Pratica di Shadowing: MIT Introduction to Deep Learning | 6.S191 - Impara a parlare inglese con i video

ALEXANDER AMINI - Good morning, everyone.

Or actually, good afternoon, everyone.

My name is Alexander Amini, and thank you all for joining us today in MIT 6S191 Introduction to Deep Learning.

We're super excited to welcome you to this class.

My name is Alexander.

I'll be your instructor along with Ava, who you'll hear from later today as well with the second lecture.

This is a one -week boot camp on everything deep learning.

So you'll go from the very beginnings of neural networks all

the way to some of the most recent LLM advances that we've had, even up to last week, even.

We'll be covering the field.

The pace of this field is really, truly remarkable, right?

And that's one thing

that we'll keep touching back on throughout the course in a lot of detail

that we want to see actually not just what the current state of the art is, but also remember how we got there.

And I think it's really easy for us to become desensitized

actually with a lot of the state of art with AI

because there are so many amazing things happening and we really are on this exponential trend.

It's really hard for us to actually remember where we were just a few years ago.

So I think what better way to do that than to see it.

So exactly one decade ago, we started teaching the class.

We created this class nine years ago.

We started teaching it nine years ago.

One decade ago, in 2015, this was the state of VR in terms of facial image generation.

So generating facial images.

Fast forward a couple years to 2018, you can already see a massive leap in quality

and then fast forward a couple more years and you see

that extend into a new dimension of time into generating videos as well.

In fact, this was a video that we generated for the class in 2020 to actually introduce the class.

If you guys haven't seen this video, you should watch it online.

It's on YouTube.

It went a bit viral a few years ago when we started the class.

Now, in the past few years, we've seen this revolution expand also into natural language as well.

So in 2022, we had ChatGPT3 released and launched to the public, followed shortly by GPT4 in 2023.

Now, the jump between 3 .5

and 4 was oftentimes regarded as this massive leap in AI capabilities when we went from GPT3 .5 to GPT4,

because really everybody felt that the capabilities were vastly improved between these two models

and everybody could feel it even non -technical folks could really

feel the capabilities come through now what you couldn't really see

or what you couldn't feel was what was happening behind all of those giant models right

so gpt4 was this incredible model

but this is what you don't see powering gpt4 are these massive and five of course as well,

are these massive data centers, GPU data centers.

They cost hundreds of millions of dollars to build and tons of infrastructure to maintain and to manage these different data centers.

Now, these are for large language models.

We've also had this revolution of AI computing capability on small form factors as well.

So just like we have large language models powered in the cloud,

we've also started to to see small language models go directly down onto edge devices.

Now traditionally this gap between large language models and small language models has been very large in a capability sense.

We couldn't really accomplish huge jumps in capability in small language world that we've seen in the large language world.

But let me show you actually what's happening right now.

So I'm gonna switch over my screen to this screen

which is actually running here

so this is actually live on this phone running right here you can see everything that's happening on the phone

so what I'll do is I'll show you actually a language model running entirely on this phone right now

so we can ask you questions this should look very familiar to what you have seen with GPT's

so how are you doing we can ask you very simple questions to start now the thing

that you'll notice here

if I swipe down from the top this model is running

entirely on airplane mode there's no Wi -Fi connected to this

phone this is entirely offline everything that's being computed is running

on the device itself now of course we can ask it more complex questions as well

so here's a question about finding the zeros in this equation it looks simple at first glance

but actually if you read it it's actually asking about it in Z7 space.

Does anybody know what Z7 space is?

Anybody?

Yes, exactly, exactly.

So it's in module seven, the finite space of a module seven.

We can ask it this question.

It can immediately know the answer.

You can check the answer.

It is indeed correct.

We can also ask it to maybe like give me more explanation.

Right.

Okay.

So it's able to do this now.

Now, let me go back out and let me actually show you another demonstration.

Now, these models are also permeating beyond text language as well.

We can ask questions directly in audio language.

Now, these are models that are not converting from text to audio, audio to text.

These are pure audio models that take audio in and directly output audio tokens out.

So I can ask you the question like, can you give me a step -by -step guide on how to give a really compelling talk?

Sure.

Start by clarifying your main message and who you're speaking to, then sketch a simple outline with an intro, a few key points, and a strong conclusion.

Next, craft a few opening and closing lines that grab attention and leave a lasting impression.

And practice your delivery so you can adjust your pace in body language.

Finally, rehearse your talk, get feedback, and tweaks any weak spots before the event.

Right.

So what have we seen?

Right.

So this is an end -to -end audio model.

It means that the latency is extremely low is what you'll probably notice there.

Because number one, you don't send anything to the cloud.

So you don't have to wait for internet times.

Everything runs on device.

And you're directly processing audio.

So it's extremely responsive.

These are models that can actually understand your emotion.

Because they don't convert audio to text.

When you convert to text, you lose emotion.

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So you directly capture the emotion in the speech, and you can directly convey the emotion back to the user.

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Now, this was a model actually developed by Liquid AI.

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It released about one week ago.

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It's multimodal.

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It can process all of these different modalities, text, audio, and vision.

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But there have been a ton of these model releases in the past few years, actually, since GPT first launched.

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And this problem that traditionally was having between large language models, the gap between large language models and small language models has always been around, right?

