CS 4803 / 7643: Deep Learning Topics: Variational Auto-Encoders - - PowerPoint PPT Presentation

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CS 4803 / 7643: Deep Learning Topics: Variational Auto-Encoders - - PowerPoint PPT Presentation

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CS 4803 / 7643: Deep Learning Topics: Variational Auto-Encoders (VAEs) Reparameterization trick Dhruv Batra Georgia Tech Administrativia HW4 Grades Released Regrade requests close: 12/03, 11:55pm Please check solutions

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SLIDE 1

CS 4803 / 7643: Deep Learning

Dhruv Batra Georgia Tech

Topics:

– Variational Auto-Encoders (VAEs) – Reparameterization trick

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SLIDE 2

Administrativia

  • HW4 Grades Released

– Regrade requests close: 12/03, 11:55pm – Please check solutions first!

  • Grade histogram: 7643

– Max possible: 100 (regular credit) + 40 (extra credit)

(C) Dhruv Batra 2

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SLIDE 3

Administrativia

  • HW4 Grades Released

– Regrade requests close: 12/03, 11:55pm – Please check solutions first!

  • Grade histogram: 4803

– Max possible: 100 (regular credit) + 40 (extra credit)

(C) Dhruv Batra 3

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SLIDE 4

Recap from last time

(C) Dhruv Batra 4

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SLIDE 5

Variational Autoencoders (VAE)

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SLIDE 6

So far...

VAEs define intractable density function with latent z: PixelCNNs define tractable density function, optimize likelihood of training data:

Slide Credit: Fei-Fei Li, Justin Johnson, Serena Yeung, CS 231n

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SLIDE 7

Variational Auto Encoders

VAEs are a combination of the following ideas:

  • 1. Auto Encoders
  • 2. Variational Approximation
  • Variational Lower Bound / ELBO
  • 3. Amortized Inference Neural Networks
  • 4. “Reparameterization” Trick

(C) Dhruv Batra 7

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SLIDE 8

Encoder Input data Features Decoder Reconstructed input data

Reconstructed data Input data

Encoder: 4-layer conv Decoder: 4-layer upconv

L2 Loss function:

Train such that features can be used to reconstruct original data Doesn’t use labels!

Autoencoders

Slide Credit: Fei-Fei Li, Justin Johnson, Serena Yeung, CS 231n

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SLIDE 9

Encoder Input data Features Decoder Reconstructed input data

Autoencoders can reconstruct data, and can learn features to initialize a supervised model Features capture factors of variation in training data. Can we generate new images from an autoencoder?

Autoencoders

Slide Credit: Fei-Fei Li, Justin Johnson, Serena Yeung, CS 231n

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SLIDE 10

Variational Autoencoders

Probabilistic spin on autoencoders - will let us sample from the model to generate data!

Image Credit: https://jaan.io/what-is-variational-autoencoder-vae-tutorial/

q𝜚 𝑨 𝑦 p𝜄 𝑦 𝑨

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SLIDE 11

Variational Auto Encoders

VAEs are a combination of the following ideas:

  • 1. Auto Encoders
  • 2. Variational Approximation
  • Variational Lower Bound / ELBO
  • 3. Amortized Inference Neural Networks
  • 4. “Reparameterization” Trick

(C) Dhruv Batra 11

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SLIDE 12

Key problem

  • P(z|x)

(C) Dhruv Batra 12

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SLIDE 13

What is Variational Inference?

  • Key idea

– Reality is complex – Can we approximate it with something “simple”? – Just make sure simple thing is “close” to the complex thing.

(C) Dhruv Batra 13

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SLIDE 14

Intuition

(C) Dhruv Batra 14

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SLIDE 15
  • Marginal likelihood – x is observed, z is missing:

The general learning problem with missing data

15 (C) Dhruv Batra

ll(θ : D) = log

N

Y

i=1

P(xi | θ) =

N

X

i=1

log P(xi | θ) =

N

X

i=1

log X

z

P(xi, z | θ)

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SLIDE 16

Jensen’s inequality

  • Use: log åz P(z) g(z) ≥ åz P(z) log g(z)

16 (C) Dhruv Batra

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SLIDE 17

Applying Jensen’s inequality

  • Use: log åz P(z) g(z) ≥ åz P(z) log g(z)

17 (C) Dhruv Batra

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SLIDE 18

Evidence Lower Bound

  • Define potential function F(q,Q):

18 (C) Dhruv Batra

ll(θ : D) ≥ F(θ, Qi) =

N

X

i=1

X

z

Qi(z) log P(xi, z | θ) Qi(z)

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SLIDE 19

ELBO: Factorization #1 (GMMs)

19 (C) Dhruv Batra

ll(θ : D) ≥ F(θ, Qi) =

N

X

i=1

X

z

Qi(z) log P(xi, z | θ) Qi(z)

