From f01d4b101f7596a35d5665597ecf447d4823af29 Mon Sep 17 00:00:00 2001 From: Morten Hjorth-Jensen Date: Mon, 15 Apr 2024 07:39:26 +0200 Subject: [PATCH] update week 13 --- doc/pub/week13/html/week13-bs.html | 317 +++++++- doc/pub/week13/html/week13-reveal.html | 320 +++++++- doc/pub/week13/html/week13-solarized.html | 307 +++++++- doc/pub/week13/html/week13.html | 307 +++++++- doc/pub/week13/ipynb/ipynb-week13-src.tar.gz | Bin 29 -> 20090 bytes doc/pub/week13/ipynb/week13.ipynb | 777 ++++++++++++++++--- doc/pub/week13/pdf/week13.pdf | Bin 403575 -> 451219 bytes doc/src/week13/week13.do.txt | 408 +++++----- 8 files changed, 2097 insertions(+), 339 deletions(-) diff --git a/doc/pub/week13/html/week13-bs.html b/doc/pub/week13/html/week13-bs.html index 25079640..6094e43d 100644 --- a/doc/pub/week13/html/week13-bs.html +++ b/doc/pub/week13/html/week13-bs.html @@ -40,6 +40,7 @@ 2, None, 'plans-for-the-week-april-15-19-2024'), + ('Readings', 2, None, 'readings'), ('Reminder from last week and layout of lecture this week', 2, None, @@ -56,6 +57,7 @@ 2, None, 'energy-based-models-and-langevin-sampling'), + ('Langevin sampling', 2, None, 'langevin-sampling'), ('Theory of Variational Autoencoders', 2, None, @@ -65,10 +67,25 @@ 2, None, 'schematic-image-of-an-autoencoder'), - ('Kullback-Leibler relative entropy', + ('Mathematics of Variational Autoencoders', 2, None, - 'kullback-leibler-relative-entropy'), + 'mathematics-of-variational-autoencoders'), + ('Using the conditional probability', + 2, + None, + 'using-the-conditional-probability'), + ('VAEs versus autoencoders', 2, None, 'vaes-versus-autoencoders'), + ('Gradient descent', 2, None, 'gradient-descent'), + ('Are VAEs just modified autoencoders?', + 2, + None, + 'are-vaes-just-modified-autoencoders'), + ('Training VAEs', 2, None, 'training-vaes'), + ('Kullback-Leibler relative entropy (notation to be updated)', + 2, + None, + 'kullback-leibler-relative-entropy-notation-to-be-updated'), ('Kullback-Leibler divergence', 2, None, @@ -88,6 +105,7 @@ ('Difference of moments', 2, None, 'difference-of-moments'), ('More observations', 2, None, 'more-observations'), ('Adding hyperparameters', 2, None, 'adding-hyperparameters'), + ('Back to VAEs', 2, None, 'back-to-vaes'), ('Code examples using Keras', 2, None, @@ -121,7 +139,8 @@ ('Exploring the Latent Space', 2, None, - 'exploring-the-latent-space')]} + 'exploring-the-latent-space'), + ('Using the KL divergence', 2, None, 'using-the-kl-divergence')]} end of tocinfo --> @@ -157,14 +176,22 @@ Contents @@ -240,11 +269,14 @@

Plans for the week A + + +

Readings

    -
  1. Reading recommendation: Goodfellow et al chapter 20.10-20-14
    2. +
  2. Reading recommendation: Goodfellow et al, for VAEs and GANs see sections 20.10-20.11
  3. To create Boltzmann machine using Keras, see Babcock and Bali chapter 4, see https://github.com/PacktPublishing/Hands-On-Generative-AI-with-Python-and-TensorFlow-2/blob/master/Chapter_4/models/rbm.py
  4. See Foster, chapter 7 on energy-based models at https://github.com/davidADSP/Generative_Deep_Learning_2nd_Edition/tree/main/notebooks/07_ebm/01_ebm
@@ -406,6 +438,11 @@

Energy-based m

That notebook is based on a recent article by Du and Mordatch, Implicit generation and modeling with energy-based models, see https://arxiv.org/pdf/1903.08689.pdf.

+ +

Langevin sampling

+ +

Note: Notes to be added

+

Theory of Variational Autoencoders

@@ -432,15 +469,113 @@

The Autoencoder again

\tilde{\boldsymbol{x}} = g(\boldsymbol{h},\boldsymbol{V})). $$ -

The goal is to minimize the construction error.

+

The goal is to minimize the construction error, often done by optimizing the means squared error.

Schematic image of an Autoencoder

- +

+
+

+
+

+ + +

Mathematics of Variational Autoencoders

+ +

We have defined earlier a probability (marginal) distribution with hidden variables \( \boldsymbol{h} \) and parameters \( \boldsymbol{\Theta} \) as

+$$ +p(\boldsymbol{x};\boldsymbol{\Theta}) = \int d\boldsymbol{h}p(\boldsymbol{x},\boldsymbol{h};\boldsymbol{\Theta}), +$$ + +

for continuous variables \( \boldsymbol{h} \) and

+$$ +p(\boldsymbol{x};\boldsymbol{\Theta}) = \sum_{\boldsymbol{h}}p(\boldsymbol{x},\boldsymbol{h};\boldsymbol{\Theta}), +$$ + +

for discrete stochastic events \( \boldsymbol{h} \). The variables \( \boldsymbol{h} \) are normally called the latent variables in the theory of autoencoders. We will also call then for that here.

+ + +

Using the conditional probability

+ +

Using the the definition of the conditional probabilities \( p(\boldsymbol{x}\vert\boldsymbol{h};\boldsymbol{\Theta}) \), \( p(\boldsymbol{h}\vert\boldsymbol{x};\boldsymbol{\Theta}) \) and +and the prior \( p(\boldsymbol{h}) \), we can rewrite the above equation as +

+$$ +p(\boldsymbol{x};\boldsymbol{\Theta}) = \sum_{\boldsymbol{h}}p(\boldsymbol{x}\vert\boldsymbol{h};\boldsymbol{\Theta})p(\boldsymbol{h}, +$$ + +

which allows us to make the dependence of \( \boldsymbol{x} \) on \( \boldsymbol{h} \) +explicit by using the law of total probability. The intuition behind +this approach for finding the marginal probability for \( \boldsymbol{x} \) is to +optimize the above equations with respect to the parameters +\( \boldsymbol{\Theta} \). This is done normally by maximizing the probability, +the so-called maximum-likelihood approach discussed earlier. +

+ + +

VAEs versus autoencoders

+ +

This trained probability is assumed to be able to produce similar +samples as the input. In VAEs it is then common to compare via for +example the mean-squared error or the cross-entropy the predicted +values with the input values. Compared with autoencoders, we are now +producing a probability instead of a functions which mimicks the +input. +

+ +

In VAEs, the choice of this output distribution is often Gaussian, +meaning that the conditional probability is +

+$$ +p(\boldsymbol{x}\vert\boldsymbol{h};\boldsymbol{\Theta})=N(\boldsymbol{x}\vert f(\boldsymbol{h};\boldsymbol{\Theta}), \sigma^2\times \boldsymbol{I}), +$$ + +

with mean value given by the function \( f(\boldsymbol{h};\boldsymbol{\Theta}) \) and a +diagonal covariance matrix multiplied by a parameter \( \sigma^2 \) which +is treated as a hyperparameter. +

+ + +

Gradient descent

+ +

By having a Gaussian distribution, we can use gradient descent (or any +other optimization technique) to increase \( p(\boldsymbol{x};\boldsymbol{\Theta}) \) by +making \( f(\boldsymbol{h};\boldsymbol{\Theta}) \) approach \( \boldsymbol{x} \) for some \( \boldsymbol{h} \), +gradually making the training data more likely under the generative +model. The important property is simply that the marginal probability +can be computed, and it is continuous in \( \boldsymbol{\Theta} \).. +

+ + +

Are VAEs just modified autoencoders?

