Gradient descent, how neural networks learn | Chapter 2, Deep learning
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- เผยแพร่เมื่อ 10 พ.ค. 2024
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Special thanks to these supporters: 3b1b.co/nn2-thanks
Written/interactive form of this series: www.3blue1brown.com/topics/ne...
This video was supported by Amplify Partners.
For any early-stage ML startup founders, Amplify Partners would love to hear from you via 3blue1brown@amplifypartners.com
To learn more, I highly recommend the book by Michael Nielsen
neuralnetworksanddeeplearning....
The book walks through the code behind the example in these videos, which you can find here:
github.com/mnielsen/neural-ne...
MNIST database:
yann.lecun.com/exdb/mnist/
Also check out Chris Olah's blog:
colah.github.io/
His post on Neural networks and topology is particular beautiful, but honestly all of the stuff there is great.
And if you like that, you'll love the publications at distill:
distill.pub/
For more videos, Welch Labs also has some great series on machine learning:
• Learning To See [Part ...
• Neural Networks Demyst...
"But I've already voraciously consumed Nielsen's, Olah's and Welch's works", I hear you say. Well well, look at you then. That being the case, I might recommend that you continue on with the book "Deep Learning" by Goodfellow, Bengio, and Courville.
Thanks to Lisha Li (@lishali88) for her contributions at the end, and for letting me pick her brain so much about the material. Here are the articles she referenced at the end:
arxiv.org/abs/1611.03530
arxiv.org/abs/1706.05394
arxiv.org/abs/1412.0233
Music by Vincent Rubinetti:
vincerubinetti.bandcamp.com/a...
Thanks to these viewers for their contributions to translations
Hebrew: Omer Tuchfeld
Italian: @teobucci
-------------------
Video timeline
0:00 - Introduction
0:30 - Recap
1:49 - Using training data
3:01 - Cost functions
6:55 - Gradient descent
11:18 - More on gradient vectors
12:19 - Gradient descent recap
13:01 - Analyzing the network
16:37 - Learning more
17:38 - Lisha Li interview
19:58 - Closing thoughts
------------------
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Part 3 will be on backpropagation. I had originally planned to include it here, but the more I wanted to dig into a proper walk-through for what it's really doing, the more deserving it became of its own video. Stay tuned!
Can you provide some link to the training set?
I'm quite new and I'm trying to learn this "Hello World" of NN,
Did you consider to do a part about phase functioned neural networks? Would be awesome!
As the part progresses,
we're getting closer to seeing that "lena.jpg" picture
You are doing very good job , lots of hard work behind this video , i salute your hard work thanks
So have you decided to do more of these videos? There is a line with CNNs and LSTMs in video series...
Dissapointed you did not animate a 13000-dimensional graph. Would make things easier
No.... No it would not....
A 2-dimensional projection of a 13000-dimensional graph would probably look like a pile of garbage.
If 3Blue1Brown cannot animate a 13000-dimensional graph, then noone can.
@@mrwalter1049 You're not thinking fourth-dimensionally!!
@@tehbonehead No-one is. We're trapped in three dimensions. That's why you could never imagine what a 4-dimensional cube looks like.
Making a 4-dimensional projection of a 13000-dimensional object isn't significantly better than 3 dimensions.
If you meant to be humorous I hope someone gets a chuckle, because I didn't. Then your effort won't be in vain.
Have a nice day 🙂
One of youtube's highest quality content channels! Chapeau
Kosio Varbenov +
True!
'chapeau'
agree
Why you write "chapeau"? Hat in French. What does hat in that sentence do?????Be honest tho
I'm an IT student, and we have an Assignment on exactly this topic. We even have to use the MNIST data set. I have to say, this is absolutely lifesaving and I can not thank you enough Grant. What you do here is something that only a handful of people on this planet can do, explain and visualize rather complicated topics beautifully and simple. So from me and A LOT of students all around the globe, thank you so so much
Yes, it is just extremely good, in an objective way. He is brilliant at it, and spends a lot of time on each video. If there is an explanation of something by 3blue1brown, you will not find anything explaining it nearly as good.
@@vsiegel bro you guys are so much smarter than i am im jealous
@@johndough510 If you are thinking about how smart you are, you are probably smarter than you think. No worries.