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SLIDE 20

ELBO: Factorization #2 (VAEs)

20 (C) Dhruv Batra

ll(θ : D) ≥ F(θ, Qi) =

N

X

i=1

X

z

Qi(z) log P(xi, z | θ) Qi(z)

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SLIDE 21

Variational Auto Encoders

VAEs are a combination of the following ideas:

  • 1. Auto Encoders
  • 2. Variational Approximation
  • Variational Lower Bound / ELBO
  • 3. Amortized Inference Neural Networks
  • 4. “Reparameterization” Trick

(C) Dhruv Batra 21

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SLIDE 22

Amortized Inference Neural Networks

(C) Dhruv Batra 22

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SLIDE 23

VAEs

(C) Dhruv Batra 23

Image Credit: https://www.kaggle.com/rvislaywade/visualizing-mnist-using-a-variational-autoencoder

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SLIDE 24

Variational Autoencoders

Probabilistic spin on autoencoders - will let us sample from the model to generate data!

Image Credit: https://jaan.io/what-is-variational-autoencoder-vae-tutorial/

q𝜚 𝑨 𝑦 p𝜄 𝑦 𝑨

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SLIDE 25

Encoder network Input Data

Putting it all together: maximizing the likelihood lower bound

Variational Auto Encoders

Slide Credit: Fei-Fei Li, Justin Johnson, Serena Yeung, CS 231n

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SLIDE 26

Encoder network Input Data

Putting it all together: maximizing the likelihood lower bound

Make approximate posterior distribution close to prior

Variational Auto Encoders

Slide Credit: Fei-Fei Li, Justin Johnson, Serena Yeung, CS 231n

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SLIDE 27

Encoder network Sample z from Input Data

Putting it all together: maximizing the likelihood lower bound

Make approximate posterior distribution close to prior

Variational Auto Encoders

Slide Credit: Fei-Fei Li, Justin Johnson, Serena Yeung, CS 231n

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SLIDE 28

Encoder network Decoder network Sample z from Input Data

Putting it all together: maximizing the likelihood lower bound

Make approximate posterior distribution close to prior

Variational Auto Encoders

Slide Credit: Fei-Fei Li, Justin Johnson, Serena Yeung, CS 231n

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SLIDE 29

Encoder network Decoder network Sample z from Sample x|z from Input Data

Putting it all together: maximizing the likelihood lower bound

Make approximate posterior distribution close to prior Maximize likelihood of

  • riginal input

being reconstructed

Variational Auto Encoders

Slide Credit: Fei-Fei Li, Justin Johnson, Serena Yeung, CS 231n

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SLIDE 30

Decoder network Sample z from Sample x|z from

Use decoder network. Now sample z from prior! Data manifold for 2-d z Vary z1 Vary z2

Kingma and Welling, “Auto-Encoding Variational Bayes”, ICLR 2014

Variational Auto Encoders: Generating Data

Slide Credit: Fei-Fei Li, Justin Johnson, Serena Yeung, CS 231n

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SLIDE 31

Vary z1 Vary z2 Degree of smile Head pose Diagonal prior on z => independent latent variables Different dimensions of z encode interpretable factors

  • f variation

Kingma and Welling, “Auto-Encoding Variational Bayes”, ICLR 2014

Variational Auto Encoders: Generating Data

Slide Credit: Fei-Fei Li, Justin Johnson, Serena Yeung, CS 231n

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SLIDE 32

Plan for Today

  • VAEs

– Reparameterization trick

(C) Dhruv Batra 32

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SLIDE 33

Variational Auto Encoders

VAEs are a combination of the following ideas:

  • 1. Auto Encoders
  • 2. Variational Approximation
  • Variational Lower Bound / ELBO
  • 3. Amortized Inference Neural Networks
  • 4. “Reparameterization” Trick

(C) Dhruv Batra 33

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SLIDE 34

Encoder network Decoder network Sample z from Sample x|z from Input Data

Putting it all together: maximizing the likelihood lower bound

Make approximate posterior distribution close to prior Maximize likelihood of

  • riginal input

being reconstructed

Variational Auto Encoders

Slide Credit: Fei-Fei Li, Justin Johnson, Serena Yeung, CS 231n

slide-35
SLIDE 35

Putting it all together: maximizing the likelihood lower bound

Variational Auto Encoders

Slide Credit: Fei-Fei Li, Justin Johnson, Serena Yeung, CS 231n

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SLIDE 36

Basic Problem

(C) Dhruv Batra 36

n Ez∼pθ(z)[f(z)]

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SLIDE 37

Basic Problem

  • Goal

(C) Dhruv Batra 37

min

θ

Ez∼pθ(z)[f(z)]

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SLIDE 38

Basic Problem

  • Goal
  • Need to compute:

(C) Dhruv Batra 38

min

θ

Ez∼pθ(z)[f(z)] rθ Ez∼pθ(z)[f(z)]

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SLIDE 39

Basic Problem

  • Need to compute:

(C) Dhruv Batra 39

rθ Ez∼pθ(z)[f(z)]

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SLIDE 40

Example

(C) Dhruv Batra 40

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SLIDE 41

Does this happen in supervised learning?