+ +

The mathematical basis of VAEs actually has relatively little to do +with classical autoencoders, for example the sparse autoencoders or +denoising autoencoders discussed earlier. +

+ +

VAEs approximately maximize the probability equation discussed +above. They are called autoencoders only because the final training +objective that derives from this setup does have an encoder and a +decoder, and resembles a traditional autoencoder. Unlike sparse +autoencoders, there are generally no tuning parameters analogous to +the sparsity penalties. And unlike sparse and denoising autoencoders, +we can sample directly from \( p(\boldsymbol{x}) \) without performing Markov +Chain Monte Carlo. +

+ + +

Training VAEs

+ +

To solve the integral or sum for \( p(\boldsymbol{x}) \), there are two problems +that VAEs must deal with: how to define the latent variables \( \boldsymbol{h} \), +that is decide what information they represent, and how to deal with +the integral over \( \boldsymbol{h} \). VAEs give a definite answer to both. +

-

Kullback-Leibler relative entropy

+

Kullback-Leibler relative entropy (notation to be updated)

When the goal of the training is to approximate a probability distribution, as it is in generative modeling, another relevant @@ -566,9 +701,12 @@

Adding hyperparameters

it is limited to the training data. Hence, in unsupervised training as well it is important to prevent overfitting to the training data. Thus it is common to add regularizers to the cost function in the same -manner as we discussed for say linear regression. +manner as discussed for say linear regression.

+ +

Back to VAEs

+

Code examples using Keras

@@ -794,10 +932,7 @@

Zero-sum game

$$ -\begin{equation} -v(\theta^{(g)}, \theta^{(d)}) -\label{_auto1} -\end{equation} $$ @@ -1710,6 +1845,164 @@

Exploring the Latent Space A pretty cool result! We see that our generator indeed has learned a distribution which qualitatively looks a whole lot like the MNIST dataset.

+ + +

Using the KL divergence

+ +

In practice, for most \( \boldsymbol{h} \), \( p(\boldsymbol{x}\vert \boldsymbol{h}; \boldsymbol{\Theta}) \) +will be nearly zero, and hence contribute almost nothing to our +estimate of \( p(\boldsymbol{x}) \). +

+ +

The key idea behind the variational autoencoder is to attempt to +sample values of \( \boldsymbol{h} \) that are likely to have produced \( \boldsymbol{x} \), +and compute \( p(\boldsymbol{x}) \) just from those. +

+ +

This means that we need a new function \( Q(\boldsymbol{h}|\boldsymbol{x}) \) which can +take a value of \( \boldsymbol{x} \) and give us a distribution over \( \boldsymbol{h} \) +values that are likely to produce \( \boldsymbol{x} \). Hopefully the space of +\( \boldsymbol{h} \) values that are likely under \( Q \) will be much smaller than +the space of all \( \boldsymbol{h} \)'s that are likely under the prior +\( p(\boldsymbol{h}) \). This lets us, for example, compute \( E_{\boldsymbol{h}\sim +Q}p(\boldsymbol{x}\vert \boldsymbol{h}) \) relatively easily. Note that we drop +\( \boldsymbol{\Theta} \) from here and for notational simplicity. +

+ +

However, if \( \boldsymbol{h} \) is sampled from an arbitrary distribution with +PDF \( Q(\boldsymbol{h}) \), which is not \( \mathcal{N}(0,I) \), then how does that +help us optimize \( p(\boldsymbol{x}) \)? The first thing we need to do is relate +\( E_{\boldsymbol{h}\sim Q}P(\boldsymbol{x}\vert \boldsymbol{h}) \) and \( p(\boldsymbol{x}) \). We will see where \( Q \) comes from later. +

+ +

The relationship between \( E_{\boldsymbol{h}\sim Q}p(\boldsymbol{x}\vert \boldsymbol{h}) \) and \( p(\boldsymbol{x}) \) is one of the cornerstones of variational Bayesian methods. +We begin with the definition of Kullback-Leibler divergence (KL divergence or \( \mathcal{D} \)) between \( p(\boldsymbol{h}\vert \boldsymbol{x}) \) and \( Q(\boldsymbol{h}) \), for some arbitrary \( Q \) (which may or may not depend on \( \boldsymbol{x} \)): +

+$$ + \mathcal{D}\left[Q(z)\|P(z|X)\right]=E_{z\sim Q}\left[\log Q(z) - \log P(z|X) \right]. +$$ + +

We can get both \( p(\boldsymbol{x}) \) and \( p(\boldsymbol{x}\vert \boldsymbol{h}) \) into this equation by applying Bayes rule to \( p(\boldsymbol{h}|\boldsymbol{x}) \)

+$$ + \mathcal{D}\left[Q(z)\|P(z|X)\right]=E_{z\sim Q}\left[\log Q(z) - \log P(X|z) - \log P(z) \right] + \log P(X). +$$ + +

Here, \( \log P(X) \) comes out of the expectation because it does not depend on \( z \). +Negating both sides, rearranging, and contracting part of \( E_{z\sim Q} \) into a KL-divergence terms yields: +

+$$ +\log P(X) - \mathcal{D}\left[Q(z)\|P(z|X)\right]=E_{z\sim Q}\left[\log P(X|z) \right] - \mathcal{D}\left[Q(z)\|P(z)\right]. +$$ + +

By Bayes rule, we have:

+$$ +E_{z\sim Q}\left[\log P(Y_i|z,X_i)\right]=E_{z\sim Q}\left[\log P(z|Y_i,X_i) - \log P(z|X_i) + \log P(Y_i|X_i) \right] +$$ + +

Rearranging the terms and subtracting \( E_{z\sim Q}\log Q(z) \) from both sides:

+$$ +\begin{array}{c} +\log P(Y_i|X_i) - E_{z\sim Q}\left[\log Q(z)-\log P(z|X_i,Y_i)\right]=\hspace{10em}\\ +\hspace{10em}E_{z\sim Q}\left[\log P(Y_i|z,X_i)+\log P(z|X_i)-\log Q(z)\right] +\end{array} +$$ + +

