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Last video I laid out the structure of a neural network
I'll give a quick recap here just so that it's fresh in our minds
And then I have two main goals for this video. The first is to introduce the idea of gradient descent,
which underlies not only how neural networks learn,
but how a lot of other machine learning works as well
Then after that we're going to dig in a little more to how this particular network performs
And what those hidden layers of neurons end up actually looking for
As a reminder our goal here is the classic example of handwritten digit recognition
the hello world of neural networks
these digits are rendered on a 28 by 28 pixel grid each pixel with some grayscale value between 0 & 1
those are what determine the activations of
784 neurons in the input layer of the network and
Then the activation for each neuron in the following layers is based on a weighted sum of
All the activations in the previous layer plus some special number called a bias
then you compose that sum with some other function like the sigmoid squishification or
a ReLu the way that I walked through last video
In total given the somewhat arbitrary choice of two hidden layers here with 16 neurons each the network has about
13,000 weights and biases that we can adjust and it's these values that determine what exactly the network you know actually does
Then what we mean when we say that this network classifies a given digit
Is that the brightest of those 10 neurons in the final layer corresponds to that digit
And remember the motivation that we had in mind here for the layered structure was that maybe
The second layer could pick up on the edges and the third layer might pick up on patterns like loops and lines
And the last one could just piece together those patterns to recognize digits
So here we learn how the network learns
What we want is an algorithm where you can show this network a whole bunch of training data
which comes in the form of a bunch of different images of handwritten digits along with labels for what they're supposed to be and
It'll adjust those
13000 weights and biases so as to improve its performance on the training data
Hopefully this layered structure will mean that what it learns
generalizes to images beyond that training data
And the way we test that is that after you train the network
You show it more labeled theta that it's never seen before and you see how accurately it classifies those new images
Fortunately for us and what makes this such a common example to start with is that the good people behind the MNIST base have
put together a collection of tens of thousands of handwritten digit images each one labeled with the numbers that they're supposed to be and
It's provocative as it is to describe a machine as learning once you actually see how it works
It feels a lot less like some crazy sci-fi premise and a lot more like well a calculus exercise
I mean basically it comes down to finding the minimum of a certain function
Remember conceptually we're thinking of each neuron as being connected
to all of the neurons in the previous layer and the weights in the weighted sum defining its activation are kind of like the
strengths of those connections
And the bias is some indication of whether that neuron tends to be active or inactive and to start things off
We're just gonna initialize all of those weights and biases totally randomly needless to say this network is going to perform
pretty horribly on a given training example since it's just doing something random for example you feed in this image of a 3 and the
Output layer it just looks like a mess
So what you do is you define a cost function a way of telling the computer: "No bad computer!
That output should have activations which are zero for most neurons, but one for this neuron what you gave me is utter trash"
To say that a little more mathematically what you do is add up the squares of the differences between
each of those trash output activations and the value that you want them to have and
This is what we'll call the cost of a single training example
Notice this sum is small when the network confidently classifies the image correctly
But it's large when the network seems like it doesn't really know what it's doing
So then what you do is consider the average cost over all of the tens of thousands of training examples at your disposal
This average cost is our measure for how lousy the network is and how bad the computer should feel, and that's a complicated thing
Remember how the network itself was basically a function one that takes in
784 numbers as inputs the pixel values and spits out ten numbers as its output and in a sense
It's parameterised by all these weights and biases
While the cost function is a layer of complexity on top of that it takes as its input
those thirteen thousand or so weights and biases and it spits out a single number describing how bad those weights and biases are and
The way it's defined depends on the network's behavior over all the tens of thousands of pieces of training data
That's a lot to think about
But just telling the computer what a crappy job, it's doing isn't very helpful
You want to tell it how to change those weights and biases so that it gets better?
To make it easier rather than struggling to imagine a function with 13,000 inputs
Just imagine a simple function that has one number as an input and one number as an output
How do you find an input that minimizes the value of this function?
Calculus students will know that you can sometimes figure out that minimum explicitly
But that's not always feasible for really complicated functions
Certainly not in the thirteen thousand input version of this situation for our crazy complicated neural network cost function
A more flexible tactic is to start at any old input and figure out which direction you should step to make that output lower
Specifically if you can figure out the slope of the function where you are
Then shift to the left if that slope is positive and shift the input to the right if that slope is negative
If you do this repeatedly at each point checking the new slope and taking the appropriate step
you're gonna approach some local minimum of the function and
the image you might have in mind here is a ball rolling down a hill and
Notice even for this really simplified single input function there are many possible valleys that you might land in
Depending on which random input you start at and there's no guarantee that the local minimum
You land in is going to be the smallest possible value of the cost function
That's going to carry over to our neural network case as well, and I also want you to notice
How if you make your step sizes proportional to the slope
Then when the slope is flattening out towards the minimum your steps get smaller and smaller and that kind of helps you from overshooting
Bumping up the complexity a bit imagine instead a function with two inputs and one output
You might think of the input space as the XY plane and the cost function as being graphed as a surface above it
Now instead of asking about the slope of the function you have to ask which direction should you step in this input space?
