This is a three. It's sloppily written and rendered at an extremely low resolution of 28 by 28 pixels.

But your brain has no trouble recognizing it as a three and I want you to take a moment to appreciate

How crazy it is that brains can do this so effortlessly?

I mean this this and this are also recognizable as threes,

even though the specific values of each pixel is very different from one image to the next.

The particular light-sensitive cells in your eye that are firing when you see this three

are very different from the ones firing when you see this three.

But something in that crazy smart visual cortex of yours

resolves these as representing the same idea while at the same time recognizing other images as their own distinct ideas

But if I told you hey sit down and write for me a program that takes in a grid of 28 by 28

pixels like this and outputs a single number between 0 and 10 telling you what it thinks the digit is

Well the task goes from comically trivial to dauntingly difficult

Unless you've been living under a rock

I think I hardly need to motivate the relevance and importance of machine learning and neural networks to the present into the future

But what I want to do here is show you what a neural network actually is

Assuming no background and to help visualize what it's doing not as a buzzword but as a piece of math

My hope is just that you come away feeling like this structure itself is

Motivated and to feel like you know what it means when you read or you hear about a neural network quote-unquote learning

This video is just going to be devoted to the structure component of that and the following one is going to tackle learning

What we're going to do is put together a neural network that can learn to recognize handwritten digits

This is a somewhat classic example for

Introducing the topic and I'm happy to stick with the status quo here because at the end of the two videos I want to point

You to a couple good resources where you can learn more and where you can download the code that does this and play with it?

on your own computer

There are many many variants of neural networks and in recent years

There's been sort of a boom in research towards these variants

But in these two introductory videos you and I are just going to look at the simplest plain-vanilla form with no added frills

This is kind of a necessary

prerequisite for understanding any of the more powerful modern variants and

Trust me it still has plenty of complexity for us to wrap our minds around

But even in this simplest form it can learn to recognize handwritten digits

Which is a pretty cool thing for a computer to be able to do.

And at the same time you'll see how it does fall short of a couple hopes that we might have for it

As the name suggests neural networks are inspired by the brain, but let's break that down

What are the neurons and in what sense are they linked together?

Right now when I say neuron all I want you to think about is a thing that holds a number

Specifically a number between 0 & 1 it's really not more than that

For example the network starts with a bunch of neurons corresponding to each of the 28 times 28 pixels of the input image

which is

784 neurons in total each one of these holds a number that represents the grayscale value of the corresponding pixel

ranging from 0 for black pixels up to 1 for white pixels

This number inside the neuron is called its activation and the image you might have in mind here

Is that each neuron is lit up when its activation is a high number?

So all of these 784 neurons make up the first layer of our network

Now jumping over to the last layer this has ten neurons each representing one of the digits

the activation in these neurons again some number that's between zero and one

Represents how much the system thinks that a given image?

Corresponds with a given digit. There's also a couple layers in between called the hidden layers

Which for the time being?

Should just be a giant question mark for how on earth this process of recognizing digits is going to be handled

In this network I chose two hidden layers each one with 16 neurons and admittedly that's kind of an arbitrary choice

to be honest I chose two layers based on how I want to motivate the structure in just a moment and

16 well that was just a nice number to fit on the screen in practice

There is a lot of room for experiment with a specific structure here

The way the network operates activations in one layer determine the activations of the next layer

And of course the heart of the network as an information processing mechanism comes down to exactly how those

activations from one layer bring about activations in the next layer

It's meant to be loosely analogous to how in biological networks of neurons some groups of neurons firing

cause certain others to fire

Now the network

I'm showing here has already been trained to recognize digits and let me show you what I mean by that

It means if you feed in an image lighting up all

784 neurons of the input layer according to the brightness of each pixel in the image

That pattern of activations causes some very specific pattern in the next layer

Which causes some pattern in the one after it?

Which finally gives some pattern in the output layer and?

The brightest neuron of that output layer is the network's choice so to speak for what digit this image represents?

And before jumping into the math for how one layer influences the next or how training works?

Let's just talk about why it's even reasonable to expect a layered structure like this to behave intelligently

What are we expecting here? What is the best hope for what those middle layers might be doing?