Note that \( X \) is fixed, and \( Q \) can be \textit{any} distribution, not +just a distribution which does a good job mapping \( X \) to the \( z \)'s +that can produce \( X \). +

+ +

Since we are interested in inferring \( P(X) \), it makes sense to +construct a \( Q \) which \textit{does} depend on \( X \), and in particular, +one which makes \( \mathcal{D}\left[Q(z)\|P(z|X)\right] \) small: +

+$$ +\log P(X) - \mathcal{D}\left[Q(z|X)\|P(z|X)\right]=E_{z\sim Q}\left[\log P(X|z) \right] - \mathcal{D}\left[Q(z|X)\|P(z)\right]. +$$ + +

Hence, during training, it makes sense to choose a \( Q \) which will make +\( E_{z\sim Q}[\log Q(z)- \) $\log P(z|X_i,Y_i)]$ (a +\( \mathcal{D} \)-divergence) small, such that the right hand side is a +close approximation to \( \log P(Y_i|X_i) \). +

+ +

This equation serves as the core of the variational autoencoder, and +it's worth spending some time thinking about what it says In two +sentences, the left hand side has the quantity we want to maximize: +\( \log P(X) \) (plus an error term, which makes \( Q \) produce \( z \)'s that +can reproduce a given \( X \); this term will become small if \( Q \) is +high-capacity). The right hand side is something we can optimize via +stochastic gradient descent given the right choice of \( Q \) (although it +may not be obvious yet how). +

+ +

So how can we perform stochastic gradient descent?

+ +

First we need to be a bit more specific about the form that \( Q(z|X) \) +will take. The usual choice is to say that +\( Q(z|X)=\mathcal{N}(z|\mu(X;\vartheta),\Sigma(X;\vartheta)) \), where +\( \mu \) and \( \Sigma \) are arbitrary deterministic functions with +parameters \( \vartheta \) that can be learned from data (we will omit +\( \vartheta \) in later equations). In practice, \( \mu \) and \( \Sigma \) are +again implemented via neural networks, and \( \Sigma \) is constrained to +be a diagonal matrix. The name variational "autoencoder" comes from +the fact that \( \mu \) and \( \Sigma \) are "encoding" \( X \) into the latent +space \( z \). The advantages of this choice are computational, as they +make it clear how to compute the right hand side. The last +term---\( \mathcal{D}\left[Q(z|X)\|P(z)\right] \)---is now a KL-divergence +between two multivariate Gaussian distributions, which can be computed +in closed form as: +

+$$ +\begin{array}{c} + \mathcal{D}[\mathcal{N}(\mu_0,\Sigma_0) \| \mathcal{N}(\mu_1,\Sigma_1)] = \hspace{20em}\\ + \hspace{5em}\frac{ 1 }{ 2 } \left( \mathrm{tr} \left( \Sigma_1^{-1} \Sigma_0 \right) + \left( \mu_1 - \mu_0\right)^\top \Sigma_1^{-1} ( \mu_1 - \mu_0 ) - k + \log \left( \frac{ \det \Sigma_1 }{ \det \Sigma_0 } \right) \right) +\end{array} +$$ + +

where \( k \) is the dimensionality of the distribution. In our case, this simplifies to:

+$$ +\begin{array}{c} + \mathcal{D}[\mathcal{N}(\mu(X),\Sigma(X)) \| \mathcal{N}(0,I)] = \hspace{20em}\\ +\hspace{6em}\frac{ 1 }{ 2 } \left( \mathrm{tr} \left( \Sigma(X) \right) + \left( \mu(X)\right)^\top ( \mu(X) ) - k - \log\det\left( \Sigma(X) \right) \right). +\end{array} +$$ + +

The first term on the right hand side is a bit more tricky. +We could use sampling to estimate \( E_{z\sim Q}\left[\log P(X|z) \right] \), but getting a good estimate would require passing many samples of \( z \) through \( f \), which would be expensive. +Hence, as is standard in stochastic gradient descent, we take one sample of \( z \) and treat \( \log P(X|z) \) for that \( z \) as an approximation of \( E_{z\sim Q}\left[\log P(X|z) \right] \). +After all, we are already doing stochastic gradient descent over different values of \( X \) sampled from a dataset \( D \). +The full equation we want to optimize is: +

+ +$$ +\begin{array}{c} + E_{X\sim D}\left[\log P(X) - \mathcal{D}\left[Q(z|X)\|P(z|X)\right]\right]=\hspace{16em}\\ +\hspace{10em}E_{X\sim D}\left[E_{z\sim Q}\left[\log P(X|z) \right] - \mathcal{D}\left[Q(z|X)\|P(z)\right]\right]. +\end{array} +$$ + +

If we take the gradient of this equation, the gradient symbol can be moved into the expectations. +Therefore, we can sample a single value of \( X \) and a single value of \( z \) from the distribution \( Q(z|X) \), and compute the gradient of: +

+$$ +\begin{equation} + \log P(X|z)-\mathcal{D}\left[Q(z|X)\|P(z)\right]. +\label{_auto1} +\end{equation} +$$ + +

We can then average the gradient of this function over arbitrarily many samples of \( X \) and \( z \), and the result converges to the gradient.

+ +

There is, however, a significant problem +\( E_{z\sim Q}\left[\log P(X|z) \right] \) depends not just on the parameters of \( P \), but also on the parameters of \( Q \). +

+ +

In order to make VAEs work, it is essential to drive \( Q \) to produce codes for \( X \) that \( P \) can reliably decode.

+$$ + E_{X\sim D}\left[E_{\epsilon\sim\mathcal{N}(0,I)}[\log P(X|z=\mu(X)+\Sigma^{1/2}(X)*\epsilon)]-\mathcal{D}\left[Q(z|X)\|P(z)\right]\right]. +$$ +
diff --git a/doc/pub/week13/html/week13-reveal.html b/doc/pub/week13/html/week13-reveal.html index eb607433..0402a66c 100644 --- a/doc/pub/week13/html/week13-reveal.html +++ b/doc/pub/week13/html/week13-reveal.html @@ -209,12 +209,15 @@

Plans for the week April 15-19, 202

+ +
+

Readings

-Readings +

    -

  1. Reading recommendation: Goodfellow et al chapter 20.10-20-14
    2. +

  2. Reading recommendation: Goodfellow et al, for VAEs and GANs see sections 20.10-20.11

  3. To create Boltzmann machine using Keras, see Babcock and Bali chapter 4, see https://github.com/PacktPublishing/Hands-On-Generative-AI-with-Python-and-TensorFlow-2/blob/master/Chapter_4/models/rbm.py

  4. See Foster, chapter 7 on energy-based models at https://github.com/davidADSP/Generative_Deep_Learning_2nd_Edition/tree/main/notebooks/07_ebm/01_ebm
@@ -381,6 +384,12 @@

Energy-based models and Lange

That notebook is based on a recent article by Du and Mordatch, Implicit generation and modeling with energy-based models, see https://arxiv.org/pdf/1903.08689.pdf.