@@vsiegel thanks for being so cool about it man, hope you have a good one
@@johndough510 I'm kill
Unlike most teachers of subjects like this, this gentleman seems to be genuinely concerned that his audience understands him, and he makes a concerted and highly successful effort to convey the ideas in a cogent, digestible and stimulating form.
TOTALLY agreed
concerted with whom? I don't think you understand how to use that word.
he is a true proffesor!
@@redflipper992 I don't think you understand what concerted means. Stop trying to act smart and think you're better than everyone here. Be humble. You are irrelevant in the big picture.
@@redflipper992I read that as "concerned"
I'm only 12 minutes into this video right now, but I just wanted to say how much I appreciate the time and spacing you give to explaining a concept. You add pauses, you repeat things with slightly different wording, and you give examples and zoom in and out, linking to relevant thought processes that might help trigger an "a-ha" moment in the viewer. Many of these "hooks" actually make me understand concepts I've had trouble grasping in Maths, all because of your videos and the way you choose to explain things. So thanks! You're helping me a lot to become a smarter person. :)
Shrooblord +
yes.
Totally agree!
Absolutely agree
Couldn't agree more
My math career is over. Once I learned about gradient descent, it was all downhill from there.
I had a similar fate, except mine went negatively uphill.
just remember there are people in an even lower minima than you are.
Yea, but making random choices makes you eventually reach the bottom.
It's all planar vectors from here.
You just need to take a few steps back and turn your life around.
3:38 you missed the chance of using the meme *"AI: I've found an output, but at what cost?"*
Not only the videos themselves are great on this channel but the lists of the supporting materials are amazing too! Drives me down a breathtaking rabbit hole every time! Thank you!
After watching your first video, I ended up drawing a "mock" neural network up on paper that would work on a 3x3 grid (after all what else are you supposed to do during a boring lecture class?). It was supposed to recognize boxes, x's, sevens, simple shapes, and I defined the 7 or so neurons that I thought it might need by hand. I did all the weighted sums and sigmoid functions on paper with calculator in hand. It took maybe an hour and a half to get everything straight but once I did, it worked. It guessed with fairly good accuracy that the little seven I "inputted" was a little seven. All that excitement because of your video.
Later that evening and the next one, I tried to program the same function taking PNGs as inputs and definitions of the neurons and it honestly was only a little more rewarding. But now that I see what the hidden neurons *actually* look like, I only want to learn so much more. I expected the patterns to be messy, but I was really surprised to see that it really does almost look like just noise.
Thank you for making these videos. I find myself suddenly motivated to go back to calculus class tomorrow and continue our less on gradients. There's just so much out there to learn and it's educators like you that are making it easier for curious individuals like me to get there.
That's so cool, thanks for sharing! I didn't expect anyone to actually go an play with it by hand, but simplifying down to a 3x3 grid seems really smart. Stay curious!
Nyhilo +
3Blue1Brown +
Were you able to do all of that by watching this video series alone? Please can you add other resources you referred? Thanks!
.....go anD play... AND AN+D
"But we can do better! Growth mindset!" at 5:18 .... a wholesome intellectual i love to see it
I just sat through a 3 day ML accelerator class and you series did a far better job at explaining them with 4 twenty minute videos. Well done mate. Really appreciate it. Thank you
this youtuber is the best in maths and engineering in general
i have never been so astounded for how easy learning machine learning can be, without having to take in bunch of complex topics that doesnt add to the discussion.
Like most of the courses try to make you understand various different complex topics and by the time you finished it, you will ahve forgotten mostly about machine learning. Why dont you just explain the catch for each concept then allow us learn it in depth afterwards
like these channels only explaining the concepts with both ease of learning and depths are the best.
i have no words to describe how thankful i am. thank you so much for such great content.
You are changing the world, shaping humanity. I wish you and your team, happy and peaceful life. This is a noble profession, god bless you guys.
Rarely seen such well explained videos that break down a complex topic into components that are so easy to understand. Perfect speed and focus on core aspects. Plus summaries and references to previously discussed correlations. Very good examples and animations, which pick up the viewer without/with little knowledge where it has to be. THANK YOU
Cant believe you explained this so easily. I thought it would take me ages to wrap my head around what neural networks basically are. This is truly amazing explanation!
That end comment with Lisha Li really points out how important it is to put a lot of effort into gathering and creating good and structured data sets. I know it's cliché to state "garbage in, garbage out", but these findings put very precise context and weight to this particular issue.