Well when you or I recognize digits we piece together various components a nine has a loop up top and a line on the right

an 8 also has a loop up top, but it's paired with another loop down low

A 4 basically breaks down into three specific lines and things like that

Now in a perfect world we might hope that each neuron in the second-to-last layer

corresponds with one of these sub components

That anytime you feed in an image with say a loop up top like a 9 or an 8

There's some specific

Neuron whose activation is going to be close to one and I don't mean this specific loop of pixels the hope would be that any

Generally loopy pattern towards the top sets off this neuron that way going from the third layer to the last one

just requires learning which combination of sub components corresponds to which digits

Of course that just kicks the problem down the road

Because how would you recognize these sub components or even learn what the right sub components should be and I still haven't even talked about

How one layer influences the next but run with me on this one for a moment

recognizing a loop can also break down into subproblems

One reasonable way to do this would be to first recognize the various little edges that make it up

Similarly a long line like the kind you might see in the digits 1 or 4 or 7

Well that's really just a long edge or maybe you think of it as a certain pattern of several smaller edges

So maybe our hope is that each neuron in the second layer of the network

corresponds with the various relevant little edges

Maybe when an image like this one comes in it lights up all of the neurons

associated with around eight to ten specific little edges

which in turn lights up the neurons associated with the upper loop and a long vertical line and

Those light up the neuron associated with a nine

whether or not

This is what our final network actually does is another question, one that I'll come back to once we see how to train the network

But this is a hope that we might have. A sort of goal with the layered structure like this

Moreover you can imagine how being able to detect edges and patterns like this would be really useful for other image recognition tasks

And even beyond image recognition there are all sorts of intelligent things you might want to do that break down into layers of abstraction

Parsing speech for example involves taking raw audio and picking out distinct sounds which combine to make certain syllables

Which combine to form words which combine to make up phrases and more abstract thoughts etc

But getting back to how any of this actually works picture yourself right now designing

How exactly the activations in one layer might determine the activations in the next?

The goal is to have some mechanism that could conceivably combine pixels into edges

Or edges into patterns or patterns into digits and to zoom in on one very specific example

Let's say the hope is for one particular

Neuron in the second layer to pick up on whether or not the image has an edge in this region here

The question at hand is what parameters should the network have

what dials and knobs should you be able to tweak so that it's expressive enough to potentially capture this pattern or

Any other pixel pattern or the pattern that several edges can make a loop and other such things?

Well, what we'll do is assign a weight to each one of the connections between our neuron and the neurons from the first layer

These weights are just numbers

then take all those activations from the first layer and compute their weighted sum according to these weights I

Find it helpful to think of these weights as being organized into a little grid of their own

And I'm going to use green pixels to indicate positive weights and red pixels to indicate negative weights

Where the brightness of that pixel is some loose depiction of the weights value?

Now if we made the weights associated with almost all of the pixels zero

except for some positive weights in this region that we care about

then taking the weighted sum of

all the pixel values really just amounts to adding up the values of the pixel just in the region that we care about

And, if you really want it to pick up on whether there's an edge here what you might do is have some negative weights

associated with the surrounding pixels

Then the sum is largest when those middle pixels are bright, but the surrounding pixels are darker

When you compute a weighted sum like this you might come out with any number

but for this network what we want is for activations to be some value between 0 & 1

so a common thing to do is to pump this weighted sum

Into some function that squishes the real number line into the range between 0 & 1 and

A common function that does this is called the sigmoid function also known as a logistic curve

basically very negative inputs end up close to zero very positive inputs end up close to 1

and it just steadily increases around the input 0

So the activation of the neuron here is basically a measure of how positive the relevant weighted sum is

But maybe it's not that you want the neuron to light up when the weighted sum is bigger than 0

Maybe you only want it to be active when the sum is bigger than say 10

That is you want some bias for it to be inactive

what we'll do then is just add in some other number like negative 10 to this weighted sum

Before plugging it through the sigmoid squishification function

That additional number is called the bias

So the weights tell you what pixel pattern this neuron in the second layer is picking up on and the bias

tells you how high the weighted sum needs to be before the neuron starts getting meaningfully active

And that is just one neuron

Every other neuron in this layer is going to be connected to all

784 pixels neurons from the first layer and each one of those 784 connections has its own weight associated with it

also each one has some bias some other number that you add on to the weighted sum before squishing it with the sigmoid and

That's a lot to think about with this hidden layer of 16 neurons

that's a total of 784 times 16 weights along with 16 biases

And all of that is just the connections from the first layer to the second the connections between the other layers

Also, have a bunch of weights and biases associated with them

All said and done this network has almost exactly

13,000 total weights and biases

13,000 knobs and dials that can be tweaked and turned to make this network behave in different ways

So when we talk about learning?

What that's referring to is getting the computer to find a valid setting for all of these many many numbers so that it'll actually solve

the problem at hand

one thought

Experiment that is at once fun and kind of horrifying is to imagine sitting down and setting all of these weights and biases by hand

Purposefully tweaking the numbers so that the second layer picks up on edges the third layer picks up on patterns etc

I personally find this satisfying rather than just reading the network as a total black box

Because when the network doesn't perform the way you

anticipate if you've built up a little bit of a relationship with what those weights and biases actually mean you have a starting place for

Experimenting with how to change the structure to improve or when the network does work?

