://vincerubinetti.bandcamp.com/album/the-music-of-3blue1brown

——————
Video timeline
0:00 – Introduction
0:38 – The Chain Rule in networks
3:56 – Computing relevant derivatives
4:45 – What do the derivatives mean?
5:39 – Sensitivity to weights/biases
6:42 – Layers with additional neurons
9:13 – Recap
——————

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46 thoughts on “Backpropagation calculus | Chapter 4, Deep learning”

  1. Two things worth adding here:
    1) In other resources and in implementations, you'd typically see these formulas in some more compact vectorized form, which carries with it the extra mental burden to parse the Hadamard product and to think through why the transpose of the weight matrix is used, but the underlying substance is all the same.

    2) Backpropagation is really one instance of a more general technique called "reverse mode differentiation" to compute derivatives of functions represented in some kind of directed graph form.

  3. Thank you very much.
    There cannot be a better tutorial to understand back propagation than this video. Thanks again for the effort to make us understand the complex mathematical theory behind this.

  9. At 5:31, why did you put the partial derivatives with respect to w and b in the same column? Shouldn't it be a two-column vector if you are accounting for 2 variables for all samples?

  10. Beautiful animation, would like to see a behind the scene of how this was done. As I understood this is a specialized software you've build yourself for math animations?

  13. Now that you have experienced the work of God in the last days, what exactly is God’s disposition? Do you dare to say that God is a God who only speaks? You dare not make such a rule. Some people also say that God is the God who opens mysteries, and God is the Lamb who breaks the seven seals. No one dares to make such a rule.

  20. Basically, like any other gradient descent in other machine learning algorythm(linreg, logreg, etc), is just

    Param := param – derrivative(cost function)

    *note
    := is update

    Just that in nn, the cost function is more massive, and looking complex
    Is that the basic most fundamental concept ? Did i get it right ?

  23. Great video! I just don't know if I'm missing something, because I thought we didn't use this cost function when working with neural networks. As we activates every neuron with the sigmoid, the cost function defined as (a – y)^2 gives us a non-convex cost function wich has multiple local minimums. So a better choice is to use the cost function from logistic regression I think.

    I'm not sure tho if I'm correct or if I'm really just missing something.

    Amazing video anyway, you made me understand neural networks so much better with these series. I'm from Brazil and I'm taking machine learning classes here at my university, and I wouldn't be able to understand backpropagation without you, thank you very much!

  27. I feel that the video could have benefitted a lot by going into the L-1 layer weights. I'm still a bit confused as to how they're computed in the backward pass

  28. Man, I really appreciate your fantastic work! Do you read the texts when you record, or do you speak spontaneously? You are very good at that, and I am very bad. 😊That's why I wanted to ask you, maybe you should make a tutorial about that, too 😬

  29. 3:42 That's not really the chain rule, because it works differently with partial derivatives. In this case, the other terms are zero, so you're left out with that formula, which coincidentally looks the same as the one from single-variable calculus.

  33. thanks a lot this helped me a lot understanding what is going on with gradiant descent , and sure yall dont hesitate to watch this video multiple times to get it right

  34. From what I understand, finding the global minima of cost function is what we need, does gradient descent ensure that. I think it would end up in local minima.

  38. Great video, missing a few things though.
    Ok, so I've calculated how sensitive each weight and bias are, but how much do I actually tweak them by?

  40. is it a mistake on the last slide that del cost/del activation is a summation with terms from layer (L+1)? i thought that it should go backwards and thus be (L-1)?

    edit: nvm i see

  42. One doesn't need synapses to say that an image of a 2 is NOT a 1, and not a 3 and not a 4 and not a 5 and so on, but one need the synapses that tells you if it can NOT be a 2.

  46. This is truly awesome, as pedegogy and as math and as programming and as machine learning. Thank you! …one comment about layers, key in your presentation is the one-neuron-per-layer, four layers. And key in the whole idea of the greater description of the ratio of cost-to-weight/cost-to-bias analysis, is your L notation (layer) and L – 1 notation. Problem, your right most neuron is output layer or "y" in your notation. So one clean up in the desction is to make some decisions: the right most layer is Y the output (no L value), because C[0]/A[L] equals 2(A[L] – y). So the right most three neurons, from right to left, should be Y (output), then L then L minus one, then all the math works. Yes?

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