The Math Behind Generative Adversarial Networks Clearly Explained!

Summary

This educational video demystifies Generative Adversarial Networks (GANs) by explaining their core components: the generative model 'G' and the discriminative model 'D'. It outlines GANs as a competitive process where 'G' creates fake data and 'D' tries to distinguish between real and fake data. The video simplifies complex concepts like the minimax game, value function, and the convergence of probability distributions, using analogies and step-by-step explanations. It aims to make viewers comfortable with the foundational math behind GANs, one of AI's most sophisticated inventions.

Takeaways

• 😎 GANs (Generative Adversarial Networks) are composed of two models: a generative model 'G' and a discriminative model 'D'.
• 🧠 Discriminative models predict the target variable given the input, like logistic regression, while generative models learn the joint probability distribution of input and output, like Naive Bayes.
• 🎭 In GANs, the generator 'G' produces fake data points, and the discriminator 'D' determines if a data point is real or generated.
• 🤼‍♂️ G and D compete in an adversarial setup, improving each other's performance over time.
• 🧮 The structure of GAN involves multi-layered neural networks with weights (theta G and theta D), chosen for their ability to approximate any function.
• 📊 The generator takes random noise as input and produces 'G of Z', attempting to mimic the original data distribution.
• 🔢 The discriminator acts as a binary classifier, outputting the probability of the input being real data, trained with labels 1 for real and 0 for generated data.
• 🎲 GANs operate on a minimax game theory, where G aims to minimize and D aims to maximize a value function, similar to the binary cross-entropy function.
• 📉 The value function for GANs is optimized using stochastic gradient descent, with the discriminator updated multiple times for every update of the generator.
• 🏁 The training goal of GANs is for the generator to replicate the original data distribution so well that the discriminator cannot distinguish between real and fake data.

Q & A

What are Generative Adversarial Networks (GANs)?

GANs are a class of artificial intelligence algorithms used in unsupervised machine learning, consisting of two neural networks, a generative model G and a discriminative model D, that are trained together in an adversarial setup. G generates new data points (fake data), and D evaluates them to determine if they are real or fake.

What is the difference between generative and discriminative models?

Generative models learn the joint probability distribution of input and output variables and can create new instances of data by learning the data distribution. Discriminative models learn the conditional probability of the target variable given the input variable and are used for prediction tasks, like logistic regression and linear regression.

How does the generator in a GAN work?

The generator in a GAN takes a random noise vector as input and transforms it into a data point that resembles the original data distribution. It does this by learning the underlying data distribution through training.

What role does the discriminator play in a GAN?

The discriminator in a GAN acts as a binary classifier that determines whether a given data point is real (from the original dataset) or fake (generated by the generator). It competes with the generator, improving its ability to distinguish between real and fake data.

What is the minimax game in the context of GANs?

The minimax game refers to the adversarial competition between the generator and discriminator in a GAN. The generator tries to minimize the loss function to produce more convincing fake data, while the discriminator tries to maximize the loss function to better distinguish between real and fake data.

How is the value function of a GAN defined?

The value function of a GAN is defined as the expectation of the log probability of the discriminator correctly classifying real data as real and fake data as fake. It is a mathematical expression that both the generator and discriminator aim to optimize in opposite directions.

What is the goal of the training process in a GAN?

The goal of the training process in a GAN is for the generator to produce data that is indistinguishable from the real data, and for the discriminator to be unable to accurately classify the generated data as fake. This is achieved when the generator's output distribution converges to the real data distribution.

Why is the training of GANs considered difficult?

Training GANs is difficult because it involves finding a balance between the generator's ability to produce realistic data and the discriminator's ability to distinguish between real and fake data. The training process can be unstable, and the models may not converge properly without careful tuning of hyperparameters and training techniques.

What is the role of the noise vector in GANs?

The noise vector, often sampled from a simple distribution like a Gaussian, serves as the input to the generator. It is a source of randomness that allows the generator to produce a diverse range of outputs, contributing to the variety of the generated data.

How does the concept of Jensen-Shannon divergence relate to GANs?

Jensen-Shannon divergence is a measure of the difference between two probability distributions. In the context of GANs, it is used to show that as the generator optimizes the value function, the distribution of generated data (PG) converges to the distribution of the real data (Pdata), minimizing the divergence between them.

