What is Automatic Differentiation?

This short tutorial covers the basics of automatic differentiation, a set of techniques that allow us to efficiently compute derivatives of functions implemented as programs. It is based in part on Baydin et al., 2018: Automatic Differentiation in Machine Learning: A Survey (arxiv.org/abs/1502.05767).
Errata:
At 6:23 in bottom right, it should be v̇6 = v̇5*v4 + v̇4*v5 (instead of "-").
Additional references:
Griewank & Walther, 2008: Evaluating Derivatives: Principles and Techniques
of Algorithmic Differentiation (dl.acm.org/doi/book/10.5555/1...)
Adams, 2018: COS 324 - Computing Gradients with Backpropagation (www.cs.princeton.edu/courses/...)
Grosse, 2018: CSC 321 - Lecture 10: Automatic Differentiation (www.cs.toronto.edu/~rgrosse/c...)
Pearlmutter, 1994: Fast exact multiplication by the Hessian (www.bcl.hamilton.ie/~barak/pap...)
Alleviating memory requirements of reverse mode:
Griewank & Walther, 2000: Algorithm 799: revolve: an
implementation of checkpointing for the reverse or adjoint mode of computational differentiation (dl.acm.org/doi/10.1145/347837...)
Dauvergne & Hascoët, 2006. The data-flow equations of checkpointing in
reverse automatic differentiation (link.springer.com/chapter/10....)
Chen, T et al., 2016: Training Deep Nets with Sublinear Memory Cost (arxiv.org/abs/1604.06174)
Gruslys et al., 2016: Memory-efficient Backpropagation
Through Time (arxiv.org/abs/1606.03401)
Siskind & Pearlmutter. Divide-and-conquer checkpointing for arbitrary programs with no user annotation (arxiv.org/abs/1708.06799)
Oktay et al., 2020: Randomized Automatic Differentiation (arxiv.org/abs/2007.10412)
Example software libraries using various implementation routes:
Source code transformation:
Tangent - github.com/google/tangent
Zygote - github.com/FluxML/Zygote.jl
Operator overloading:
Autograd - github.com/HIPS/autograd
Jax - github.com/google/jax
PyTorch - pytorch.org/
Graph-based w/ embedding mini lanugage:
TensorFlow - www.tensorflow.org
Special thanks to Ryan Adams, Alex Beatson, Geoffrey Roeder, Greg Gundersen, and Deniz Oktay for feedback on this video.
Some of the animations in this video were created with 3Blue1Brown's manim library (github.com/3b1b/manim).
Music: Trinkets by Vincent Rubinetti
Links:
KZbin: / ariseffai
Twitter: / ari_seff
Homepage: www.ariseff.com
If you'd like to help support the channel (completely optional), you can donate a cup of coffee via the following:
Venmo: venmo.com/ariseff
PayPal: www.paypal.me/ariseff

[@anjelpatel7918]

I like how more and more people are adopting 3b1b's style. Makes the content much better and easier to understand. This slowly converts a lot of the more complicated topics into easy-to-digest modules.

[@Artaxerxes.]

It literally uses manim

[@tomerzilbershtein849]

3B1B’s creator Grant Sanderson created an animation library for himself to use to make videos. People forked that library (made a copy of it) and now there is a community supported version of it for creators, while he continues to use his own ( as well as the community one). Pretty cool stuff!

[@atotoole21]

@@Artaxerxes. Nice! I didn't know about manim or that 3B1B's animation technic was python based. I assumed it was done by hand using Illustrator or something.

[@umbraemilitos]

Yes, though I don't think 3B1B wants their videos to be a template to copy. I think he's happy to inspire, but doesn't think that his Manim program is the right tool for most cases. He released a video explaining the SOME criteria, and it allows for lots of creative expression in teaching.

[@andreypopov6166]

3b1b or any other style on its own doesn't mean that the content is easier to understand.

[@stathius]

Class act, being concise and clear at the same time is no easy feat. Thank you.

[@andrewbeatty5912]

Best summary I've ever seen !