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This was actually a very special release for the community because of this context on the left -hand side.

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So I'll read it out just for everybody to see.

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So it's pretty crazy when you think about it.

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One of the biggest wow moments of GPT was the update from 3 .5 to 4.

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And now this open source model that anybody can download

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and put on their phone with only two billion parameters is

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better than the original GPT -4 model in terms of every benchmark basically that comes out, right?

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And this is a model, again, that you can run entirely on your phone.

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It's open source.

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You can fine -tune it on your own prem, right?

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So that's the pace of the field.

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That's why this class is happening at such an exciting time.

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So let's dive into it, right?

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This is what we'll learn in the class.

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Let's dive in by, first of all, laying some foundation of what exactly is deep learning and to do

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that we need to first understand what is intelligence

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because deep learning at the heart of deep learning is ai right

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so what is ai first of all ai is the practice

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of building artificial algorithms to process information in order to inform

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some future decisions right at its core that's what we call intelligence in fact even

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And artificial intelligence is just the ability for a machine to do that same ability, process information to inform a future decision.

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Now, machine learning is nothing more than a subset of AI

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that specifically does this without explicitly programming a computer on the steps of how to process that information.

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It's able to learn this from data, right?

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That's the differentiating point between AI and machine learning.

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And now deep learning is nothing more than a subset of machine learning, which focuses on the use of neural networks,

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specifically deep neural networks, to do this task of learning from data to inform future decisions and other tasks.

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So we'll circle around this theme of teaching computers to learn a task directly from raw data.

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That really is what this whole class is all about, and we'll learn about different models to do that, different ways, and different algorithms to accomplish that main objective.

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This class is split between both technical lectures as well as hands -on project software labs.

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We'll have several new updates this year, especially towards the second half of the week, where we start to cover a lot more of the new advances in AI

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and deep learning and we'll conclude also with some guest lectures

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and some project presentation competitions that will give everybody the opportunity to win some awesome prizes.

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Do you want to come up with the TED Talk?

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Yes.

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Also, John has graciously offered to host the top winners of the project presentation for TEDxMIT talks.

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So who has not given a TEDxMIT talk?

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Probably most of this audience.

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So this is your opportunity to win a spot on the big stage

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and give a TEDxMIT talk as part of the project pitch competitions.

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the software labs are an excellent way to get hands -on

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and actually learn how to deploy

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and build what we learn about in the technical lectures directly with your own code

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and programming we offer software labs in both tentaflow and pytorch we'll talk more about this later today

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okay so how do the project labs work right

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so every day we have a dedicated software lab that that helps mirror exactly what you learn in the technical lectures.

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Also, you learn this in the project labs as well.

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And every day covers on both sides.

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So starting with lab one today, where you'll be learning how to build a music generation model that can learn how to generate a certain style of music.

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Lab two, which is tomorrow, is going to be focused on computer vision and facial detection systems.

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And then lab three is a lab dedicated on large language modeling.

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You'll fine -tune your own large language model.

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You'll evaluate it.

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You'll see the whole process end -to -end from training all the way to evaluation and deployment.

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And finally, on the last day of this class, we'll cover a final project pitch competition.

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Three to five minutes, very Shark Tank -like style.

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You'll get on stage.

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You can work individually or in groups.

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We'll talk more about the details for this a bit later throughout the class.

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This class has many amazing resources to help throughout the week.

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Please post to Piazza.

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I think everybody here should have received a link for the Piazza, but please reach out to us if you haven't.

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We also have some incredible TAs who you'll meet throughout the week and will help support if there's any questions.

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All of these slides, by the way, are online on the website, so feel free to grab any of these links or details from online as well.

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And, of course, we have an incredible team myself and Ava are your main instructors, but then we also have an incredible team of TAs

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and we also have some incredible guest lectures lined up for Thursday

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and Friday as well that you'll definitely not want to miss.

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And finally, just a huge thanks and call out to our sponsors who without their support, this class is not possible at all.

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Okay, so now let's start with some of the fun stuff and let's start by asking ourselves,

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first of all, a question that probably is pretty self -explanatory at this point, right? which is, why do we actually care about deep learning so much, right?

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Why are you all here?

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Why are you so excited about this topic?

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And the answer really boils down to what I believe is one thing, right?

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Traditional machine learning algorithms really require a level of hand engineering,

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human -guided hand engineering to teach computers how to perform a particular task.

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Deep learning is so exciting because it enables us to learn those rules

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that traditionally are driven by human engineering now by computers and by data, right?

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So the key idea here is to learn those features directly from data and to do that in a hierarchical fashion, right?

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So if we want to learn how to detect faces, for example, right, you may do that as a human, how would you detect faces, right?

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And think from a first principles way, you would probably start by looking at an image

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and first detecting the lines in the image

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and composing those lines together to see

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if there are certain corners in certain parts of the image

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that resemble a face and then putting those corners

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and different configurations together to see if you can configure an eye

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or a nose or a mouth and

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if you detect any of those things then it's even more inkling

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that you probably are looking at a face right deep learning

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algorithms are extremely hierarchical in the same way they build up features from very low level features like lines

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and edges to higher -level shapes like corners and curves, all the way up to mid -level objects and higher -level objects as well.

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Now, applying the fundamental building blocks of how to learn those hierarchical representations is actually something

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that is not new and not very recent at all.