+
+

Langevin sampling

+ +

Note: Notes to be added

+
+

Theory of Variational Autoencoders

@@ -412,17 +421,129 @@

The Autoencoder again

$$

 
-

The goal is to minimize the construction error.

+

The goal is to minimize the construction error, often done by optimizing the means squared error.

Schematic image of an Autoencoder

- +

+
+

+
+

+
+ +
+

Mathematics of Variational Autoencoders

+ +

We have defined earlier a probability (marginal) distribution with hidden variables \( \boldsymbol{h} \) and parameters \( \boldsymbol{\Theta} \) as

+

 
+$$ +p(\boldsymbol{x};\boldsymbol{\Theta}) = \int d\boldsymbol{h}p(\boldsymbol{x},\boldsymbol{h};\boldsymbol{\Theta}), +$$ +

 
+ +

for continuous variables \( \boldsymbol{h} \) and

+

 
+$$ +p(\boldsymbol{x};\boldsymbol{\Theta}) = \sum_{\boldsymbol{h}}p(\boldsymbol{x},\boldsymbol{h};\boldsymbol{\Theta}), +$$ +

 
+ +

for discrete stochastic events \( \boldsymbol{h} \). The variables \( \boldsymbol{h} \) are normally called the latent variables in the theory of autoencoders. We will also call then for that here.

+
+ +
+

Using the conditional probability

+ +

Using the the definition of the conditional probabilities \( p(\boldsymbol{x}\vert\boldsymbol{h};\boldsymbol{\Theta}) \), \( p(\boldsymbol{h}\vert\boldsymbol{x};\boldsymbol{\Theta}) \) and +and the prior \( p(\boldsymbol{h}) \), we can rewrite the above equation as +

+

 
+$$ +p(\boldsymbol{x};\boldsymbol{\Theta}) = \sum_{\boldsymbol{h}}p(\boldsymbol{x}\vert\boldsymbol{h};\boldsymbol{\Theta})p(\boldsymbol{h}, +$$ +

 
+ +

which allows us to make the dependence of \( \boldsymbol{x} \) on \( \boldsymbol{h} \) +explicit by using the law of total probability. The intuition behind +this approach for finding the marginal probability for \( \boldsymbol{x} \) is to +optimize the above equations with respect to the parameters +\( \boldsymbol{\Theta} \). This is done normally by maximizing the probability, +the so-called maximum-likelihood approach discussed earlier. +

+
+ +
+

VAEs versus autoencoders

+ +

This trained probability is assumed to be able to produce similar +samples as the input. In VAEs it is then common to compare via for +example the mean-squared error or the cross-entropy the predicted +values with the input values. Compared with autoencoders, we are now +producing a probability instead of a functions which mimicks the +input. +

+ +

In VAEs, the choice of this output distribution is often Gaussian, +meaning that the conditional probability is +

+

 
+$$ +p(\boldsymbol{x}\vert\boldsymbol{h};\boldsymbol{\Theta})=N(\boldsymbol{x}\vert f(\boldsymbol{h};\boldsymbol{\Theta}), \sigma^2\times \boldsymbol{I}), +$$ +

 
+ +

with mean value given by the function \( f(\boldsymbol{h};\boldsymbol{\Theta}) \) and a +diagonal covariance matrix multiplied by a parameter \( \sigma^2 \) which +is treated as a hyperparameter. +

+
+ +
+

Gradient descent

+ +

By having a Gaussian distribution, we can use gradient descent (or any +other optimization technique) to increase \( p(\boldsymbol{x};\boldsymbol{\Theta}) \) by +making \( f(\boldsymbol{h};\boldsymbol{\Theta}) \) approach \( \boldsymbol{x} \) for some \( \boldsymbol{h} \), +gradually making the training data more likely under the generative +model. The important property is simply that the marginal probability +can be computed, and it is continuous in \( \boldsymbol{\Theta} \).. +

+
+ +
+

Are VAEs just modified autoencoders?

+ +

The mathematical basis of VAEs actually has relatively little to do +with classical autoencoders, for example the sparse autoencoders or +denoising autoencoders discussed earlier. +

+ +

VAEs approximately maximize the probability equation discussed +above. They are called autoencoders only because the final training +objective that derives from this setup does have an encoder and a +decoder, and resembles a traditional autoencoder. Unlike sparse +autoencoders, there are generally no tuning parameters analogous to +the sparsity penalties. And unlike sparse and denoising autoencoders, +we can sample directly from \( p(\boldsymbol{x}) \) without performing Markov +Chain Monte Carlo. +

+
+ +
+

Training VAEs

+ +

To solve the integral or sum for \( p(\boldsymbol{x}) \), there are two problems +that VAEs must deal with: how to define the latent variables \( \boldsymbol{h} \), +that is decide what information they represent, and how to deal with +the integral over \( \boldsymbol{h} \). VAEs give a definite answer to both. +

-

Kullback-Leibler relative entropy

+

Kullback-Leibler relative entropy (notation to be updated)

When the goal of the training is to approximate a probability distribution, as it is in generative modeling, another relevant @@ -560,10 +681,14 @@

Adding hyperparameters

it is limited to the training data. Hence, in unsupervised training as well it is important to prevent overfitting to the training data. Thus it is common to add regularizers to the cost function in the same -manner as we discussed for say linear regression. +manner as discussed for say linear regression.

+
+

Back to VAEs

+
+

Code examples using Keras

@@ -803,10 +928,7 @@

Zero-sum game

 
$$ -\begin{equation} -v(\theta^{(g)}, \theta^{(d)}) -\tag{1} -\end{equation} $$

 

@@ -1731,6 +1853,186 @@

Exploring the Latent Space

+
+

Using the KL divergence

+ +

In practice, for most \( \boldsymbol{h} \), \( p(\boldsymbol{x}\vert \boldsymbol{h}; \boldsymbol{\Theta}) \) +will be nearly zero, and hence contribute almost nothing to our +estimate of \( p(\boldsymbol{x}) \). +

+ +

The key idea behind the variational autoencoder is to attempt to +sample values of \( \boldsymbol{h} \) that are likely to have produced \( \boldsymbol{x} \), +and compute \( p(\boldsymbol{x}) \) just from those. +

+ +

This means that we need a new function \( Q(\boldsymbol{h}|\boldsymbol{x}) \) which can +take a value of \( \boldsymbol{x} \) and give us a distribution over \( \boldsymbol{h} \) +values that are likely to produce \( \boldsymbol{x} \). Hopefully the space of +\( \boldsymbol{h} \) values that are likely under \( Q \) will be much smaller than +the space of all \( \boldsymbol{h} \)'s that are likely under the prior +\( p(\boldsymbol{h}) \). This lets us, for example, compute \( E_{\boldsymbol{h}\sim +Q}p(\boldsymbol{x}\vert \boldsymbol{h}) \) relatively easily. Note that we drop +\( \boldsymbol{\Theta} \) from here and for notational simplicity. +