[@arkasaha4412]

Man this is pure gold. We all use this stuff but hardly have a clear idea about it's nitty-gritties. Thanks for thre awesome content and presentation, keep it up! :)

[@raminbohlouli1969]

I knew basically 0 about AD and didn't know where to start since all the articles, websites ,books etc that I have looked into, explained everything in a really comlicated way. I would like to thank you immensly for this very informative yet simple video! Now I know enough to dive deeper into the concept. This video was all I needed. Keep up the great work! You got yourself a new follower.

[@abhishek.shenoy]

This is so well explained! I love the quality of your videos!

[@jaf7979]

Well done, superbly explained in context of other differentiation methods. Exactly what I needed!

[@koushik7604]

This is highly motivated by Andrej Karpathy's lecture, but very clear explanation. It is indeed a good addition to my resource list.

[@pulusound]

very well explained video with lovely calm background music. i need to brush up on my vector calculus and come back but this gave me a good intuition. hope you make more of these!

[@TheLokiGT]

Very good job. One of the very few good videos I've seen around about autodiff.

[@pandatory1108]

Excellent video Ari. Thanks for such a great explanation! Also, your animations were really well done. I suspected you might be using manim based on the style and then I read the description :)

[@chandank5266]

Your way of explanation is outstanding.....love from india sir♥️

[@BrianAmedee]

Excellent presentation mate. That was an awesome explanation and a nice trip down memory lane (university days).

[@esaliya]

This is a neat summary that's hard to find in a single place!

[@jorgeanicama8625]

Thank you Ari. I used symbolic computation in the past but this novel way of calculating derivatives is quite interesting. Learnt lots by watching your video. For sure, I will follow up with the recommended literature

[@YorkiePP]

Fantastic video on autodiff, really cleared up a lot of things I wasn't sure about.

[@prydt]

Amazing explanation of Autograd and wonderful visualizations!!! Thank you so much.

[@jkkang9666]

Thanks for the great summary and the nice video.

[@weinansun9321]

more videos please, this is amazing!

[@aldaszarnauskas27]

Great video, well presented, clearly explained, nice visualisation... Thank you!

[@SohailKhan-zb5td]

Thanks a lot. This kind of videos are really a lot of hardwork to produce. Thanks a lot

[@ccgarciab]

Looking forward to your future videos

[@user-kl1xv8in2q]

Thanks you so much. This video really helps me to understand a little more what is automatic differentiation is.

[@Roshan-xd5tl]

Brilliant video, Ari. Thank you!

[@arnold-pdev]

Went from complete ignorance to understanding in 15 min. Thank you!

[@asdf56790]

Exactly what I was looking for! Thank you :)

[@AJ-et3vf]

Awesome presentation! I understand autodiff a little bit more. I'll rewatch several more times in the future to understand it better till I completely understand it :)

[@thivinanandh4430]

Awesome Explanation..!!!!! Keep rocking..!!!

[@setsunakevin6861]

Amazing video! Very well explained.

[@halneufmille]

Thanks! I never understood this before, but it became obvious in one second.

[@Vaporizer41]

Great video!, I love your content, hope you will keep making many more :)

[@jishnuak3000]

Very intuitive explanation, thanks

[@BrianBin]

I like your tutorial video because it is short and good

[@nathanielscreativecollecti6392]

Bravo! I have a final today and now I get it!

[@vijaymaraviya9443]

Awesome summary👌

[@sandropollastrini2707]

Beautiful and clear!

[@datamike7457]

Ari, this is great content! I used to call symbolic differentiation 'analytical'. It is obnoxious to track all of the coefficients.

[@tom_verlaine_again]

Great lesson! Thank you.

[@hadik4497]

Thanks! This is phenomenal!

[@andersgadlauridsen1533]

So is so great content, please keep making more :)

[@KulvinderSingh-pm7cr]

This is exceptionally well explained.

[@KulvinderSingh-pm7cr]

And thanks a lot for references too, they're very useful.

[@kong1397]

Wow, that's great explanation.

[@juandavidnavarro]

Excellent video!! thank you so much. I have a question: is there any AD reverse mode based on dual numbers?

