The Math Behind Generative Adversarial Networks Clearly Explained!
Summary
TLDRThis 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
🧠 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.
🎲 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.
🔍 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.
📈 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.
Mindmap
Keywords
💡Generative Adversarial Networks (GANs)
💡Generative Model
💡Discriminative Model
💡Neural Networks
💡Adversarial Setup
💡Minimax Game
💡Value Function
💡Stochastic Gradient Descent
💡Jensen-Shannon Divergence
💡Global Minimum
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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