So as to decrease the output of the function most quickly in other words. What's the downhill direction?
And again it's helpful to think of a ball rolling down that hill
Those of you familiar with multivariable calculus will know that the gradient of a function gives you the direction of steepest ascent
Basically, which direction should you step to increase the function most quickly
naturally enough taking the negative of that gradient gives you the direction to step that decreases the function most quickly and
Even more than that the length of this gradient vector is actually an indication for just how steep that steepest slope is
Now if you're unfamiliar with multivariable calculus
And you want to learn more check out some of the work that I did for Khan Academy on the topic
Honestly, though all that matters for you and me right now
Is that in principle there exists a way to compute this vector. This vector that tells you what the
Downhill direction is and how steep it is you'll be okay if that's all you know and you're not rock solid on the details
because if you can get that the algorithm from minimizing the function is to compute this gradient direction then take a small step downhill and
Just repeat that over and over
It's the same basic idea for a function that has 13,000 inputs instead of two inputs imagine organizing all
13,000 weights and biases of our network into a giant column vector
The negative gradient of the cost function is just a vector
It's some Direction inside this insanely huge input space that tells you which
nudges to all of those numbers is going to cause the most rapid decrease to the cost function and
of course with our specially designed cost function
Changing the weights and biases to decrease it means making the output of the network on each piece of training data
Look less like a random array of ten values and more like an actual decision that we want it to make
It's important to remember this cost function involves an average over all of the training data
So if you minimize it it means it's a better performance on all of those samples
The algorithm for computing this gradient efficiently which is effectively the heart of how a neural network learns is called back propagation
And it's what I'm going to be talking about next video
There I really want to take the time to walk through
What exactly happens to each weight and each bias for a given piece of training data?
Trying to give an intuitive feel for what's happening beyond the pile of relevant calculus and formulas
Right here right now the main thing. I want you to know independent of implementation details
is that what we mean when we talk about a network learning is that it's just minimizing a cost function and
Notice one consequence of that is that it's important for this cost function to have a nice smooth output
So that we can find a local minimum by taking little steps downhill
This is why by the way
Artificial neurons have continuously ranging activations rather than simply being active or inactive in a binary way
if the way that biological neurons are
This process of repeatedly nudging an input of a function by some multiple of the negative gradient is called gradient descent
It's a way to converge towards some local minimum of a cost function basically a valley in this graph
I'm still showing the picture of a function with two inputs of course because nudges in a thirteen thousand dimensional input
Space are a little hard to wrap your mind around, but there is actually a nice non-spatial way to think about this
Each component of the negative gradient tells us two things the sign of course tells us whether the corresponding
Component of the input vector should be nudged up or down, but importantly the relative magnitudes of all these components
Kind of tells you which changes matter more
You see in our network an adjustment to one of the weights might have a much greater
impact on the cost function than the adjustment to some other weight
Some of these connections just matter more for our training data
So a way that you can think about this gradient vector of our mind-warpingly
massive cost function is that it encodes the relative importance of each weight and bias
That is which of these changes is going to carry the most bang for your buck
This really is just another way of thinking about direction
To take a simpler example if you have some function with two variables as an input and you
Compute that its gradient at some particular point comes out as (3,1)
Then on the one hand you can interpret that as saying that when you're standing at that input
moving along this direction increases the function most quickly
That when you graph the function above the plane of input points that vector is what's giving you the straight uphill direction
But another way to read that is to say that changes to this first variable
Have three times the importance as changes to the second variable that at least in the neighborhood of the relevant input
Nudging the x value carries a lot more bang for your buck
All right
Let's zoom out and sum up where we are so far the network itself is this function with
784 inputs and 10 outputs defined in terms of all of these weighted sums
the cost function is a layer of complexity on top of that it takes the
13,000 weights and biases as inputs and spits out a single measure of lousyness based on the training examples and
The gradient of the cost function is one more layer of complexity still it tells us
What nudges to all of these weights and biases cause the fastest change to the value of the cost function
Which you might interpret is saying which changes to which weights matter the most
So when you initialize the network with random weights and biases and adjust them many times based on this gradient descent process
How well does it actually perform on images that it's never seen before?
Well the one that I've described here with the two hidden layers of sixteen neurons each chosen mostly for aesthetic reasons
well, it's not bad it classifies about 96 percent of the new images that it sees correctly and
Honestly, if you look at some of the examples that it messes up on you kind of feel compelled to cut it a little slack
Now if you play around with the hidden layer structure and make a couple tweaks
You can get this up to 98% and that's pretty good. It's not the best
You can certainly get better performance by getting more sophisticated than this plain vanilla Network
But given how daunting the initial task is I just think there's something?