Haha, "weight".....get it?
StiffWood True. This also becomes ethically important in medical applications of AI, where poor input can create racist AI and the like.
I wish someone would have introduced this to me at a young age back in the 90s. I had no idea neural network have existed for so long
Now it's easier to explain. He couldn't have made a video like this back then, both because TH-cam didn't exist, and all the relevant stuff would be in technical papers...
@@lopezb Also, it was a really niche field that didn't show that much promise.
Also, from my understanding, modern neural networks are very different from the one in the 90s
It's a college graduate course.
In the 90s NN fell into disfavor.
This video is truly a work of art. The animations are mesmerizing. Thank you for all your work, Grant!
Keep coming back to this series and sharing it with so many people. This whole channel is truly a gift. Thank you so much for making these!
"I'm more of a multiple choice guy" LOL
Did anybody else feel bad for the network after he called the output utter trash?😢
I know, especially with those sad computer eyes. Tragic
Yes, specially when he called it bad computer :(
we are all utter trash for future robots
Bruh there's literally Reinforcement Learning where the Network is tortured by the researchers when it gets a wrong answer and the torture continues until it gets the right answer lol
Lmao.
my professor has explained this in 3 lectures for about 6-7 hours. 3B1B explained it in 30 mins and it is much more clearer. I can now visualize and understand the what/why/how behind the basic deep learning algorithms. Really appreciate it!!!
Same!
This man is a living legend. I really sincerely believe he's one of the best "explainers" for many complex mathematical topics. I found your channel because of linear algebra, and now I'm relieved whenever I search for a topic and see one of your videos. You truly are the master of your league
We need AI to recognise what the doctors write
whoever designs the algorithm and engineers the software deserves a nobel peace prize
@@frankchen4229 LOL
Who writes any more??? That horse left the barn a long time ago. Besides, we are no longer doctors. Courtesy of insurance companies we are "providers". (The same strategy of devalued identities has long been used by invading armies to anonymize those being conquered, an apropos metaphor.)
@@johnbarbuto5387 why so mad
@@johnbarbuto5387 not courtesy of insurance companies; courtesy of the fact that healthcare needs to be paid for. State systems are also payer systems.
How confident I feel in my current knowledge of neural networks:
15:41 - *"Uh...I'm really more of a multiple choice kinda guy."*
The neural networks themselves generally feel the same, so at least we won't be alone in that sentiment.
Assuming thinking rocks and metal count as good companionship at any rate.
Math courses in my college are basically trash compared to your videos , finally now I understand how math is being applied in computer science . Thank you so much for teaching in such an illustrative way .
Thank you soo much.. 3Blue1Brown! For this video and this platform.. that has made learning of such complexity.. much more easier. Even for people, like beginners, who are not from the specific background. You guys, make such topics, so fun to learn. I wish you guys all the best for you channel, and wish you get much more appreciation for the beautiful change.. you bring to the minds of this generation.
"It's actually just calculus."
"Even worse!"
i'm dead
Actually, AI just lie to people of the world
so far
calculus is the easiest form of math
@@the.abhiram.r
I wish you are joking
Watching 3b1b
Step 3b. Drop Everything
Step 1b. Watch religiously.
oh the accuracy of this comment
My god I relate
for me it is.
step 3b: make sure you are ready. you need to be 100% focused.
step 1b: Watch it critically, be sure not to strengthen your miss believes. If it seems simpel and obvious I am probably misunderstanding it.
+
counting level: Parker
I saw this video when i know nothing and i had a lots of intuitions, I'm rewatching after studied a lot more and I'm still learning a lot. You're a great teacher
I want to thank you for the great effort that you put in simplifying such complex topics.
Universities really lag behind in terms of making concepts crystal clear to student like you do. Evern MIT and other fancy universities which post their lectures online.
Keep up the great work!!
Hands down, I have never seen anyone explain mathematics so beautifully the way he does. Kudos!
You sir deserve a noble, or an oscar...what an incredibly effective method of teaching. thank you so very much!!!
NO! BAD COMPUTER! made me crack up! lol
A balon di or too
@@shakhaoathossain5032 XD good one.