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A lot of these core algorithms have existed since the 1950s, 1970s.

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They really took off a lot.

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So why are we really now seeing this resurgence?

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And there are really three answers to this.

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One is big data.

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The data in the world has never been more prevalent than today.

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And these algorithms live off of data.

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This is what they are feeding off of, right?

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So they really need and require massive amounts of data that the world has now been able to provide.

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The second is hardware.

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Companies like AMD and NVIDIA, of course, as well, are providing new GPUs that are accelerating parallel computing.

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and accelerating the development and the deployment of these algorithms

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and finally software that you'll get hands -on experience throughout this class

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that actually democratizes the ability to create train and deploy any of these models even at relatively large scales

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okay

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so let's start with the fundamental the most fundamental building block of what makes up pretty much every neural network

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and that is the single neuron right also known as the perceptron Right,

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so this is really the most basic building block that's very important to understand when starting deep learning.

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Now, what is the idea of a single perceptron?

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At its core, it's actually extremely simple.

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Let's start by first talking about the forward propagation of information through this neuron.

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So if you have information and you're passing it through a neuron, we can define a set of the inputs to that neuron as X1 to Xn,

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Or here we call it XM, excuse me.

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Okay, so we have M inputs, one to M, and this one neuron takes all of these inputs and it has to create some output.

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This is the goal of this neuron.

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Now, how do we do this?

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How does this neuron actually take those M inputs and compute its output?

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It does this by taking each of those inputs, multiplying every input with a corresponding weight, what's called a weight.

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Every input has a corresponding weight, So x1 corresponds to weight 1, x2 corresponds to weight 2, you multiply every weight with its corresponding input,

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and then you add up all of the answers, and then you pass this through to the output.

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With one more exception, which is that you also pass it through what's called an activation function.

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Here it's denoted as g.

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So you multiply your inputs with your weights, add up everything into one number, and then pass it through some transformation,

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which we'll talk about in a second and that is your final answer.

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That's one number that comes out of this neuron from M inputs that go in.

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Okay.

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There's one more detail I left out of this, which is that we can also have a, what's called a bias term to this neuron as well.

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So in addition to our M inputs on the left, we also have one more weight, W0, which is called our bias.

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Now our bias is what can allow us to shift shift this function up or down, right?

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And I'll show a graphical example of this in a second.

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But the equation is still the same.

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We take every input, multiply it by corresponding weights, add a bias, pass through our non -linearity.

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So now that x and w are actually just lists of numbers, each of which are m -dimensional,

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we can actually write this in vector form, in linear algebra form.

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So the output now, y, is just obtained by taking the dot product between x and w,

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adding this bias, and passing through a non -linearity.

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This simplifies the equation, at least notation -wise, quite a lot from the previous slide.

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Now, you might be wondering about this activation function that I mentioned previously.

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I said this is a non -linear function.

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What does that mean?

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It just means that you're transforming something from the x -axis that can be any real number, anything from negative infinity to infinity,

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into a new real number that sometimes is bounded, sometimes not, but that transformation is not linear.

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So here's one example of a nonlinear activation function.

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This is called the sigmoid activation function.

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It transforms any number on the x -axis into a new number between 0 and 1, right?

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In fact, there are many types of nonlinear activation functions that you'll get exposure with throughout this class.

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The sigmoid activation, which you see on the left -hand side, is just one example of very common activation functions in neural networks.

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The central theme that links all of these together, though, is that they're all nonlinear.

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The activation function with sigmoid on the left is very commonly used for things like probabilities, because it always outputs between 0 and 1, which is the space that probabilities live in.

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On the right -hand side, you have activation functions that output between 0 and positive infinity,

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which is great for introducing non -negativity constraints into your models as well.

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And it's very simple because it's actually just two linear functions with a non -linearity between them.

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Okay, so now why do we actually even need activation functions?

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Why not just directly use the dot product that comes from our weights and our inputs?

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The point of an activation function is precisely to introduce nonlinearities into our model.

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And why do we care about this?

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It's because in real life...

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Real life is highly nonlinear.

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It's highly complex and dynamic.

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And let me show you an example.

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So here's an example of a data set, just a two -dimensional data set, not even extremely complex.

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But if I asked you to draw a line separating the green points from the red points in this data set, it would actually be really hard.

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In fact, there's no line that perfectly separates the green points from the red points.

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And that's because this data is nonlinear, even though it's just in two -dimensional space.

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It's not in high -dimensional space, but it's still nonlinear

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and very complex now imagine you're dealing with real world data that's extremely high dimensional

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and also non -linear definitely you need non -linearities in your model to handle this

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if i tell you you can make a curved line this problem becomes very easy right

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and that's what it means to have non -linear activation functions

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we're trying to learn a mapping a decision to classify between red points

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and green points but we allow our model to draw curves in the edges in this decision space between the two points.

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Let's understand this with maybe a simple example, and we can walk through it together.

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So let's assume now that we have a neural network that was trained with two weights.

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So W1 and W2 is 3 and negative 2, just as an example.

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We trained it.

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This is the result that we got.

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And we want to pass in a new input to this model.

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So we have two inputs to this model, X1 and X2 we obtain the output of this neuron by,

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as we saw before, computing a dot product between our X and our W.