+ +

However, if \( \boldsymbol{h} \) is sampled from an arbitrary distribution with +PDF \( Q(\boldsymbol{h}) \), which is not \( \mathcal{N}(0,I) \), then how does that +help us optimize \( p(\boldsymbol{x}) \)? The first thing we need to do is relate +\( E_{\boldsymbol{h}\sim Q}P(\boldsymbol{x}\vert \boldsymbol{h}) \) and \( p(\boldsymbol{x}) \). We will see where \( Q \) comes from later. +

+ +

The relationship between \( E_{\boldsymbol{h}\sim Q}p(\boldsymbol{x}\vert \boldsymbol{h}) \) and \( p(\boldsymbol{x}) \) is one of the cornerstones of variational Bayesian methods. +We begin with the definition of Kullback-Leibler divergence (KL divergence or \( \mathcal{D} \)) between \( p(\boldsymbol{h}\vert \boldsymbol{x}) \) and \( Q(\boldsymbol{h}) \), for some arbitrary \( Q \) (which may or may not depend on \( \boldsymbol{x} \)): +

+

 
+$$ + \mathcal{D}\left[Q(z)\|P(z|X)\right]=E_{z\sim Q}\left[\log Q(z) - \log P(z|X) \right]. +$$ +

 
+ +

We can get both \( p(\boldsymbol{x}) \) and \( p(\boldsymbol{x}\vert \boldsymbol{h}) \) into this equation by applying Bayes rule to \( p(\boldsymbol{h}|\boldsymbol{x}) \)

+

 
+$$ + \mathcal{D}\left[Q(z)\|P(z|X)\right]=E_{z\sim Q}\left[\log Q(z) - \log P(X|z) - \log P(z) \right] + \log P(X). +$$ +

 
+ +

Here, \( \log P(X) \) comes out of the expectation because it does not depend on \( z \). +Negating both sides, rearranging, and contracting part of \( E_{z\sim Q} \) into a KL-divergence terms yields: +

+

 
+$$ +\log P(X) - \mathcal{D}\left[Q(z)\|P(z|X)\right]=E_{z\sim Q}\left[\log P(X|z) \right] - \mathcal{D}\left[Q(z)\|P(z)\right]. +$$ +

 
+ +

By Bayes rule, we have:

+

 
+$$ +E_{z\sim Q}\left[\log P(Y_i|z,X_i)\right]=E_{z\sim Q}\left[\log P(z|Y_i,X_i) - \log P(z|X_i) + \log P(Y_i|X_i) \right] +$$ +

 
+ +

Rearranging the terms and subtracting \( E_{z\sim Q}\log Q(z) \) from both sides:

+

 
+$$ +\begin{array}{c} +\log P(Y_i|X_i) - E_{z\sim Q}\left[\log Q(z)-\log P(z|X_i,Y_i)\right]=\hspace{10em}\\ +\hspace{10em}E_{z\sim Q}\left[\log P(Y_i|z,X_i)+\log P(z|X_i)-\log Q(z)\right] +\end{array} +$$ +

 
+ +

Note that \( X \) is fixed, and \( Q \) can be \textit{any} distribution, not +just a distribution which does a good job mapping \( X \) to the \( z \)'s +that can produce \( X \). +

+ +

Since we are interested in inferring \( P(X) \), it makes sense to +construct a \( Q \) which \textit{does} depend on \( X \), and in particular, +one which makes \( \mathcal{D}\left[Q(z)\|P(z|X)\right] \) small: +

+

 
+$$ +\log P(X) - \mathcal{D}\left[Q(z|X)\|P(z|X)\right]=E_{z\sim Q}\left[\log P(X|z) \right] - \mathcal{D}\left[Q(z|X)\|P(z)\right]. +$$ +

 
+ +

Hence, during training, it makes sense to choose a \( Q \) which will make +\( E_{z\sim Q}[\log Q(z)- \) $\log P(z|X_i,Y_i)]$ (a +\( \mathcal{D} \)-divergence) small, such that the right hand side is a +close approximation to \( \log P(Y_i|X_i) \). +

+ +

This equation serves as the core of the variational autoencoder, and +it's worth spending some time thinking about what it says In two +sentences, the left hand side has the quantity we want to maximize: +\( \log P(X) \) (plus an error term, which makes \( Q \) produce \( z \)'s that +can reproduce a given \( X \); this term will become small if \( Q \) is +high-capacity). The right hand side is something we can optimize via +stochastic gradient descent given the right choice of \( Q \) (although it +may not be obvious yet how). +

+ +

So how can we perform stochastic gradient descent?

+ +

First we need to be a bit more specific about the form that \( Q(z|X) \) +will take. The usual choice is to say that +\( Q(z|X)=\mathcal{N}(z|\mu(X;\vartheta),\Sigma(X;\vartheta)) \), where +\( \mu \) and \( \Sigma \) are arbitrary deterministic functions with +parameters \( \vartheta \) that can be learned from data (we will omit +\( \vartheta \) in later equations). In practice, \( \mu \) and \( \Sigma \) are +again implemented via neural networks, and \( \Sigma \) is constrained to +be a diagonal matrix. The name variational "autoencoder" comes from +the fact that \( \mu \) and \( \Sigma \) are "encoding" \( X \) into the latent +space \( z \). The advantages of this choice are computational, as they +make it clear how to compute the right hand side. The last +term---\( \mathcal{D}\left[Q(z|X)\|P(z)\right] \)---is now a KL-divergence +between two multivariate Gaussian distributions, which can be computed +in closed form as: +

+

 
+$$ +\begin{array}{c} + \mathcal{D}[\mathcal{N}(\mu_0,\Sigma_0) \| \mathcal{N}(\mu_1,\Sigma_1)] = \hspace{20em}\\ + \hspace{5em}\frac{ 1 }{ 2 } \left( \mathrm{tr} \left( \Sigma_1^{-1} \Sigma_0 \right) + \left( \mu_1 - \mu_0\right)^\top \Sigma_1^{-1} ( \mu_1 - \mu_0 ) - k + \log \left( \frac{ \det \Sigma_1 }{ \det \Sigma_0 } \right) \right) +\end{array} +$$ +

 
+ +

where \( k \) is the dimensionality of the distribution. In our case, this simplifies to:

+

 
+$$ +\begin{array}{c} + \mathcal{D}[\mathcal{N}(\mu(X),\Sigma(X)) \| \mathcal{N}(0,I)] = \hspace{20em}\\ +\hspace{6em}\frac{ 1 }{ 2 } \left( \mathrm{tr} \left( \Sigma(X) \right) + \left( \mu(X)\right)^\top ( \mu(X) ) - k - \log\det\left( \Sigma(X) \right) \right). +\end{array} +$$ +