Outlines

00:00

🧠 Introduction to Generative Adversarial Networks (GANs)

The video begins with an introduction to GANs, emphasizing that they are not a single model but a combination of two models: a generative model (G) and a discriminative model (D). The presenter clarifies the difference between generative and discriminative models in machine learning, explaining that generative models learn the joint probability distribution of input and output variables, while discriminative models learn the conditional probability of the target variable given the input. The video then delves into the adversarial nature of GANs, where the generator creates fake data points, and the discriminator determines whether data points are original or generated. The presenter uses the universal approximation theorem to justify the use of neural networks in GANs, highlighting their ability to approximate any function.

05:02

🎲 The Minimax Game and Value Function of GANs

The second paragraph explains the concept of a minimax game, which is a two-player game where one player tries to maximize their chances of winning while the other tries to minimize the first player's chances. This is analogous to the adversarial setup in GANs, where the generator (G) aims to minimize a certain expression, and the discriminator (D) aims to maximize it. The value function for GANs is introduced, which is similar to the binary cross-entropy function used in binary classification tasks. The presenter then demonstrates how the value function relates to the binary cross-entropy function and how it is used in the training process of GANs. The concept of expectations in probability theory is also introduced to explain how the value function is applied to the entire training dataset.

10:04

🔍 Proving Convergence of GANs to the Original Data Distribution

In this segment, the presenter discusses the theoretical guarantee of GANs' ability to replicate the original data distribution. The goal is to show that the generator's output distribution (PG) will converge to the original data distribution (Pdata) if the generator can find the global minimum of the value function. The presenter uses calculus to find the conditions under which the discriminator (D) maximizes the value function and then substitutes these conditions back into the value function to analyze the generator's objective. The concept of Jensen-Shannon divergence is introduced as a measure of the difference between two probability distributions, and it is shown that the generator's objective is to minimize this divergence, which leads to the conclusion that PG will equal Pdata at the global minimum of the value function.

15:05

📈 Training Phases and Challenges in GANs

The final paragraph describes the different phases of training in GANs and the challenges involved. Initially, neither the generator nor the discriminator performs well, as the generator's output distribution (PG) does not replicate the original data distribution (Pdata), and the discriminator cannot accurately classify data points. As training progresses, the discriminator improves and can distinguish between real and fake data, while the generator's output distribution becomes closer to the original data distribution. Eventually, if training is successful, the generator's output distribution becomes indistinguishable from the original data distribution, and the discriminator cannot tell the difference, outputting 0.5 for every input. The presenter concludes by acknowledging the complexity of training GANs in practice and encourages viewers to appreciate the mathematical concepts behind this advanced AI technology.

Keywords

💡Generative Adversarial Networks (GANs)

GANs refer to a class of machine learning models that consist of two neural networks, a generator and a discriminator, which are trained together in an adversarial fashion. The generator creates new data points, while the discriminator evaluates them to determine if they are real or fake. In the context of the video, GANs are used to produce new data points that closely resemble the original data distribution, with the ultimate goal of the generator being to 'fool' the discriminator into thinking the generated data is real.

💡Generative Model

A generative model is a type of machine learning model that learns the joint probability distribution of the input and output variables. It can be used to generate new data instances by sampling from the learned distribution. In the video, the generator in a GAN is a generative model that produces fake data points, aiming to replicate the distribution of the original data.

💡Discriminative Model

Discriminative models, in contrast to generative models, learn the conditional probability of the target variable given the input variable. They are typically used for classification tasks. In the video, the discriminator in a GAN is a discriminative model that tries to distinguish between real and generated data points.

💡Neural Networks

Neural networks are a set of algorithms modeled loosely after the human brain that are designed to recognize patterns. They interpret sensory data through a kind of machine perception, labeling, or clustering raw input. In the video, both the generator and discriminator are implemented as neural networks due to their ability to approximate any function, as per the universal approximation theorem.

💡Adversarial Setup

In the context of GANs, an adversarial setup refers to the competitive training process where the generator and discriminator are pitted against each other. The generator aims to create increasingly realistic data, while the discriminator aims to improve its ability to identify real from fake data. This competition leads to the improvement of both models.

💡Minimax Game

A minimax game is a two-player game where one player aims to maximize their chances of winning while the other tries to minimize the first player's chances. In the video, the training of GANs is likened to a minimax game, where the generator tries to minimize the discriminator's ability to correctly classify data, and the discriminator tries to maximize its accuracy.