[@amadlover]

timely information about source code manipulation and google tangent. It was a kind of confirmation for me that it was indeed possible. I started to learn meta programming hoping to generate code for the differentials, based on the function, without actually knowing if it was possible., basically a shot in the dark. cheers

[@SuperDonalByrne]

Great video!

[@manumerous]

This video is genius! love it.

[@tom-sz]

Great video! Where can I learn more about the rounding and truncation errors plot at 2:06? I need to make an analysis of these errors for a project. Thanks :)

[@amirrezarezayan8121]

great great great , Thanks a million 😃

[@francois__]

Love it!

[@bitahasheminezhad2887]

That was awesome, thank you

[@stansilverman1901]

In order to explain this to my wife, I differentiated voter rights-the analog process humans decide who should be allowed to vote, someone who looks like me, or everyone?. I think she got it. Brilliant Ari

[@jbl4174]

Thanks for putting out such a great video. Im still a bit confused why forward mode AD requires a separate forward pass for each input variable. In Bayden et al. it says "Conversely, in the other extreme of f : R^n → R, forward mode AD requires n evaluations to compute the gradient". But I dont see why you couldnt compute the primal table and then the tangent table for each n variables, unless "n evaluations" means n evaluations of the tangent table and not forward passes.

[@newbie8051]

Beautiful video but I lost track quite a few times, is there any pre-requisite topics/stuff I should know before trying to understand this

[@dullyvampir83]

Great video, thank you! Just a question, you said a main problem with symbolic differentiation is that no control flow operations can be part of the function. Is that in any way different for Automatic differentiation?

[@superagucova]

Loved this video! Are you using 3b1b's Manim?

[@ariseffai]

Yep! Manim is awesome

[@UnnamedThe]

12:26 May I ask where you got that c

[@ariseffai]

Baydin (arxiv.org/abs/1502.05767) references this bound in Sec. 3.2. I don't have the exact location for it in Griewank and Walther.

[@UnnamedThe]

@@ariseffai Thank you a lot! That is already very helpful.

[@jianwang7433]

thanks for sharing

[@alfcnz]

@Ari, this is really great! 🤩🤩🤩

[@ariseffai]

Thanks Alfredo!

[@sofa33]

Thank you so much!

[@gabrielmccartney7975]

Hello! Can we use dual numbers for integration?

[@rtcoffee1235]

thanks for this!

[@GordonWade-kw2gj]

Wonderful video. The detailed example helps tremendously. And I think there's an error: At t=6.24, sInce $v_6 = v_5\times v_4$, in $\dot{v}_6$ shouldn't there be a plus sign where you've got a minus sign?

[@jorgeanicama8625]

One more note ARI. I think there is a small typo. From minute 7:36 until 7:46 the derivative of V6 should be a "+" instead of a "-".

[@bryanbischof4351]

This is quite good. I’m wondering if a part 2 digging deeper yet into how the implementation takes advantage of the concept you introduce here would be possible?

[@ariseffai]

Thanks Bryan. That's a possibility. It would certainly be interesting to dig deeper into the implementation schemes, which were only briefly described here. In the meantime, check out some of the links for further information on implementations.

[@PahenPWNZ]

Awesome explanation, thanks! But I still have one question, can someone explain please, at 12:05, right column (Adjoints) I don't understand how did we get these values (f. e. v bar 5 = v4 * v bar 6, etc...) From where did these values come from? If we use the formula at the previous slide with sum of children nodes, I get different values..

[@MarkKrebs]

Hi I have same Q. The moment when adjoints are defined is a break to me. vbar5 = v4 * vbar6 seems "backwards." I see it matches the formula given on the prior graph page, but not the intuition for it. "The sum of the output values, weighted by my leverage in creating them," is as close as I can get.

[@abhaysolanki9284]

I know when he said children I automatically thought of v3 and v4. But instead the children in the case v5 is only v6. And children for v4 are v5 and v6. Children are the nodes that the node is pointing to.

[@sirallen2591]

Thanks!

[@germangonzalez3063]

Very useful

[@ktugee]

slight type : @6.29 : v6' = v5'v4 + v4'v5. ( there should a + instead of - )

[@paulpassek6118]

Thanks for the superb video. I think you made a little mistake in the forward mode example at 6:24. Shouldn't it be v̇_6 = v̇_5*v_4 + v̇_4*v5 ?