 
+ +

The first term on the right hand side is a bit more tricky. +We could use sampling to estimate \( E_{z\sim Q}\left[\log P(X|z) \right] \), but getting a good estimate would require passing many samples of \( z \) through \( f \), which would be expensive. +Hence, as is standard in stochastic gradient descent, we take one sample of \( z \) and treat \( \log P(X|z) \) for that \( z \) as an approximation of \( E_{z\sim Q}\left[\log P(X|z) \right] \). +After all, we are already doing stochastic gradient descent over different values of \( X \) sampled from a dataset \( D \). +The full equation we want to optimize is: +

+ +

 
+$$ +\begin{array}{c} + E_{X\sim D}\left[\log P(X) - \mathcal{D}\left[Q(z|X)\|P(z|X)\right]\right]=\hspace{16em}\\ +\hspace{10em}E_{X\sim D}\left[E_{z\sim Q}\left[\log P(X|z) \right] - \mathcal{D}\left[Q(z|X)\|P(z)\right]\right]. +\end{array} +$$ +

 
+ +

If we take the gradient of this equation, the gradient symbol can be moved into the expectations. +Therefore, we can sample a single value of \( X \) and a single value of \( z \) from the distribution \( Q(z|X) \), and compute the gradient of: +

+

 
+$$ +\begin{equation} + \log P(X|z)-\mathcal{D}\left[Q(z|X)\|P(z)\right]. +\tag{1} +\end{equation} +$$ +

 
+ +

We can then average the gradient of this function over arbitrarily many samples of \( X \) and \( z \), and the result converges to the gradient.

+ +

There is, however, a significant problem +\( E_{z\sim Q}\left[\log P(X|z) \right] \) depends not just on the parameters of \( P \), but also on the parameters of \( Q \). +

+ +

In order to make VAEs work, it is essential to drive \( Q \) to produce codes for \( X \) that \( P \) can reliably decode.

+

 
+$$ + E_{X\sim D}\left[E_{\epsilon\sim\mathcal{N}(0,I)}[\log P(X|z=\mu(X)+\Sigma^{1/2}(X)*\epsilon)]-\mathcal{D}\left[Q(z|X)\|P(z)\right]\right]. +$$ +

 
+

+ diff --git a/doc/pub/week13/html/week13-solarized.html b/doc/pub/week13/html/week13-solarized.html index c53b24d2..c83dfcec 100644 --- a/doc/pub/week13/html/week13-solarized.html +++ b/doc/pub/week13/html/week13-solarized.html @@ -67,6 +67,7 @@ 2, None, 'plans-for-the-week-april-15-19-2024'), + ('Readings', 2, None, 'readings'), ('Reminder from last week and layout of lecture this week', 2, None, @@ -83,6 +84,7 @@ 2, None, 'energy-based-models-and-langevin-sampling'), + ('Langevin sampling', 2, None, 'langevin-sampling'), ('Theory of Variational Autoencoders', 2, None, @@ -92,10 +94,25 @@ 2, None, 'schematic-image-of-an-autoencoder'), - ('Kullback-Leibler relative entropy', + ('Mathematics of Variational Autoencoders', 2, None, - 'kullback-leibler-relative-entropy'), + 'mathematics-of-variational-autoencoders'), + ('Using the conditional probability', + 2, + None, + 'using-the-conditional-probability'), + ('VAEs versus autoencoders', 2, None, 'vaes-versus-autoencoders'), + ('Gradient descent', 2, None, 'gradient-descent'), + ('Are VAEs just modified autoencoders?', + 2, + None, + 'are-vaes-just-modified-autoencoders'), + ('Training VAEs', 2, None, 'training-vaes'), + ('Kullback-Leibler relative entropy (notation to be updated)', + 2, + None, + 'kullback-leibler-relative-entropy-notation-to-be-updated'), ('Kullback-Leibler divergence', 2, None, @@ -115,6 +132,7 @@ ('Difference of moments', 2, None, 'difference-of-moments'), ('More observations', 2, None, 'more-observations'), ('Adding hyperparameters', 2, None, 'adding-hyperparameters'), + ('Back to VAEs', 2, None, 'back-to-vaes'), ('Code examples using Keras', 2, None, @@ -148,7 +166,8 @@ ('Exploring the Latent Space', 2, None, - 'exploring-the-latent-space')]} + 'exploring-the-latent-space'), + ('Using the KL divergence', 2, None, 'using-the-kl-divergence')]} end of tocinfo --> @@ -206,11 +225,14 @@

Plans for the week April 15-19, 202 + +









+

Readings

-Readings +

    -
  1. Reading recommendation: Goodfellow et al chapter 20.10-20-14
    2. +
  2. Reading recommendation: Goodfellow et al, for VAEs and GANs see sections 20.10-20.11
  3. To create Boltzmann machine using Keras, see Babcock and Bali chapter 4, see https://github.com/PacktPublishing/Hands-On-Generative-AI-with-Python-and-TensorFlow-2/blob/master/Chapter_4/models/rbm.py
  4. See Foster, chapter 7 on energy-based models at https://github.com/davidADSP/Generative_Deep_Learning_2nd_Edition/tree/main/notebooks/07_ebm/01_ebm
@@ -371,6 +393,11 @@

Energy-based models and Lange

That notebook is based on a recent article by Du and Mordatch, Implicit generation and modeling with energy-based models, see https://arxiv.org/pdf/1903.08689.pdf.

+









+

Langevin sampling

+ +

Note: Notes to be added

+









Theory of Variational Autoencoders

@@ -397,15 +424,113 @@

The Autoencoder again

\tilde{\boldsymbol{x}} = g(\boldsymbol{h},\boldsymbol{V})). $$ -

The goal is to minimize the construction error.

+

The goal is to minimize the construction error, often done by optimizing the means squared error.











Schematic image of an Autoencoder

- +

+
+

+
+

+ +









+

Mathematics of Variational Autoencoders

+ +

We have defined earlier a probability (marginal) distribution with hidden variables \( \boldsymbol{h} \) and parameters \( \boldsymbol{\Theta} \) as

+$$ +p(\boldsymbol{x};\boldsymbol{\Theta}) = \int d\boldsymbol{h}p(\boldsymbol{x},\boldsymbol{h};\boldsymbol{\Theta}), +$$ + +

for continuous variables \( \boldsymbol{h} \) and

+$$ +p(\boldsymbol{x};\boldsymbol{\Theta}) = \sum_{\boldsymbol{h}}p(\boldsymbol{x},\boldsymbol{h};\boldsymbol{\Theta}), +$$ + +

for discrete stochastic events \( \boldsymbol{h} \). The variables \( \boldsymbol{h} \) are normally called the latent variables in the theory of autoencoders. We will also call then for that here.