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Our W is already trained, so we can plug in those two weights directly into this vector here, adding our bias, and then passing through non -linearity.

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Now, if you look at this, what's inside of the activation function, this is nothing more than a two -dimensional line as a function of x1 and x2 are two inputs.

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We can plot this line, right?

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This is a trained neural network, or it's a trained neuron at least.

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We can plot this line and observe what this decision boundary looks like in the space between x1 and x2.

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So for any new point, x1 and x2, if I want to pass in an input to this neuron, I can actually plot this point.

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So let's say I pass in a new input, negative 1 positive 2 is the xy coordinate of this point.

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I can plot it in this space and I can see

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that it actually lies on the left side of this decision boundary.

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What does it mean to lie on the left side of the decision boundary?

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Well, let's actually compute it mathematically first and then we'll see.

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So if we plug in those numbers to the equation, we'll get 1 plus 3 times negative one

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which is negative three minus two times two

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which is negative four right add up all those together you get negative six right that's a negative number

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that means that we're on the left side of the decision boundary

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when we pass that through our non -linearity

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which is our sigmoid function being a very negative number

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that means that we're going to be further on the left on the

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under 50 under 0 .5 on the sigmoid function so where it's a very small number close to zero 0 .00.

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In fact, we can actually draw a hyperplane between these two parts of the space

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and say anything that lies on the left side of the decision boundary will always be negative, right?

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It will always be negative before the activation function.

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It will always be less than 0 .5 after the activation function, if it's sigmoid activation.

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And anything on the right is going to be positive before the activation function.

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Now, this is really nice and convenient

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that we can visualize this trained neuron in this way here

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that for any input we can directly visualize exactly where it lives in this decision space

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but of course this is only a two -dimensional space it

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has only two neurons in our neuron excuse me two parameters

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in our neuron in our neuron right it's extremely small right modern neural networks have billions of parameters

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so imagine this plot now with a billion dimensional space this is what we would be looking at

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so of course that's not possible right

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so this is why this is a helpful exercise to explain

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but of course we need to actually scale without having this exact visual

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but building up more mathematical intuition so let's do exactly

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that let's scale this idea now into larger networks

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and let's start by taking that one neuron

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and actually building a neural network out of it to seeing how this all comes together, right?

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So this is our original neuron picture, okay?

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Now, if there's a few things that you remember from this class, this is probably the most important thing, is that you should remember how a single neuron works,

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and that's by doing a dot product, adding a bias, and applying a non -linearity.

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It's really three steps, dot product, add a bias, and apply a non -linearity.

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That equation is the core to so many different parts of deep learning.

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Now, let's simplify the diagram a little bit now that we got that foundation settled, right?

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I'll remove all of the weight labels and we'll just have every line, you can remember

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that every line will have both an input coming in as well as a weight associated to that line, okay?

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Now, z here, what's shown as z, is the result of that dot product plus a bias, right?

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This is what's happening before the non -linearity, right?

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So the final output is simply Y of G, which is our non -linearity, of Z.

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OK, now let's define a multi -output neural network now.

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To do this, we just have two neurons instead of one.

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They will both see the same inputs, but now they will have their own independent weights.

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So each neuron has its own three weights.

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It sees the same three inputs, but it now can output two different answers because it has two different sets of weights.

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Okay, so now with this mathematical understanding, let's see if we can now build our first layer, neural network layer, entirely from scratch.

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So let's do this by first initializing a matrix called W.

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This is going to be a matrix that contains all of our weights for all of our neurons in that one layer.

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Okay, so we can create now this matrix of weights

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which is going to be dimensionality of the input space by

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the dimensionality of our output space right now the bias is also important to initialize here and then

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when we create our call function our forward pass function this

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is going to show actually how we can multiply do the

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dot product of our inputs with our weights okay right here add the bias

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apply the non -linearity and that's our answer right it's the same equation again

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okay we can do this in tensorflow we can also do it in pytorch

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and you'll notice actually between these two it's very very similar

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type of language a lot of parallels basically just different names for a lot of things

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but the overall frameworks are extremely similar

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same foundations we'll initialize the weights

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and the biases in the forward pass we'll compute the output

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of this layer by computing a matrix multiplication of our inputs with our weights adding a bias

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and then applying our non -linearity okay now luckily both tensorflow

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and pytorch have actually defined that exact layer for us already

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so you don't actually need to to write

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that code yourself you would just call the layer in tensorflow

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it's called the dense layer in pytorch it's called a linear

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layer they do the exact same thing they do the matrix multiply add a bias apply a non -linearity

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so we can just call it instead right we call it with the number of units

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or the number of neurons in each layer okay let's keep

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building up this abstraction step by step that's a single layer neural network.

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This is one where we can have a single hidden layer now.

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So this is actually two layers.

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Now we can see how we can do this transformation from input space to the hidden layer space with one layer, and then hidden layer space to output space with another layer.

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So now we actually have two weight matrices, right?

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The first weight matrix on the left -hand side, W1, converts the inputs to the hidden layer.

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Second weight matrix is W2, which goes to the output, Each of these are separate weight matrices, and they actually have different sizes as well,

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because your output shape is different than your hidden layer shape.

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Now, if we look at a single unit, let's zoom back in now to a single unit within the hidden layer.

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Let's take, for example, this one, Z2.