💡Value Function

In the context of GANs, the value function is a mathematical expression that represents the objective of the minimax game. It is what the generator and discriminator aim to minimize or maximize, respectively. The video explains that the value function for GANs is similar to the binary cross-entropy function, which is used to measure the performance of the discriminator.

💡Stochastic Gradient Descent

Stochastic Gradient Descent (SGD) is an optimization algorithm used to update the parameters of a model in a way that minimizes a loss function. In the video, SGD is used to optimize the value function for both the generator and discriminator during the training of a GAN.

💡Jensen-Shannon Divergence

Jensen-Shannon Divergence is a method for measuring the difference between two probability distributions. It is symmetric and always finite, unlike the more commonly used Kullback-Leibler divergence. In the video, it is mentioned as a way to understand the optimization goal of a GAN, where the generator aims to minimize the divergence between the distribution of generated data and the original data.

💡Global Minimum

The global minimum refers to the lowest possible value of a function, which in the context of GANs, the generator aims to achieve for the value function. Achieving the global minimum means that the generator has successfully learned to produce data that is indistinguishable from the original data distribution. The video discusses the theoretical underpinnings of why and how this can be achieved.

Highlights

Generative Adversarial Networks (GANs) are not a single model but a combination of two models: a generative model (G) and a discriminative model (D).

Discriminative models learn the conditional probability of the target variable given the input variable, like logistic regression.

Generative models learn the joint probability distribution of the input and output variables, allowing for new data instance generation.

In GANs, the generator (G) produces new data points, while the discriminator (D) determines if data points are original or generated.

GANs operate in an adversarial setup where the generator and discriminator compete, improving each other's performance.

The structure of GANs involves multi-layered neural networks for G and D, with weights represented by theta G and theta D.

The generator takes random noise as input and produces data points that mimic the original data distribution.

The discriminator acts as a binary classifier, determining the likelihood of the input data being from the original dataset.

GAN training involves a minimax game where the generator tries to minimize the loss function while the discriminator tries to maximize it.

The value function for GANs is derived, which is similar to the binary cross-entropy function but with expectations over the entire dataset.

The training loop for GANs includes an outer loop for the entire training process and an inner loop for discriminator updates.

For every K updates of the discriminator, the generator is updated once, following the principle of alternating training.

The theoretical guarantee of GANs is discussed, aiming to prove that the generator's distribution will converge to the original data distribution.

Jensen-Shannon divergence is introduced as a method to measure the difference between two probability distributions.

The global minimum of the GAN value function is analyzed, showing that the generator's distribution matches the original data's distribution at this point.

Different phases of GAN training are described, from initial setup to the point where the generator successfully replicates the data distribution.

The video concludes with a simplified view of GANs, emphasizing the difficulty of training and the importance of understanding the underlying math.

