Generative Adversarial NetworksAdvance Machine Learning

Sourish Das and Madhavan Mukund

Chennai Mathematical Institute

Aug-Nov, 2020

Introduction

1. ”Generative Adversarial Networks” (Nov 2020) by Goodfellowetal . Communications of the ACM, Vol 63, No. 11,DOI:10.1145/3422622 (original version: NIPS2014)

2. ”Hands-on Machine Learning with Scikit-Learn, Keras &TensorFlow”, by Aurélien Géron - Page 592–606

The Idea

I Problem Statement:1. Suppose x is the data generated from p(x)

2. End goal is to learn about unknown p( )

I Main Idea of GAN:

1. Two neural networks compete against each other

2. The competition will push them to excel and learn the datagenerating process p(x)

Structure of GAN

I Generator Network1. A network that generate data (for example decoder part of

VAE) - often known as fake data

2. Goal: Trick the Discriminator

I Discriminator Network1. Takes either a fake data from the generator or a real data from

the training set as input

2. guess whether the input data is fake or real.

3. Goal: Tell real or fake data

Generative Adversarial Network

source: Hands-on Machine Learning with Scikit-Learn, Keras &TensorFlow, by Aurélien Géron, page - 592

Example of GAN

Source: Goodfellow etal . 2020

Example of GAN

Source: Goodfellow etal . 2020

Unique Feature of GAN

I It is based on Game Theory,while other generative models are based on optimization

I GAN avoid the entire issue of designing a tractable densityfunction and learn only a tractable sample generation process.

I These are called implicit generative models.

Let’s look inside GAN

I Suppose G is the generator model such that

x = G (z),

where

1. G is differentiable function

2. z is the latent variable sampled from some simple priordistribution

3. x is drawn from pmodel

I This idea is common in Probability and Statistics

I If we want to simulate from pmodel = Exponential(λ),simulate z ∼ Unif (0, 1) then pass it to G (z) = − lnzλ = x

GAN is about solving Game

I The generator function G (z,θG ) = x uses θG as parameters

I D(x,θD) is the discriminator network model where x is inputand θD is the parameters of the D

I Both players have cost functions that are defined in terms ofboth players’ parameters.

I Discriminator wishes to minimize J(D)(θ(D),θ(G)) such that

minθ(D)

J(D)(θ(D),θ(G))

I Generator wishes to minimize J(G)(θ(D),θ(G)) such that

minθ(G)

J(D)(θ(D),θ(G))

Solution of GAN is Nash equilibrium

I The solution to a game is a Nash equilibrium, more preciselyof local differential Nash equilibria.

I In this context, a Nash equilibrium is a tuple (θ(D),θ(G)) thatis a local minimum of JD wrt θ(D) and local minimum of JG

wrt θ(G)

Discriminator’s cost

I The cost used for the discriminator is:

JD(θD ,θG ) = −12Ex∼pdata log D(x)−

1

2Ez log(1− D(G (z)))

I standard cross-entropy cost that is minimized when training astandard binary classifier with a sigmoid output

I the classifier is trained on two minibatches of data:

1. one coming from the dataset, where the label is 1 for allexamples

2. other coming from the generator, where the label is 0 for allexamples.

I All versions of the GAN game encourage thediscriminator to minimize the above cost function.

Generator’s cost and Minimax

I The simplest version of the game is a zero-sum game.

I In this version of the game

JG = −JD

I Because JG is tied directly to JD , value function specifyingthe discriminator’s payoff:

V (θD ,θG ) = −JD(θD ,θG )

I Zero-sum games are also called minimax:

θG∗ = arg minθG

maxθD

V (θD ,θG )

More on Value FunctionsLet’s look into the value function:

V (θD ,θG ) = Ex∼p̂ log D(x) + Ez∼pz (z) log(1− D(G (z)))

=

∫x

p̂(x) log(D(x))dx +

∫z

pz(z) log(1− D(G (z)))dz

Now

x = G (z) =⇒ z = G−1(x) =⇒ dz = (G−1)′(x)dx

=

∫x

p̂(x) log(D(x))dx +

∫x

pz(G−1(x)) log(1− D(x))(G−1)′(x)dx

=

∫x

p̂(x) log(D(x))dx +

∫x

pg (x)(1− D(x))dx,

where pg (x) = pz(G−1(x))(G−1)′(x)

=

∫x{p̂(x) log(D(x)) + pg (x)(1− D(x))}dx

Understanding the Value Functions

maxD

V (D,G ) = maxD

∫x{p̂(x) log(D(x)) + pg (x)(1− D(x))}dx

∂

∂D(x){p̂(x) log(D(x)) + pg (x)(1− D(x))} = 0

=⇒ p̂(x)D(x)

− pg (x)1− D(x)

= 0

=⇒ D∗G (x) =p̂(x)

p̂(x) + pg (x)

So this the optimal D for given G

Understanding the Cost function of G

C (G ) = maxD

V (D,G )

= maxD

∫x

p̂(x) log(D(x)) + pg (x) log(1− D(x))dx

=

∫x

p̂(x) log(D∗G (x)) + pg (x) log(1− D∗G (x))dx

=

∫x

p̂(x) log(p̂(x)

p̂(x) + pg (x)) + pg (x) log(

pg (x)

p̂(x) + pg (x))dx

=

∫x

p̂(x) log(p̂(x)

p̂(x)+pg (x)2

) + pg (x) log(pg (x)

p̂(x)+pg (x)2

)dx− log(4)

= KL(p̂(x)|| p̂(x) + pg (x)2

) + KL(pg (x)||p̂(x) + pg (x)

2)

− log(4)

Understanding of Cost Function G

C (G ) = KL(p̂(x)|| p̂(x) + pg (x)2

) + KL(pg (x)||p̂(x) + pg (x)

2)− log(4)

C (G ) will be minimum when

KL(p̂(x)|| p̂(x)+pg (x)2 ) = 0 and KL(pg (x)||p̂(x)+pg (x)

2 ) = 0, i.e.,

=⇒ p̂(x) = p̂(x) + pg (x)2

=⇒ pg (x) =p̂(x) + pg (x)

2=⇒ p̂(x) = pg (x)

Example of GAN

Source: ”Generative Adversarial Networks” (Nov 2020) byGoodfellow etal . Communications of the ACM, Vol 63, No. 11,DOI:10.1145/3422622 (original version: NIPS2014)

Convergence of GANS

I In many different models (other than Goodfelow etal . 2014and some other models), whether a Nash equilibrium exists

I whether any spurious Nash equilibria exist !!

I whether the learning algorithm converges to a Nashequilibrium? If so then how quickly?

I ”In many cases of practical interest, these theoretical questionsare open, and the best learning algorithms seem empirically tooften fail to converge.”

Summary

I GANs are generative model based on game theory.

I They have had great practical success in terms of generatingrealistic data, especially images.

I It is currently still difficult to train them.

I For GANs to become a more reliable technology, it will benecessary to design models, costs, or training algorithms forwhich it is possible to find good Nash equilibria consistentlyand quickly

Thank You

sourish@cmi.ac.in

sourish@cmi.ac.in