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This is just the second unit in our hidden layer of this model.

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This is the same perceptron that we saw before, nothing special here.

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We compute its output, Z2, by taking a dot product of all inputs with the weights corresponding to that neuron,

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adding its bias, and applying a non -linearity.

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If we took a different node, like Z3, for example, it would look also the exact same, except its weights would be different than Z2's weights.

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The inputs are the same, but they see different weights.

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Okay, so this picture looks a bit messy, so let's abstract even further so we can keep building up this intuition.

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So now I'm going to remove all the lines

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and just put these symbols here in cases wherever we have these fully connected layers.

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Everything that connects from input to output with the matrix multiply in between.

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And again, we can actually stack these fully connected

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or linear layers on top of each other with nonlinearities between them to make sure that we have the nonlinearities.

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across our network as well

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and here's an example of how to do this you basically stack the same layers

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that we saw before now in these sequential blocks sequential just

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basically allows you to stack them okay now finally the the point

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that we've been waiting for is how to go from these

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neural networks to deep neural networks there's nothing more to this

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than just stacking a bunch of linear layers on top of

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each other a deep neural network is nothing more than a neural network

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that has more than usually three layers is what people can conceptually say, right?

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So this is a network where the final output is computed by going deeper

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and deeper and deeper across each of these layers to compute its final output at the end of the network.

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And again, to no surprise, this is done in the same abstraction code as before, just having a sequential block and a bunch of linear layers within that block.

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Okay, this is awesome.

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So now we have this good intuition of how to build

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up from a single neuron all the way to a layer to a neural network

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and to compose these in the forward pass.

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Let's take a quick look of how we can apply this to a particular problem

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that it's it's a good starting problem for everybody in this class

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which is should will I pass this class right

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so this is a question it has a yes no answer

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and there's a lot of data for this question actually

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because we've taught this class for a long time

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so we have a lot of data from past students who have taken this class here's an example

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so we have an x and a y -axis

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x is the number of lectures that you attend

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and the y -axis is the number of hours

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that you spend on the final project and we also have individual data points

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which represent past students who have taken this class you can see where they live in this space

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and then we have you you fall right here at four or five, right?

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You've attended four lectures and you spent five hours on your final project.

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And the question is, will you pass this class based off of all of the other data that you've seen, right?

420

Now, how can we do this?

421

We have a neural network that can do this.

422

Let's try and actually feed it into our neural network.

423

We have two inputs, which are the exact two axes that you saw before, number of classes, number of lectures, the number of hours one input is four the other input

424

is five we can feed these two inputs into our neural network

425

and we can see that it predicts an answer of 0 .1

426

or 10 probability

427

that you'll pass this class very bad i know okay who

428

knows why the the network is wrong in this case yes exactly okay Okay, so this is a randomly initialized network.

429

It is effectively the same as a brand new baby, right?

430

It has never seen the world before.

431

It has never seen any of the data before.

432

So the step that's missing here is that we've just done a forward pass through the model.

433

Now we have to actually teach the model, right?

434

And the key part of this is that every time that we have a prediction, the model predicts point one, there's also a ground truth answer for every prediction that we have in our training set.

435

And the ground truth answer here is true, right?

436

You did actually pass this class, right?

437

So there's a deviation here.

438

There's a difference between the 0 .1 that the model predicted and the 1 that it should be answering.

439

Now, to train this network, we need to show it this difference.

440

We need to tell it when it gets the answer wrong

441

so it can learn to move closer towards the correct answer the next time it sees a 4 .5, right?

442

If it sees 4 .5 again, it should now improve on its answer and should predict something closer to a 1.

443

We do this by quantifying what's called a loss.

444

A loss is nothing more than just the error, this deviation term between the prediction and the ground truth.

445

Smaller losses means that the model is outputting a higher quality answer.

446

Larger losses means that the answer is more erroneous.

447

So let's assume, of course, that the data is not just from one student, but we have a data set of many, many students right?

448

This is coming from now a loss over the average of each of those losses

449

that we have in our data set, right?

450

So when training a neural network, oftentimes we're not going to actually minimize the loss on a single data point, but minimize it across the entire data set, right?

451

And that's to have higher quality outputs.

452

Now, if we look at this problem in the case of classification

453

a yes no type of problem then this is where the

454

answers will typically predict a probability a probability of being a yes versus probability of being a no

455

and that's why we predict between zero

456

and one this ties us back to the sigmoid function from before right

457

because those outputs are zero and one outputs

458

so we can train it in this way right

459

and we can use losses like the cross entropy loss that actually support matching those two distributions of probability against each other.

460

Now, you might be asking yourselves, okay, what if you didn't want to predict a probability distribution, like a yes -no question, but rather a continuous value, like what's the temperature going to be?

461

Or what score will I actually get on the class rather than will I just pass or fail it?

462

In those cases, you care about predicting continuous values instead, not probability distributions.

463

So you actually want to output and have losses that encourage deviations that are also continuous.

464

In those cases, you may consider things like mean squared errors or L1 errors,

465

things like this, that simply compare the predicted value minus the ground truth value and just minimizing that absolute difference.

466

Okay, now let's put all of this loss information together into the problem of actually finding the neural network weights

467

that are optimally suited for any particular data set or any particular task.