But not for the reasons you might expect

Digging into what the weights and biases are doing is a good way to challenge your assumptions and really expose the full space of possible

solutions

By the way the actual function here is a little cumbersome to write down. Don't you think?

So let me show you a more notationally compact way that these connections are represented. This is how you'd see it

If you choose to read up more about neural networks

Organize all of the activations from one layer into a column as a vector

Then organize all of the weights as a matrix where each row of that matrix

corresponds to the connections between one layer and a particular neuron in the next layer

What that means is that taking the weighted sum of the activations in the first layer according to these weights?

Corresponds to one of the terms in the matrix vector product of everything we have on the left here

By the way so much of machine learning just comes down to having a good grasp of linear algebra

So for any of you who want a nice visual understanding for matrices and what matrix vector multiplication means take a look at the series I did on linear algebra

especially chapter three

Back to our expression instead of talking about adding the bias to each one of these values independently we represent it by

Organizing all those biases into a vector and adding the entire vector to the previous matrix vector product

Then as a final step

I'll rap a sigmoid around the outside here

And what that's supposed to represent is that you're going to apply the sigmoid function to each specific

component of the resulting vector inside

So once you write down this weight matrix and these vectors as their own symbols you can

communicate the full transition of activations from one layer to the next in an extremely tight and neat little expression and

This makes the relevant code both a lot simpler and a lot faster since many libraries optimize the heck out of matrix multiplication

Remember how earlier I said these neurons are simply things that hold numbers

Well of course the specific numbers that they hold depends on the image you feed in

So it's actually more accurate to think of each neuron as a function one that takes in the

outputs of all the neurons in the previous layer and spits out a number between zero and one

Really the entire network is just a function one that takes in

784 numbers as an input and spits out ten numbers as an output

It's an absurdly

Complicated function one that involves thirteen thousand parameters in the forms of these weights and biases that pick up on certain patterns and which involves

iterating many matrix vector products and the sigmoid squish evocation function

But it's just a function nonetheless and in a way it's kind of reassuring that it looks complicated

I mean if it were any simpler what hope would we have that it could take on the challenge of recognizing digits?

And how does it take on that challenge? How does this network learn the appropriate weights and biases just by looking at data? Oh?

That's what I'll show in the next video, and I'll also dig a little more into what this particular network we are seeing is really doing

Now is the point I suppose I should say subscribe to stay notified about when that video or any new videos come out

But realistically most of you don't actually receive notifications from YouTube, do you ?

Maybe more honestly I should say subscribe so that the neural networks that underlie YouTube's

Recommendation algorithm are primed to believe that you want to see content from this channel get recommended to you

anyway stay posted for more

Thank you very much to everyone supporting these videos on patreon

I've been a little slow to progress in the probability series this summer

But I'm jumping back into it after this project so patrons you can look out for updates there

To close things off here I have with me Lisha Li

Lee who did her PhD work on the theoretical side of deep learning and who currently works at a venture capital firm called amplify partners

Who kindly provided some of the funding for this video so Lisha one thing

I think we should quickly bring up is this sigmoid function

As I understand it early networks used this to squish the relevant weighted sum into that interval between zero and one

You know kind of motivated by this biological analogy of neurons either being inactive or active (Lisha) - Exactly

(3B1B) - But relatively few modern networks actually use sigmoid anymore. That's kind of old school right ? (Lisha) - Yeah or rather

ReLU seems to be much easier to train (3B1B) - And ReLU really stands for rectified linear unit

(Lisha) - Yes it's this kind of function where you're just taking a max of 0 and a where a is given by

what you were explaining in the video and what this was sort of motivated from I think was a

partially by a biological

Analogy with how

Neurons would either be activated or not and so if it passes a certain threshold

It would be the identity function

But if it did not then it would just not be activated so be zero so it's kind of a simplification

Using sigmoids didn't help training, or it was very difficult to train

It's at some point and people just tried relu and it happened to work

Very well for these incredibly

Deep neural networks. (3B1B) - All right

Thank You Lisha

for background amplify partners in early-stage VC invests in technical founders building the next generation of companies focused on the

applications of AI if you or someone that you know has ever thought about starting a company someday

Or if you're working on an early-stage one right now the Amplify folks would love to hear from you

they even set up a specific email for this video 3blue1brown@amplifypartners.com

so feel free to reach out to them through that