  • Goal

(C) Dhruv Batra 41

min

θ

Ez∼pθ(z)[f(z)]

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SLIDE 42

But what about other kinds of learning?

  • Goal

(C) Dhruv Batra 42

min

θ

Ez∼pθ(z)[f(z)]

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SLIDE 43

Two Options

  • Score Function based Gradient Estimator

aka REINFORCE (and variants)

  • Path Derivative Gradient Estimator

aka “reparameterization trick”

(C) Dhruv Batra 43

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SLIDE 44

Option 1

  • Score Function based Gradient Estimator

aka REINFORCE (and variants)

(C) Dhruv Batra 44

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SLIDE 45

45

= rθ Z πθ(τ)R(τ)dτ

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Recall: Policy Gradients

rθJ(θ) = rθEτ∼pθ(τ)[R(τ)]

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= Z rθπθ(τ)R(τ)dτ

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= Z rθπθ(τ) · πθ(τ) πθ(τ) · R(τ)dτ

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= Z πθ(τ)rθ log πθ(τ)R(τ)dτ

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= Eτ∼pθ(τ)[rθ log πθ(τ)R(τ)]

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Expand expectation Exchange integration and expectation

rθ log π(τ) = rθπ(τ) π(τ)

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slide-46
SLIDE 46

Example

(C) Dhruv Batra 46

slide-47
SLIDE 47

Mental Break!

  • VAE Demo

– https://www.siarez.com/projects/variational-autoencoder

(C) Dhruv Batra 47

slide-48
SLIDE 48

Two Options

  • Score Function based Gradient Estimator

aka REINFORCE (and variants)

(C) Dhruv Batra 48

  • Path Derivative Gradient Estimator

aka “reparameterization trick”

slide-49
SLIDE 49

Option 2

(C) Dhruv Batra 49

  • Path Derivative Gradient Estimator

aka “reparameterization trick”

slide-50
SLIDE 50

Option 2

(C) Dhruv Batra 50

  • Path Derivative Gradient Estimator

aka “reparameterization trick”

slide-51
SLIDE 51

Reparameterization Intuition

(C) Dhruv Batra 51

z = µ + 2✏i ✏i ∼ p(✏) σ2

Figure Credit: http://blog.shakirm.com/2015/10/machine-learning-trick-of-the-day-4-reparameterisation-tricks/

slide-52
SLIDE 52

Reparameterization Intuition

(C) Dhruv Batra 52

Image Credit: https://www.kaggle.com/rvislaywade/visualizing-mnist-using-a-variational-autoencoder

slide-53
SLIDE 53

Example

(C) Dhruv Batra 53

slide-54
SLIDE 54

Two Options

  • Score Function based Gradient Estimator

aka REINFORCE (and variants)

(C) Dhruv Batra 54

  • Path Derivative Gradient Estimator

aka “reparameterization trick”

slide-55
SLIDE 55

Example

(C) Dhruv Batra 55

Figure Credit: http://gokererdogan.github.io/2016/07/01/reparameterization-trick/

slide-56
SLIDE 56

Example

(C) Dhruv Batra 56

Figure Credit: http://gokererdogan.github.io/2016/07/01/reparameterization-trick/

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SLIDE 57

Variational Auto Encoders

VAEs are a combination of the following ideas:

  • 1. Auto Encoders
  • 2. Variational Approximation
  • Variational Lower Bound / ELBO
  • 3. Amortized Inference Neural Networks
  • 4. “Reparameterization” Trick

(C) Dhruv Batra 57

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SLIDE 58

Encoder network Input Data

Putting it all together: maximizing the likelihood lower bound

Variational Auto Encoders

Slide Credit: Fei-Fei Li, Justin Johnson, Serena Yeung, CS 231n

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SLIDE 59

Encoder network Input Data

Putting it all together: maximizing the likelihood lower bound

Make approximate posterior distribution close to prior

Variational Auto Encoders

Slide Credit: Fei-Fei Li, Justin Johnson, Serena Yeung, CS 231n

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SLIDE 60

Encoder network Sample z from Input Data

Putting it all together: maximizing the likelihood lower bound

Make approximate posterior distribution close to prior

Variational Auto Encoders

Slide Credit: Fei-Fei Li, Justin Johnson, Serena Yeung, CS 231n

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SLIDE 61

Encoder network Decoder network Sample z from Input Data

Putting it all together: maximizing the likelihood lower bound

Make approximate posterior distribution close to prior

Variational Auto Encoders

Slide Credit: Fei-Fei Li, Justin Johnson, Serena Yeung, CS 231n