468

So we know that we want to find a neural network that looks roughly like this form, right?

469

We want it to minimize the loss on our entire training dataset.

470

And this means that we want to find the Ws, the weights, that minimize the loss here,

471

the J, the J of W, right?

472

J of W is our empirical average loss across our entire dataset of N data points.

473

okay so remember again

474

that w is nothing more than just a collection of many

475

many weights across all of the different layers in your neural

476

network everything from layer zero the first layer all the way to layer n

477

which is your last layer now our loss function is also nothing more than just a function

478

that outputs a single scaler at the end right we take

479

the average loss the average error across every data point compute its average

480

and that is our loss it's just one number at the end of the day

481

and that one number is a function of our weights

482

so on on here you can see a visualization of let's

483

say again a two parameter neural network just two parameters w0

484

and w1 for any configuration of those two parameters we have a particular value of the loss which is the height, okay?

485

There are some weights that encourage a really good loss, very low.

486

There are other weights that are really bad losses, right?

487

Very high on the curve here, right?

488

And the goal is that we want to find the neural network that has the two weights, W0 and W1, that have the lowest loss.

489

Now, how do we do that?

490

So, this is called training the neural network.

491

We start by training the neural network.

492

We just randomly initialize it.

493

We pick a W0 and a W1 to start with.

494

Let's say we start here at this black point.

495

We compute its loss.

496

And then we compute the gradient.

497

The gradient tells us the direction of the slope at that location, right?

498

It doesn't tell us the slope everywhere.

499

It just tells us at this very local location, which way is pointing up.

500

But of course we want to go down.

501

So we actually take the negative of the gradient and take a small step down.

502

And then we repeat this process.

503

We compute the new gradient at the new point, and we take a negative step in the opposite direction of

504

which way the gradient is pointing until we finally get to the bottom.

505

And this will actually be guaranteed to converge to a bottom.

506

May not be the lowest bottom, right?

507

Because it actually depends on where you start.

508

You may not find yourself in the absolute lowest, but you will find a bottom and you'll eventually converge there.

509

So we can summarize this algorithm.

510

This is known as gradient descent

511

and we can write it roughly in pseudocode it looks like this we start by initializing our weights randomly

512

and then we repeat the following process until convergence we compute the gradient at the weights

513

that we're currently in

514

and then we take a small negative step of our gradient with our weights

515

so we change the gradients to step in the opposite way

516

that the gradient tells us

517

and then we keep repeating this until there's convergence until our weights are not really changing anymore

518

and then finally we call those final weights the trained network

519

of the model um in code right this also is is uh is doable in code we can do this

520

while looping in a

521

while loop where we compute the loss we can compute the gradient of our loss with respect to our weights

522

and then compute that same update formula

523

again now a very important line here is to look at this term right here the gradient right now i

524

glossed over this this point

525

but actually the gradient this is a key part of this algorithm

526

because computing this tells us exactly how to update our weights on every part of the iteration

527

but I never actually told you how to compute this so let's talk about actually computing the gradient

528

for a neural network

529

and this is a process known as back propagation we already

530

learned about forward propagation of information through the model now we'll learn about back propagation

531

which is how to actually compute the gradient of the model

532

we'll start with a very simple neural network to illustrate this example

533

and this is actually probably the simplest neural network

534

that you can have which is just a single neuron with a single output as well

535

so x goes to a single neuron goes to a single output goes to a loss.

536

So let's visualize what this would look like

537

if we wanted to compute the gradient of our loss with respect to our weight w2,

538

which tells us how a small change in w2 affects our loss j.

539

Okay, so let's write it out as a derivative.

540

So this is the derivative of j with respect to w2, right?

541

Now to compute this, we can use the chain rule, We can decompose dj, dw2 into two parts,

542

the first part being dj, dy, times dy, dw2.

543

This is just an application of the chain rule in linear algebra.

544

Now, this is only possible because y is only dependent on the previous layer.

545

Let's suppose now we wanted to compute the gradients of the weight before

546

that, which is w1 not w2

547

so let's actually change this to dj of d w1 now

548

here this is dy dw1 we all we again need to apply a chain rule again to compute this right

549

so we expand it out one more time

550

that part turns into dy dz1 followed by dz1 dw1 right

551

and if you had a deeper neural network you would just keep repeating this process as you go deeper

552

and deeper into the model and

553

and it just repeats like this right you can repeat this process by propagating those gradients

554

that you compute from the end from the output all the

555

way back in in the in the topology of the neural network all the way back to the input

556

and this will allow you to determine how every single weight

557

impacts your final loss that's exactly what the gradient shows you it shows you

558

if you move in some direction how will the loss change?

559

Will it go up or down?

560

Now, that's the backprop algorithm.

561

In theory, it's just an application of the chain rule, right?

562

It's nothing more than the chain rule. In practice, this is a lot more complex

563

because computing the gradient of neural networks does not look as clean as what we saw in that previous slide, right?

564

In practice, neural network loss topologies look highly non -convex, right?

565

There are a lot of different minimum, and it's highly dependent on the initializations that you pick in these models

566

and the regularizations that you apply to make sure that you can find actually this minimum.

567

So this was a really cool paper from many years ago, 2017, where they actually tried to visualize what these high dimensional neural network loss landscapes actually look like.