+ +









+

Using the conditional probability

+ +

Using the the definition of the conditional probabilities \( p(\boldsymbol{x}\vert\boldsymbol{h};\boldsymbol{\Theta}) \), \( p(\boldsymbol{h}\vert\boldsymbol{x};\boldsymbol{\Theta}) \) and +and the prior \( p(\boldsymbol{h}) \), we can rewrite the above equation as +

+$$ +p(\boldsymbol{x};\boldsymbol{\Theta}) = \sum_{\boldsymbol{h}}p(\boldsymbol{x}\vert\boldsymbol{h};\boldsymbol{\Theta})p(\boldsymbol{h}, +$$ + +

which allows us to make the dependence of \( \boldsymbol{x} \) on \( \boldsymbol{h} \) +explicit by using the law of total probability. The intuition behind +this approach for finding the marginal probability for \( \boldsymbol{x} \) is to +optimize the above equations with respect to the parameters +\( \boldsymbol{\Theta} \). This is done normally by maximizing the probability, +the so-called maximum-likelihood approach discussed earlier. +

+ +









+

VAEs versus autoencoders

+ +

This trained probability is assumed to be able to produce similar +samples as the input. In VAEs it is then common to compare via for +example the mean-squared error or the cross-entropy the predicted +values with the input values. Compared with autoencoders, we are now +producing a probability instead of a functions which mimicks the +input. +

+ +

In VAEs, the choice of this output distribution is often Gaussian, +meaning that the conditional probability is +

+$$ +p(\boldsymbol{x}\vert\boldsymbol{h};\boldsymbol{\Theta})=N(\boldsymbol{x}\vert f(\boldsymbol{h};\boldsymbol{\Theta}), \sigma^2\times \boldsymbol{I}), +$$ + +

with mean value given by the function \( f(\boldsymbol{h};\boldsymbol{\Theta}) \) and a +diagonal covariance matrix multiplied by a parameter \( \sigma^2 \) which +is treated as a hyperparameter. +

+ +









+

Gradient descent

+ +

By having a Gaussian distribution, we can use gradient descent (or any +other optimization technique) to increase \( p(\boldsymbol{x};\boldsymbol{\Theta}) \) by +making \( f(\boldsymbol{h};\boldsymbol{\Theta}) \) approach \( \boldsymbol{x} \) for some \( \boldsymbol{h} \), +gradually making the training data more likely under the generative +model. The important property is simply that the marginal probability +can be computed, and it is continuous in \( \boldsymbol{\Theta} \).. +

+ +









+

Are VAEs just modified autoencoders?

+ +

The mathematical basis of VAEs actually has relatively little to do +with classical autoencoders, for example the sparse autoencoders or +denoising autoencoders discussed earlier. +

+ +

VAEs approximately maximize the probability equation discussed +above. They are called autoencoders only because the final training +objective that derives from this setup does have an encoder and a +decoder, and resembles a traditional autoencoder. Unlike sparse +autoencoders, there are generally no tuning parameters analogous to +the sparsity penalties. And unlike sparse and denoising autoencoders, +we can sample directly from \( p(\boldsymbol{x}) \) without performing Markov +Chain Monte Carlo. +

+ +









+

Training VAEs

+ +

To solve the integral or sum for \( p(\boldsymbol{x}) \), there are two problems +that VAEs must deal with: how to define the latent variables \( \boldsymbol{h} \), +that is decide what information they represent, and how to deal with +the integral over \( \boldsymbol{h} \). VAEs give a definite answer to both. +











-

Kullback-Leibler relative entropy

+

Kullback-Leibler relative entropy (notation to be updated)

When the goal of the training is to approximate a probability distribution, as it is in generative modeling, another relevant @@ -531,9 +656,12 @@

Adding hyperparameters

it is limited to the training data. Hence, in unsupervised training as well it is important to prevent overfitting to the training data. Thus it is common to add regularizers to the cost function in the same -manner as we discussed for say linear regression. +manner as discussed for say linear regression.

+









+

Back to VAEs

+









Code examples using Keras

@@ -759,10 +887,7 @@

Zero-sum game

$$ -\begin{equation} -v(\theta^{(g)}, \theta^{(d)}) -\label{_auto1} -\end{equation} $$ @@ -1675,6 +1800,164 @@

Exploring the Latent Space

A pretty cool result! We see that our generator indeed has learned a distribution which qualitatively looks a whole lot like the MNIST dataset.

+ +









+

Using the KL divergence

+ +

In practice, for most \( \boldsymbol{h} \), \( p(\boldsymbol{x}\vert \boldsymbol{h}; \boldsymbol{\Theta}) \) +will be nearly zero, and hence contribute almost nothing to our +estimate of \( p(\boldsymbol{x}) \). +

+ +

The key idea behind the variational autoencoder is to attempt to +sample values of \( \boldsymbol{h} \) that are likely to have produced \( \boldsymbol{x} \), +and compute \( p(\boldsymbol{x}) \) just from those. +

+ +

This means that we need a new function \( Q(\boldsymbol{h}|\boldsymbol{x}) \) which can +take a value of \( \boldsymbol{x} \) and give us a distribution over \( \boldsymbol{h} \) +values that are likely to produce \( \boldsymbol{x} \). Hopefully the space of +\( \boldsymbol{h} \) values that are likely under \( Q \) will be much smaller than +the space of all \( \boldsymbol{h} \)'s that are likely under the prior +\( p(\boldsymbol{h}) \). This lets us, for example, compute \( E_{\boldsymbol{h}\sim +Q}p(\boldsymbol{x}\vert \boldsymbol{h}) \) relatively easily. Note that we drop +\( \boldsymbol{\Theta} \) from here and for notational simplicity. +

+ +

However, if \( \boldsymbol{h} \) is sampled from an arbitrary distribution with +PDF \( Q(\boldsymbol{h}) \), which is not \( \mathcal{N}(0,I) \), then how does that +help us optimize \( p(\boldsymbol{x}) \)? The first thing we need to do is relate +\( E_{\boldsymbol{h}\sim Q}P(\boldsymbol{x}\vert \boldsymbol{h}) \) and \( p(\boldsymbol{x}) \). We will see where \( Q \) comes from later. +

+ +

The relationship between \( E_{\boldsymbol{h}\sim Q}p(\boldsymbol{x}\vert \boldsymbol{h}) \) and \( p(\boldsymbol{x}) \) is one of the cornerstones of variational Bayesian methods. +We begin with the definition of Kullback-Leibler divergence (KL divergence or \( \mathcal{D} \)) between \( p(\boldsymbol{h}\vert \boldsymbol{x}) \) and \( Q(\boldsymbol{h}) \), for some arbitrary \( Q \) (which may or may not depend on \( \boldsymbol{x} \)): +

+$$ + \mathcal{D}\left[Q(z)\|P(z|X)\right]=E_{z\sim Q}\left[\log Q(z) - \log P(z|X) \right]. +$$ + +

We can get both \( p(\boldsymbol{x}) \) and \( p(\boldsymbol{x}\vert \boldsymbol{h}) \) into this equation by applying Bayes rule to \( p(\boldsymbol{h}|\boldsymbol{x}) \)

+$$ + \mathcal{D}\left[Q(z)\|P(z|X)\right]=E_{z\sim Q}\left[\log Q(z) - \log P(X|z) - \log P(z) \right] + \log P(X). +$$ + +