@@shakhaoathossain5032 add a grammy in there too hahaha
The fact it was recorded digitally meant he said that to a real computer.😭
AND SO DID YOU!!! 😠
First of all, thank you infinitely for how beautifully and elegantly you have taught this, as with everything. Oftentimes my classes in uni bore me with factdumping, but you never fail to help me fall in love with maths and cs again. Second, i didn't know you did the KA course on multivariate calc! Yet another time you saved my life!!
Probably, even a 5 year old would master deep learning when taught in this way. What a video man ! Awesomeeeeeeeee
When I watch your videos, all I want to do is keep going. Thanks for motivating me, as well as others, with your amazing content. I really appreciate it.
I love you so much. I'm taking multivariate calculus and I'm doing some neural network work right now, and none of my teachers have the passion nor the capability to teach as well as you. You help me keep my passion for learning alive
Incompetence galore.
So uhhhh, did you just say that this was the "Hello World!" of neural networking.....
@Winston Mcgee "p r e t t y m u c h i t"
Yeah its scary to think but many programmers can easily code stuff like this as a simple AI tool. Just a bunch of linear algebra and statistics. It gets more interesting when you try to do more than minimizing a cost function and really make it 'learn'. Even after I say all this, it took me quite a bit to fully digest the "hello world" haha.
Well, the data are the hello world, because it is the first image set that people take to test or learn about an algorithm.
Because you don't need all those knowledge to build a network that can read digits. It's easy with Keras even my grandmother can do it
Yes it is, because the dataset is veerry clean and CNN through Keras is very easy to implement, though you need to have huge background in math and code-writing (I'm a statistician so I have a little bit of both) :)
When something is amazing, it will look like
with your work. Even your bg voice is totally understandable, legit and direct
3:40 "NO! Bad computer!"
AJ Nelson Bad boy!
I felt so sorry for the computer...
"To say that more mathematically..."
i heard eric cartman say it.
+
how can anyone dislike these videos? very detailed, accurate explanations and cool animations. Keep up the good work!!
stupids are everywhere.
Possibly one of the best mini-courses on ML anywhere. Clearly explained concepts, beautiful post-production. kudos
I'm doing my Masters in applied Econ right now and we briefly went over Neural Networks in my advanced econometrics class. Some of the terminology was a bit different and I felt like I could understand it decently in office hours with my professor, but this was still a great resource to solidify my understanding of the concept. (Also we looked at the MNIST dataset in class too) :D
This is mindblowingly interesting and extremely well explained. Thank you!
The best explanation so far. Viewed over 50 videos. No one could explained in simple terms like you did. Thank you.
I've watched many videos and done some reading on how neural networks work (learn), but I couldn't find a satisfactory explanation until I watched this video. Your examples, analogies, visuals... were just perfect. Thank you so much.
Hi,
I just started learning machine algorithms including neural networks and I find what you are publishing on this channel extremely useful, perfectly explained and quite helpful and I thank you for that from the bottom of my heart 🙏🏻
Continue and may God bless you all
Love the eye/ pi animations! :) Gets me smiling amidst all the complexities of Neural Networks 😲
The graphics of this video is absolutely stunning! Thank you for your work ♡
This series is a masterpiece of education. All the time and effort shows. Kudos!
Easily one of the best videos I've seen. To explain such a complex concept so clearly is such an amazing skill. Thank you.
amazing videos! I just discovered them, and I'm so hooked into this, all this algebra and math I've tried learning, starts to make sense to me.
thanks, and keep on the good work!
This series is the best neural network explanation I have ever encountered as a beginner!
You are the best science teacher I have every seen. If anyone upstairs is serious about our education system they should use your videos as baseline for how to teach properly; you never use a term that wasn't clearly defined prior, you use analogies perfectly, and you tie new technical info back to the original concept thereby making sense how the new info fits together in the larger picture. If my high-school and college teachers were like you I would have done infinitely better at school.
You do such a great job at explaining complex topics in a "simple" way, great job
Your videos are priceless. Thank you very much for your work and also a big thank you to your Patreon supporters.
At the 16 minute mark, I got sucker punched. After having gone through this and the previous video on machine learning and just loving how an art student like myself is enjoying math such as this and feeling like I'm making some progress..... You tell me that this is all stuff that was figured out in the 80s and I'm like...... Oh Come On! Lol!