568

Of course, this is a two dimensional representation of something that is, you know, millions of dimensions.

569

So you need to take it with a grain of salt.

570

But even projecting it down into two dimensions, you can see that this is a highly nonlinear landscape.

571

Now, recall the update equation that we defined during gradient descent.

572

And remember this parameter that we also didn't talk so much about, right?

573

I said we take a small step in the opposite direction of the gradient.

574

What I didn't mention is, you know, how big is this step?

575

This step here is referred to as eta, right, this small symbol here.

576

This is called the learning rate this is the rate at

577

which you follow the gradient in practice setting this learning rate it's just a number

578

that you set in practice setting this number is also quite difficult right

579

because if you set something

580

that is too small then you actually never really reach some

581

of the the big minimums right this is a minimum

582

but the model gets stuck in it even though it's not the biggest one right you would actually

583

and if you set something too large you can overshoot

584

and your loss can also explode and you never find the minimum, either minimum.

585

So ideally you want to use learning rates

586

that are basically just large enough that you can skip over these kind of fake local minimums,

587

but get stuck in these global or close to global minimums to converge there.

588

So how do we deal with this?

589

How do we actually set the learning rates?

590

One option is just to try a bunch of different learning rates and see what works best.

591

And actually this is not as crazy of an option as it sounds like

592

but can you do something smarter than this how about we design adaptive learning rates

593

that adapt to the shape of our lost landscape

594

that actually is a much smarter idea because it means

595

that we can now increase or decrease our learning rate depending on how large the gradient is at that location.

596

When you have a very large gradient, maybe you wanna take a bit of a smaller step.

597

If you have a small gradient, maybe you want to adapt or have some momentum that can carry you through these local minimums and pass over them.

598

And that's exactly what a lot of different optimizers have come out with.

599

So we saw the most basic version of stochastic gradient descent previously, but there are a bunch of other variations of gradient descent

600

algorithms called atom at a delta at a grad rms prop etc

601

that you'll get a lot of practice with throughout your software labs

602

and in general these are actually much better types of optimizers for the reasons especially around setting learning rates

603

and making sure that you can actually find these minimum

604

and these are widely studied in practice right in general you will very rarely use the vanilla stochastic gradient descent,

605

SGD algorithm, you'd probably almost always use

606

and change this line to something like Atom or another adaptive learning rate scheduler or optimizer.

607

Awesome.

608

Okay.

609

I'll continue now in just talking about some more practical considerations, especially now towards the end of the technical part of the lecture, this first lecture.

610

I want talk about some of the practical considerations

611

that you should have when actually training neural networks

612

so let's go back to the gradient descent algorithm for a

613

second we just saw this this algorithm here we saw how we can compute the gradient

614

but remember this is done through back propagation this is a really expensive process

615

because you have to go entirely back

616

and compute derivatives for every single parameter in your network going

617

from the output all the way back to the input

618

and you have to do this for every single data point in your data set right not just one data point, of course.

619

Now, in most real -life problems, it's not feasible to do this on every training iteration.

620

You simply would not want to do that over every data point in your data set.

621

So instead, we're going to define what's called stochastic gradient descent, SGD.

622

So instead of computing the gradient over your entire data set, let's compute the gradient over just one point.

623

Pick a random point

624

and compute the gradient with respect to that one point and step in the opposite direction of that one point.

625

Now, that is much, much faster to compute, obviously, on big data sets.

626

Computing a gradient on one point is going to be way more efficient than the entire data set, but it's also going to be way more noisy.

627

So there's a trade -off there, of course.

628

Now, the middle ground, what is the middle ground?

629

Is that you do this over what's called a mini -batch.

630

So you compute some batch size of data points, and you compute the gradient with respect to

631

that batch size batch sizes can range from anything very small

632

to like a few data points all the way up to you know

633

like most commonly people use things like 32 is a very

634

common batch size you can scale it up in large language

635

models we have batch sizes of you know millions right it depends on the type of data sets

636

that you're using

637

but in general you always want to pick something much smaller than the size of your entire data set

638

because you don't want to be going through your entire data set every time.

639

Now, B is normally not that large for most problems.

640

Like I said, this is usually on the order of tens, hundreds, and even this gives you a pretty good estimate of the gradient for most problems.

641

This increases the accuracy that's going to be much more accurate than SGD, stochastic gradient descent, that's only looking at one point,

642

but it will also be much more efficient than computing it on the whole data set.

643

So it's this nice balance, and you can actually control the balance yourself by controlling the size of the batch.

644

Now, the last topic I want to touch through is overfitting, right?

645

This is known, a very well -known problem in the field of machine learning, but it extends also, obviously, into deep learning as well.

646

Now, what is overfitting?

647

Overfitting is nothing more than the process of learning too much into your data, right?

648

Looking too much into it and overfitting on the details of

649

that data set that do not generalize outside of the particular training data that you've seen.

650

So here's a good example, right?

651

If you start on the right -hand side and here's your training data set,

652

the optimal line that fits this data is not going to follow all of the intricate details of the data.