Incredible about any network doing this well on images that it's never seen before
Given that we never specifically told it what patterns to look for
Originally the way that I motivated this structure was by describing a hope that we might have
That the second layer might pick up on little edges
That the third layer would piece together those edges to recognize loops and longer lines and that those might be pieced together to recognize digits
So is this what our network is actually doing? Well for this one at least
Not at all
remember how last video we looked at how the weights of the
Connections from all of the neurons in the first layer to a given neuron in the second layer
Can be visualized as a given pixel pattern that that second layer neuron is picking up on
Well when we actually do that for the weights associated with these transitions from the first layer to the next
Instead of picking up on isolated little edges here and there. They look well almost random
Just put some very loose patterns in the middle there it would seem that in the unfathomably large
13,000 dimensional space of possible weights and biases our network found itself a happy little local minimum that
despite successfully classifying most images doesn't exactly pick up on the patterns that we might have hoped for and
To really drive this point home watch what happens when you input a random image
if the system was smart you might expect it to either feel uncertain maybe not really activating any of those 10 output neurons or
Activating them all evenly
But instead it
Confidently gives you some nonsense answer as if it feels as sure that this random noise is a 5 as it does that an actual
image of a 5 is a 5
phrase differently even if this network can recognize digits pretty well it has no idea how to draw them a
Lot of this is because it's such a tightly constrained training setup
I mean put yourself in the network's shoes here from its point of view the entire universe consists of nothing
But clearly defined unmoving digits centered in a tiny grid and its cost function just never gave it any
Incentive to be anything, but utterly confident in its decisions
So if this is the image of what those second layer neurons are really doing
You might wonder why I would introduce this network with the motivation of picking up on edges and patterns
I mean, that's just not at all what it ends up doing
Well, this is not meant to be our end goal, but instead a starting point frankly
This is old technology
the kind researched in the 80s and 90s and
You do need to understand it before you can understand more detailed modern variants and it clearly is capable of solving some interesting problems
But the more you dig in to what those hidden layers are really doing the less intelligent it seems
Shifting the focus for a moment from how networks learn to how you learn
That'll only happen if you engage actively with the material here somehow
One pretty simple thing that I want you to do is just pause right now and think deeply for a moment about what
Changes you might make to this system
And how it perceives images if you wanted it to better pick up on things like edges and patterns?
But better than that to actually engage with the material
I
Highly recommend the book by Michael Nielsen on deep learning and neural networks
In it you can find the code and the data to download and play with for this exact example
And the book will walk you through step by step what that code is doing
What's awesome is that this book is free and publicly available
So if you do get something out of it consider joining me in making a donation towards Nielsen's efforts
I've also linked a couple other resources that I like a lot in the description including the
phenomenal and beautiful blog post by Chris Ola and the articles in distill
To close things off here for the last few minutes
I want to jump back into a snippet of the interview that I had with Leisha Lee
You might remember her from the last video. She did her PhD work in deep learning and in this little snippet
She talks about two recent papers that really dig into how some of the more modern image recognition networks are actually learning
Just to set up where we were in the conversation the first paper took one of these particularly deep neural networks
That's really good at image recognition and instead of training it on a properly labeled data
Set it shuffled all of the labels around before training
Obviously the testing accuracy here was going to be no better than random since everything's just randomly labeled
But it was still able to achieve the same training accuracy as you would on a properly labeled dataset
Basically the millions of weights for this particular network were enough for it to just memorize the random data
Which kind of raises the question for whether minimizing this cost function actually corresponds to any sort of structure in the image?
Or is it just you know?
memorize the entire
Data set of what the correct classification is and so a couple of you know half a year later at ICML this year
There was not exactly rebuttal paper paper that addressed some asked like hey
Actually these networks are doing something a little bit smarter than that if you look at that accuracy curve
if you were just training on a
Random data set that curve sort of went down very you know very slowly in almost kind of a linear fashion
So you're really struggling to find that local minima of possible
you know the right weights that would get you that accuracy whereas if you're actually training on a structured data set one that has the
Right labels. You know you fiddle around a little bit in the beginning, but then you kind of dropped very fast to get to that
Accuracy level and so in some sense it was easier to find that
Local maxima and so it was also interesting about that is it caught brings into light another paper from actually a couple of years ago
Which has a lot more
simplifications about the network layers
But one of the results was saying how if you look at the optimization landscape the local minima that these networks tend to learn are
Actually of equal quality so in some sense if your data set is structure, and you should be able to find that much more easily
My thanks as always to those of you supporting on patreon
I've said before just what a game-changer patreon is but these videos really would not be possible without you I
Also want to give a special. Thanks to the VC firm amplifi partners in their support of these initial videos in the series
They focus on very early stage machine learning and AI companies
and I feel pretty confident in the
Probabilities that some of you watching this and even more likely some of the people that you know are
right now in the early stages of getting such a company off the ground and
The amplifi folks would love to hear from any such founders
and they even set up an email address just for this video that you can reach out to them through three blue one brown at
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