Honestly, the 80s is actually very recent compared to other stuff. In math undergrad I was learning concepts from the 1800s lmao
@@apuapustaja2047 yup! Math is older than all of us... it's our discoveries that are "new"~
Honestly, don’t feel discouraged. General Relativity was formulated almost a century ago, but that doesn’t mean it is easier to understand.
@Dark Aether some definitely could say that~
@Dark Aether What do you even mean by that? Right now, we are living in a moment in which scientific knowledge is being acquired at the fastest rate we have ever seen. The number of active scientists right now, as you might expect, is the largest in history. Now, if you are talking about "raw" intelligence... well, I'm pretty sure evolving into creatures with a noticeable higher intelligence is going to take a long, long, long time.
Me in part 1: Ah I think I'm starting to understand this whole thing.
Me in part 2: Nevermind...
excatly
Once you learn more math it will be meaningful. I guess if you dont understand this video then something is missing in your calculus and/or linear algebra
Thank you very much for your in sights, every second of your video is worth watching without skipping. Your way of explaining the concepts are very clear and easy to grasp, it starts a fire to learn more on the topic. Thank you once again for your efforts. Lookin forward to more of your contents
WOO BEEN WAITING FOR THIS
This series is fantastic and just the right level of being complex but understandable. Thanks!
This lecture series (and channel overall) is phenomenal! I tutor some students in various math and science subjects, and I have to start recommending these videos more.
This is a gem to practically understand all the topics we learn at school and see the beauty of maths
Hi 3B1B, as a technical physics student, beeing interested into this topic for quite a while now, and also enjoying your content for quit a while, i really wanna thank you for not only going into this topic particulary, but also for educating a relatively large audience with your informative videos and improving the interest into mathematical sciences for a lot of people including me, which is pretty important in my opinion! Keep it up!!
Thank you for making these videos. They are truly amazing because anyone can learn something from them. All the way from people who are currently studying in this domain who have the mathematical toolbox to truly grasp every moment to people who are just curious and generally end up learning the intuitive basics ! Eagerly waiting for the next video :)
I've never studied calculus but I'm learning how to use NN in Python and I'm reading a book about it and have watched some videos, so I have the rough idea of how it works. This video showed up on my recommended page while I was studying, and the explanation made some grey areas seem more clear now! And the crazy thing is a neural network helped me find this video to understand more about neural networks! and it doesn't know what it's doing, it's just minimizing some hidden cost function!
"Our growth mindset is reflected when we think always if we can do better!" You are always awesome. Thanks for all the informative videos that imparts a lot of fundamental knowledge to people like me.
You wouldn't believe what my prof showed to us in class today! Only a day after this video was released. You're videos are incredible. Thank you! :)
Absolutely love your videos! I'm trying to show this video to my friends who doesn't know English so well, and I decided to make subtitles. Hope you approve them, I really want to spread word about your work.
The visualizations are so great. Appreciate the thoughtfulness of these videos
Wow, what a brilliant job you do in your videos, i can barely imagine the hardwork and efforts necessary to summarize all this deep knowledge and create meaningful animations, congrats and pleas keep it up!!!!
Some thoughts on the results:
1. 14:01 The weights for the first layer _seem_ to be meaningless patterns when viewed individually, but combined, they do encode some kind of sophisticated pattern detection. That particular pattern detection isn't uniquely specified or constrained by this particular set of weights on the first layer; rather, there are infinitely many ways that the pattern detection scheme can be encoded in the weights of this single layer.
These infinite other solutions can be thought of as the set of matrices that are row-equivalent to the 16x700ish matrix where each row is the set of weights for each of the neurons on this layer. You can rewrite each of the rows as a linear combination of the set of current rows, while possibly still preserving the behavior of the whole NN by performing a related transformation to the weights of the next layer. In this way, you can rewrite the patterns of the first layer to try and find an arrangement that tells you something about the reasoning. Row reduction in particular might produce interesting results!
2. 15:10 I think I understand the reason why your NN produces a confident result - it's because it was never trained to understand what a number _doesn't_ look like. All of the training data, from what I can tell, is numbers associated with 100% confident outputs. You'd want to train it on illegible handwriting, noise, whatever you expect to feed it later, with a result vector that can be interpreted as 0% confidence, by having small equal weights, having all weights to zero, or maybe an additional neuron that the NN uses to report "no number".