Transcripts

hello people from the future welcome to

normalise nerd in this video I'm gonna

explain the gang yes the famous

generative adversarial networks I know

that this is one of those topics if you

don't approach it properly then this

might feel really intimidating but trust

me by the end of this video you will

feel very comfortable with gangs now I

put a lot of effort in making these

videos so if you like my content please

subscribe and hit the bell icon let's

get started okay the first thing that

you need to know is gang is not a single

model it's a combination of two models

the first one is a generative model

called G and the second one is a

discriminative model called D now what

the hell are discriminative and

generative models well in machine

learning we have two main methods for

building predictive models the most

famous one is the discriminative method

well in this case the model learns the

conditional probability of the target

variable given the input variable most

common examples are logistic regression

linear regression etc on the other hand

in a generative model the model learns

the joint probability distribution of

the input variable and the output

variable if the model wants to predict

something then it uses Bayes theorem and

it computes the conditional probability

of the target variable given the input

variable the most common example is the

naive Bayes model

the biggest advantage of generative

models over discriminative models is

that we can use generative model to make

new instances of data because in this

case we are learning the distribution

function of the data itself which is

simply not possible using a

discriminator in our Gantz we are using

this generative model to produce new

data points that is we are producing

fake data points using our generator and

we are using this discriminator to tell

if a given data point is an original one

or it has been produced by our generator

now these two models work in an

adversarial setup that

means they compete with each other and

eventually both of them gets better and

better in their job let me show you the

structure of this thing okay so here's

the high-level view of our gang this G

and D are nothing but multi-layered

neural networks and this theta G and

theta D are just the weights okay we are

using neural networks here because they

can approximate any function we know

that from the universal approximation

theorem now look at here suppose this is

our distribution function of the

original data now in reality we can't

really draw that or even mathematically

compute that because we input images we

mean put voices we input videos and they

are higher dimensional complex data so

this is only for mathematical analysis

okay and look at here this is a noise

distribution and you can see that this

is just the normal distribution I am

taking and I am gonna sample randomly

some data from this distribution and

we'll feed that to our generator will to

get something from the generator

we must input something right and we are

inputting here noise that means this Z

contains no information and after

passing this Z to our model generator it

will produce something called G of Z now

look at that I have described the

distribution of the G of Z with the same

X that I have written for the earlier

data well I am doing this because the

domain of our original data is same as

the range of G of Z this is important

because we are trying to replicate our

original data so just remember the short

forms when I say P data this represents

the probability distribution of our

original data

when I say PZ it represents the

distribution of the noise and when I say

PG it represents the distribution

function for the output of our generator

and we are going to pass reconstructed

data and original data to our

discriminator and this will provide us a

single number and the single

will tell the probability of the input

belonging to the original data so you

can see this discriminator is just a

simple binary classifier and for the

training purpose when you are putting

the original data to the discriminator

we will say Y is equal to 1 and when we

are going to pass reconstructed data we

will say the level is 0 and the D will

try to maximize the chances of

predicting correct classes but G will

try to fool D so we can say that G and D

are playing a two-player minimax game

what the hell is that

well a minimax game is just a two-player

game like tic-tac-toe where we can

interpret the objective as one player is

trying to maximize its probability of

winning what the other player is trying

to minimize the probability of winning

of the first player okay now we are

saying about maximizing and minimizing

but what should they maximize or

minimize we need a mathematical

expression right and that's called as

value function let me show you the value

function for this minimax game here well

this is the value function for gaen and

here mean and Max simply represents that

G wants to minimize this expression and

D wants to maximize this expression now

I know that at first this might feel

gibberish but if you look here closely

you will find that this expression is

surprisingly similar to the binary cross

Centauri function and if you are feeling

like that then you are absolutely

correct

let me show you why so this is our

ordinary binary cross interval function

for a moment just ignore the negative

sign and the summation so this is just

the binary kazantip function for one

input right why is the ground truth that

is the label and Y hat is just the

prediction of the model when y is equal

to 1 that is when we are passing the

original data the wipe read is equal to

D of X so if you just replace these

things in the formula you will get lost

to Ln of D of X now when we are giving

the

data as our input the wipe red will be d

of G of Z because obviously first we

have passed the noise to our generator

and it has produced something and then

we are giving the produced fake data to

the model B and if you replace these

things in the function you will get Ln

of 1 minus D of G of Z now let's combine

them so I have just added them together

and we get this does it look similar to

the value function yes but here we are

missing the capital E's at the front

well they are just expectations

understand that this expression is valid

for only one data point but we have to

do this for the entire training data set

right and to represent that

mathematically we need to use

expectation well expectation is just the

average value of some experiment if you

perform this experiment a large number

of times

suppose you are playing a game where you

need to roll a die and your score is the

number on the upper face so if you play

this game for a really long time then

the expected score is 3.5 the formula is

very simple you just need to add all the

possible outcomes multiplied with their

probability so it's kind of a weighted

mean so let's apply the expectation on

this equation and look at here that we