653

But on the same side, on the other, excuse me, on the other side, there's also underfitting as well, where you don't learn a function that's expressive enough

654

that captures all of the intricacies of the data

655

so for example a linear function on the left side is going to be underfitting on this problem

656

because the problem is a non -linear type of relationship

657

but on the right hand side it's also too high the

658

the learned solution on the right hand side is too high dimensional

659

and too overfitting to to capture the generalization of the of

660

the problem the ideal is somewhere in the middle right where you don't have too much complexity

661

and you don't read into too much of the details so that when you see brand new data

662

a test set or in deployment you can actually extend your solution to have accurate predictions in

663

that regime as well so to address this problem how do you go

664

and toggle between these two sides of the coin it's called regularization right regularization is the technique

665

that allows us to constrain how much we want to underfit versus overfit on a particular problem

666

And this is effectively allowing us to discourage the learning of very complex models.

667

This is how we're able to learn billions of parameter models on actually data sets,

668

but not actually memorizing all of the details in the data sets or allowing us to generalize beyond those data sets.

669

And this will allow us to improve the ability of our model to perform even on unseen data.

670

The most popular form of regularization, there are many different ways to regularize depending on the type of model, but I'll talk about one of the most popular ways.

671

This is the idea of dropout.

672

Dropout is a stochastic method.

673

It means that it's probabilistic.

674

Let's revisit this chart, this schematic that we saw earlier in the lecture from a neural network.

675

In dropout, all we do is

676

that we randomly drop out some activations of the hidden layers with some probability that we define.

677

Okay, so let's assume that we set a probability like 0 .5, 50%.

678

All this will mean is

679

that we will randomly pick 50 % of the neurons in our two hidden layers

680

and randomly kill the outputs from coming out of those neurons.

681

So what does that mean?

682

That this neuron here has to be resilient to sometimes receiving inputs from this neuron, but sometimes not receiving it from this neuron, right?

683

And then on other iterations, it may receive it from the other set of neurons.

684

So what this encourages is actually the ability that on every iteration, every neuron is seeing different pathways through the network.

685

This is forcing it to not rely on any one pathway

686

and not memorize any one pathway too much through its forward propagation.

687

And that's actually able to allow it to generalize as well

688

because it has to find these relationships that span across different pathways.

689

Yes?

690

Are the same activations set to zero for the entire mini -batch?

691

Same activations are set to zero for the entire mini -batch, but then on the next mini -batch, you have a new set of activations, a new set of neurons that are set to zero.

692

So you reset on every batch.

693

Awesome.

694

Okay.

695

So, you repeat that in every iteration.

696

Let me show you one more technique for regularization as well that goes beyond dropout.

697

Dropout is an architectural level of regularization.

698

This is going to be beyond architectural level.

699

So, let's assume for any architecture, you can have the ability to stop training once you start to overfit.

700

It sounds easy.

701

Let's see how you might actually do this, right?

702

This is called early stopping.

703

So we already have this definition of overfitting,

704

which is simply that we start to perform better on our training set than we are on our testing set, right?

705

That's effectively what it means to overfit.

706

So we're no longer generalizing our learnings from training into testing, right?

707

Our training is continuing to improve, but our testing data is just getting worse, right?

708

So we can do this by actually having two separate data sets, one that we train on and one that we test on.

709

And constantly throughout training, we're evaluating both data sets.

710

Now, in the beginning, both lines just decreased together, which is excellent.

711

It means our model is learning.

712

It's moving from that underfit regime to a more fit regime.

713

And it continues to have this trend.

714

Eventually, though, the network loss will start to plateau

715

and the testing loss will start to increase right this is actually the regime where we start to see overfitting

716

and this pattern typically will continue for the rest of training

717

now the question here is where would you actually you train

718

the model for all of these training iterations you save the

719

model at these checkpoints across the x -axis you can now

720

take this checkpoint right you've saved all the checkpoints you look at the curve

721

and you actually say okay this is the checkpoint

722

that I want to use

723

because after this checkpoint yes my training loss continues to get better

724

but it's not actually translating to my testing data set

725

so this is the one that i actually should be using

726

and we can see

727

that you know after the early stopping checkpoint things do get worse on your testing data

728

so you wouldn't actually want to apply those checkpoints this is such a very powerful idea

729

because it's very general this approach you can really apply this to any type of model

730

and you can really track these these curves right it doesn't

731

require architectural modifications uh you just monitor the loss of your

732

neural network over time on two different data sets one that you train on one that you test done.

733

Awesome.

734

Okay.

735

I'll conclude the first lecture now, just summarize quickly what we've covered.

736

So we started with the most fundamental building blocks of neural networks, which was just a single neuron.

737

We've scaled that up to single layers and multi -layer perceptrons, multi -layered neural networks.

738

And we've learned how we could build complex hierarchical learning machines that go from input to output across a hierarchy of abstractions.

739

And finally, we addressed a lot of the practical sides of actually training these models, evaluating them, picking the best model and so on.

740

In the next lecture, we're going to be hearing from Ava on covering sequence models, right?

741

This is a really exciting lecture, especially in today's world, because sequence models power GPTs, right?

742

Everything, a lot of things are sequences in today's world, right?

743

You think of text as a sequence of words, audio is a sequence of waveforms, video is a sequence of images, right?

744

A lot of things are sequences, and what we saw today is only covering non -sequential data so far in my lecture.

745

In the next lecture, we'll see how we can extend that into sequences of data.

746

So we'll take a five -minute break, and then we'll just set up, and then we'll continue from there.

747

Thank you.

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