2 is a painfully easy mistake to make, because it requires the human assembling the programming data to think outside the box.
Quite hard to read your comment, but it seems that we have the same idea: the neural network _is_ detecting smaller elements like "edges and loops" (as video author puts it), assuming those pictures 14:01 are of the actual results. The next layer then starts combining these elements, and it seems that if you stare at it long enough you can almost start guessing what it might be doing, like combining row 1 column 2 = strong + row 3 column 1 = strong + row 3 column 4 = strong + row 2 column 4 = weak + row 1 column 4 = weak, you might be going toward an 8 on those counts, and depending on some other combination of pattern strengths it might be a 6 or 9 if there is absence of signal upper/right or lower/left given by some of those patterns which are matched with the input. This is almost certainly not accurate as an example, but it seems to be the theme of how it works. 16:05 _"... picking up on edges and ... not at all what the network is doing."_ This statement in the video seems wrong.
P.S. If this first part above is true, then the neural network might be capable of drawing a 5 (15:39). You just have to extract that answer in the way that it is in there, which is a bit more involved that following it's normal operation for which it is built. If you look into what combination of patterns from the first layer output, in what strengths, leads to a number (5 for example), than you could superimpose those patterns unto each other, and that would be what this neural network thinks is that number. It shouldn't be too hard to write a function to the already trained network, to draw this out.
You explained in less than 4 minutes something that took me several days to understand from other sources. You are awesome!
I think what puts this material apart from the competition is the authors intuition of the focal points where the audience might loose the plot. Then he takes a patient and systematic turn to reiterate what have been learned so far to reinforce the basics to decrease the cognitive leap needed to grasp the next step. This ability is in my experience pretty unique.
I am so happy I discovered this today. I ignored all (well, most) of the math, and I still came away with a pretty solid understanding of how it works. Your explanations are ridiculously clear; you have a gift.
I love how he has people like Desmos and Markus Persson supporting him on patreon
Thanks for the part 2, I was waiting for it impatiently.
man the animations are super helpful it's amazing how much effort was put into this
You have the gift of teaching things that you know ! Not all teacher have this!
Most brilliant educator I've ever had the please of seeing work. Thank you, you give me some semblance of hope for the human race.
I was looking forward to this video all week.
Your video is outstanding. I'm truly happy to have found your channel. I love the teaching method and constant visual examples.
I'd like to applaud 3Blue1Brown for these chapters and the work. Bravo, you achieved eloquent concise examples and 'splainin. Bravo (y). I'm learning!
You're an amazing explainer of amazing things and this particular thing you explained is particularly amazing!
Wow, incredible! You really do know how to explain complex things the simple way.
Thanks sounds simple when you explain it but when I’ve looked at books all I see was formulas now I kind of understand what’s going on. Thanks you sir!
Thank you for the extraordinary effort to present this in a simple and elegant and beautiful way. Thanks a lot.
Would just take a moment here to appreciate the sheer brilliance of Grant on this series. I would not have reached a decent level of NN w/o these explanations and so would so many other people. Single best series on NN / Math out there in general.
Once I graduate and start working, I’m gonna send you the money I owe you for watching all these videos. I’m doing BSEE for control systems so hopefully it works out.
You explain things very well and very patiently. Thank you.
The production effort of your work ---- amazing
@ 3:39
Wow... I didn't realize 3B1B likes to bully neural networks. That was ruthless.
Yes, it's a jungle in there.
@- RedBlazerFlame -
' Oversimplified' reference
There's a training method called Reinforcement Learning where you literally torture the Network when it gets the wrong output lol
Pause and ponder?! Every 10 seconds I stop for a minute of thinking! And on all of your videos!
This time I had a time I thought for like half an hour.
This is the best lesson on the basics of NN. Awesome content and delivery
Love you for this sort of fully clearcut explaination with logic
BEST VISUAL THERAPY IN MY LIFE
Fantastic series! Doing a PhD in neural networks and I had never seen them so well explained! Thank you!
I’m curious to know why you’d want a PhD in only neural networks and not some broader major that covers neural networks in it such as like a data analytics or CSE. I didn’t think they had that much content in them, unless you’re doing more biological imitations
Your work is much much appreciated my good sir. We can't thank you enough.
This is the first video I found about AI that actually explains the how, instead of just dumbing everything down and leaving just as confused as before watching. Thank you!