are adding all the scores with their

probability same thing goes for here but

this is only true for a discrete

distribution if we assume that our P

data and PZ are actually continuous

distribution then the integral sign will

replace the summation and we have to

place the DX and DZ accordingly and this

whole thing is written in the short form

as e ok so now you know the value

function for Gann does it look

intimidating now I don't think so now

I'm gonna tell you how we optimize this

function in practice well this is our

big training loop and just like every

other neural network we have to optimize

the loss function using some stochastic

process

I am using here the stochastic gradient

descent okay so first we enter our big

training loop and we fix the learning of

G and then we are entering the inner

loop for B well this loop will continue

for K steps okay and in this loop first

we take m data points from the original

distribution and M data points from the

fake data okay and then we update the

parameters of our discriminator by

gradient ascent why because remember

that our discriminator is trying to

maximize the value function so after we

have performed K updates of D we get out

of this loop and we fix the learning of

D now we are going to train our

generator for this case we take only M

fake data samples and update the

parameters of our generator by gradient

descent why because remembers generator

is trying to minimize the value function

now you might ask why I haven't taken

this portion in the update step of

generator well look closely does this

expression contains any term

corresponding to the generator no so the

partial derivative of this term with

respect to theta GU will be zero that's

why we are taking only this portion one

important thing you should note that for

every key updates of the discriminator

we are updating the generator once okay

if you have understood this video so far

then you know what is the value function

for Gann and how we optimize this in

practice but if you are like me and want

to know what is the guarantee that our

generator will surely replicate the

original distribution then take a deep

breath and continue watching okay just

to be clear we want to prove that PG

will converge to P data if our generator

is able to find the global minimum for

the value function in other words we

want to show that PG is equal to P data

at the global minimum of the value

function okay this is a two-step process

first of all we are fixing the G

we wanna see for which value of the

discriminator the value function is

maximum look here that I have replaced G

of Z with X well we can do this because

the domain of both of them are same now

if you differentiate this then you will

see that the maximum value of this

expression will occur if the d of x

attains this expression P data over P

data plus P G well obviously one can

differentiate that and attain this

expression but let us look into it ibly

so we can represent our value function

like this formula a ln x plus b ln 1

minus x and we want to find the value of

x for which this expression is maximum

so if I take B is equal to 0.6 and a is

equal to 0.45 then you will see that the

graph looks something like this and the

Maxima occurs at point 4 to 9 which is

nothing but a Upon A plus B now let's

fix the BX as this and replace that into

our value function so after fixing D and

substituting that in the value function

we get this and after a little

modification we are getting this long

expression and here

mimsey just represents that G will try

to minimize this thing now understand

what we want to do here

well we want to prove that probability

distribution of generator will be

exactly same as the probability

distribution of the data so it makes

sense to talk about some of the methods

to measure the difference between two

distributions and one of the most famous

methods are G is divergence that is

Jenson Shannon divergence now if you

look at the formula for J's divergence

then it looks surprisingly close to this

long expression isn't it just for a

refresher this e here just represents

the expectation in the first portion to

find the expectation of this value we

are using the probabilities from the

first distribution but in the second

portion we are using the probabilities

from the second distribution okay now

let's see if we can somehow get to the J

is divergence from this thing okay so

after the little modification we are

getting this so what have we done here

we have just multiplied two in these two

logarithms and for this we need to

subtract two times the Ln two here all

right and if we look closely here then

this whole portion is actually equals to

two times the J is divergence of P data

and PG and obviously we have the

negative two Ln two here so G wants to

minimize this what is the minimum value

of this expression

well the J is divergence between any two

distribution cannot be negative the

minimum it can get is zero and it will

attend zero only when p1 is equal to p2

that is if P data is equal to P G then

only this term will be zero and the

whole expression will attain its minimum

that is minus 2 Ln 2 so voila now you we

have proved that add the global minimum

of our value function the P G will be

exactly same as P data and our generator

is actually trying to attain that state

now let me show you how G achieves that

state that is different phases of

training so at the beginning neither the

discriminator nor the generator knows

what they are doing so the P G is not

replicating the P data and the

classifier discriminator is not

classifying as well after updating the

theta D that is when the discriminator

has learned something so the classifier

will be better so now the discriminator

can actually distinguish between the

real data and the fake data now after

the generator has learned something look

at that the distribution P G

is now closer to the P data and the

discriminator is trying to predict the

true level of the data points but it is

not performing as well now at the end

when the generator has attained the

minimum of the value function then it

has successfully replicated the

distribution function of the data point

so now PG is indistinguishable from P

data so now it is impossible for the

discriminator to tell which data point

is an original one and which data point

is a generated one so the discriminator

will output 0.5 for every input and that

is what we want to achieve well this is

a very simplistic view of the gaen in

reality training the Gann is really hard

the goal of this video was to make you

understand

Gantz I hope you are now very

comfortable with the concept of ganz and

if you have understood everything that I

have talked about in this video then do

congratulate yourself because now you

know the math behind one of the finest

inventions in AI I hope you have liked

this video please share this video and

subscribe to my channel

stay safe and thanks for watching

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