[@ariseffai]

Thanks Paul, good catch-placed this under errata.

[@chnlior]

Great summary, Ari. Thank you. I think there is small error in 6:23. v6' = v5'v4 + v4'v5 and not "-".

[@ariseffai]

Thanks Lior, good catch-placed this under errata.

[@user-vm9hl3gl5h]

어쨌든 요점은, 모든 것을 다 closed form으로 저장해서 gradient를 매번 구하는 게 아니라는 점이다. 한 번 계산할 때마다, output value와 더불어 gradient value도 함께 계산해두어, 나중에 forward / backward 할 때 사용한다.

[@9888622400]

thanks bro!

[@deepanshuchoudhary4598]

Please reply to my Question. Where do you learn these and how are you able to grasp them completely, I'm a data science student and i need to know it badly. Pls share insights.

[@ariseffai]

I found the survey by Baydin et al. to be particularly helpful. See the description for links!

[@proweiqi]

this is very good. but some of the stuff moves too fast and not explaining things like the primal part clearly enough

[@rachelellis6655]

Derivative at 0:43 would actually be: f' (x) = (2x)e^(2x-1)- 3x^2 ... would it not? Great video.. I've subscribed! I'm just learning derivative and chain rule so I want to be sure I'm understanding the concept/rules/procedures correctly. I'm probably wrong though, that's why I'm asking for verification... thanks!

[@diodin8587]

not mention *dual number*?

[@zappist751]

THANK YOU LORD THANK YOU JESUS AND THANK YOU SIR

[@bokibogi]

4:27 automatic differentiation ...

[@yavarjn2055]

Wooow

[@Rems766]

chain rule rules

[@Manishsingh-dl6ho]

Fking Great!!!

[@sarvasvarora]

Reddit gang?

[@user-rr7uz9hd4m]

Do you get paid to make such videos? Definitely should

[@maxyazhbin826]

please no music, fantastic otherwise

[@MariaFernandez-pv9hn]

You should point on the screen what you are talking about when doing examples.

[@ollllj]

on expression-swell: one of my proudest computations (and hard to debug code) is the automated differentiation 3rd derivative of the general quotient rule within [shadertoy ... /WdGfRw ReTrAdUi39] , with identical parts already pre-multiplied out by how much it is constantly repeated. webgl code: Struct d000{float a;float b;float c;float d;};//1 domains t,dt,dt²,dt³ , sure, this could just be a vec4, but i REALLY needed my custom labels for debugging. d000 di(d000 a,d000 b){return d000( //autodiff up to 3 derivatives for division , up to 3 iterations of; quotient rule within chain rule) a.a/b.a //0th derivative, simple division ,(a.b*b.a-a.a*b.b)/(b.a*b.a) //dx first derivative ,((a.c*b.a+a.b*b.b-a.b*b.b-a.a*b.c)*(b.a*b.a)-2.*(a.b*b.a-a.a*b.b)*(b.a*b.b))/(b.a*b.a*b.a*b.a) //dxdx second derivative ,((((a.d*b.a+a.c*b.b+a.c*b.b+a.b*b.c-a.c*b.b-a.b*b.c-a.b*b.c-a.a*b.d)*(b.a*b.a) +(a.c*b.a+a.b*b.b-a.b*b.b-a.a*b.c)*(b.b*b.a*b.a*b.b)) +(-2.*(a.c*b.a+a.b*b.b-a.b*b.b-a.a*b.c)*(b.a*b.b) +(a.b*b.a-a.a*b.b)*(b.b*b.b+b.a*b.c)))*(b.a*b.a*b.a*b.a) -((a.c*b.a+a.b*b.b-a.b*b.b-a.a*b.c)*(b.a*b.a) -2.*(a.b*b.a-a.a*b.b)*(b.a*b.b)) *4.*(b.b*b.a*b.a*b.a))/(b.a*b.a*b.a*b.a*b.a*b.a*b.a*b.a)) //dxdxdx //3rd derivative quotient rule sure is something ;}

[@a.osethkin55]

Thanks!!!