Here, \( \log P(X) \) comes out of the expectation because it does not depend on \( z \). +Negating both sides, rearranging, and contracting part of \( E_{z\sim Q} \) into a KL-divergence terms yields: +

+$$ +\log P(X) - \mathcal{D}\left[Q(z)\|P(z|X)\right]=E_{z\sim Q}\left[\log P(X|z) \right] - \mathcal{D}\left[Q(z)\|P(z)\right]. +$$ + +

By Bayes rule, we have:

+$$ +E_{z\sim Q}\left[\log P(Y_i|z,X_i)\right]=E_{z\sim Q}\left[\log P(z|Y_i,X_i) - \log P(z|X_i) + \log P(Y_i|X_i) \right] +$$ + +

Rearranging the terms and subtracting \( E_{z\sim Q}\log Q(z) \) from both sides:

+$$ +\begin{array}{c} +\log P(Y_i|X_i) - E_{z\sim Q}\left[\log Q(z)-\log P(z|X_i,Y_i)\right]=\hspace{10em}\\ +\hspace{10em}E_{z\sim Q}\left[\log P(Y_i|z,X_i)+\log P(z|X_i)-\log Q(z)\right] +\end{array} +$$ + +

Note that \( X \) is fixed, and \( Q \) can be \textit{any} distribution, not +just a distribution which does a good job mapping \( X \) to the \( z \)'s +that can produce \( X \). +

+ +

Since we are interested in inferring \( P(X) \), it makes sense to +construct a \( Q \) which \textit{does} depend on \( X \), and in particular, +one which makes \( \mathcal{D}\left[Q(z)\|P(z|X)\right] \) small: +

+$$ +\log P(X) - \mathcal{D}\left[Q(z|X)\|P(z|X)\right]=E_{z\sim Q}\left[\log P(X|z) \right] - \mathcal{D}\left[Q(z|X)\|P(z)\right]. +$$ + +

Hence, during training, it makes sense to choose a \( Q \) which will make +\( E_{z\sim Q}[\log Q(z)- \) $\log P(z|X_i,Y_i)]$ (a +\( \mathcal{D} \)-divergence) small, such that the right hand side is a +close approximation to \( \log P(Y_i|X_i) \). +

+ +

This equation serves as the core of the variational autoencoder, and +it's worth spending some time thinking about what it says In two +sentences, the left hand side has the quantity we want to maximize: +\( \log P(X) \) (plus an error term, which makes \( Q \) produce \( z \)'s that +can reproduce a given \( X \); this term will become small if \( Q \) is +high-capacity). The right hand side is something we can optimize via +stochastic gradient descent given the right choice of \( Q \) (although it +may not be obvious yet how). +

+ +

So how can we perform stochastic gradient descent?

+ +

First we need to be a bit more specific about the form that \( Q(z|X) \) +will take. The usual choice is to say that +\( Q(z|X)=\mathcal{N}(z|\mu(X;\vartheta),\Sigma(X;\vartheta)) \), where +\( \mu \) and \( \Sigma \) are arbitrary deterministic functions with +parameters \( \vartheta \) that can be learned from data (we will omit +\( \vartheta \) in later equations). In practice, \( \mu \) and \( \Sigma \) are +again implemented via neural networks, and \( \Sigma \) is constrained to +be a diagonal matrix. The name variational "autoencoder" comes from +the fact that \( \mu \) and \( \Sigma \) are "encoding" \( X \) into the latent +space \( z \). The advantages of this choice are computational, as they +make it clear how to compute the right hand side. The last +term---\( \mathcal{D}\left[Q(z|X)\|P(z)\right] \)---is now a KL-divergence +between two multivariate Gaussian distributions, which can be computed +in closed form as: +

+$$ +\begin{array}{c} + \mathcal{D}[\mathcal{N}(\mu_0,\Sigma_0) \| \mathcal{N}(\mu_1,\Sigma_1)] = \hspace{20em}\\ + \hspace{5em}\frac{ 1 }{ 2 } \left( \mathrm{tr} \left( \Sigma_1^{-1} \Sigma_0 \right) + \left( \mu_1 - \mu_0\right)^\top \Sigma_1^{-1} ( \mu_1 - \mu_0 ) - k + \log \left( \frac{ \det \Sigma_1 }{ \det \Sigma_0 } \right) \right) +\end{array} +$$ + +

where \( k \) is the dimensionality of the distribution. In our case, this simplifies to:

+$$ +\begin{array}{c} + \mathcal{D}[\mathcal{N}(\mu(X),\Sigma(X)) \| \mathcal{N}(0,I)] = \hspace{20em}\\ +\hspace{6em}\frac{ 1 }{ 2 } \left( \mathrm{tr} \left( \Sigma(X) \right) + \left( \mu(X)\right)^\top ( \mu(X) ) - k - \log\det\left( \Sigma(X) \right) \right). +\end{array} +$$ + +

The first term on the right hand side is a bit more tricky. +We could use sampling to estimate \( E_{z\sim Q}\left[\log P(X|z) \right] \), but getting a good estimate would require passing many samples of \( z \) through \( f \), which would be expensive. +Hence, as is standard in stochastic gradient descent, we take one sample of \( z \) and treat \( \log P(X|z) \) for that \( z \) as an approximation of \( E_{z\sim Q}\left[\log P(X|z) \right] \). +After all, we are already doing stochastic gradient descent over different values of \( X \) sampled from a dataset \( D \). +The full equation we want to optimize is: +

+ +$$ +\begin{array}{c} + E_{X\sim D}\left[\log P(X) - \mathcal{D}\left[Q(z|X)\|P(z|X)\right]\right]=\hspace{16em}\\ +\hspace{10em}E_{X\sim D}\left[E_{z\sim Q}\left[\log P(X|z) \right] - \mathcal{D}\left[Q(z|X)\|P(z)\right]\right]. +\end{array} +$$ + +

If we take the gradient of this equation, the gradient symbol can be moved into the expectations. +Therefore, we can sample a single value of \( X \) and a single value of \( z \) from the distribution \( Q(z|X) \), and compute the gradient of: +

+$$ +\begin{equation} + \log P(X|z)-\mathcal{D}\left[Q(z|X)\|P(z)\right]. +\label{_auto1} +\end{equation} +$$ + +

We can then average the gradient of this function over arbitrarily many samples of \( X \) and \( z \), and the result converges to the gradient.

+ +

There is, however, a significant problem +\( E_{z\sim Q}\left[\log P(X|z) \right] \) depends not just on the parameters of \( P \), but also on the parameters of \( Q \). +

+ +

In order to make VAEs work, it is essential to drive \( Q \) to produce codes for \( X \) that \( P \) can reliably decode.

+$$ + E_{X\sim D}\left[E_{\epsilon\sim\mathcal{N}(0,I)}[\log P(X|z=\mu(X)+\Sigma^{1/2}(X)*\epsilon)]-\mathcal{D}\left[Q(z|X)\|P(z)\right]\right]. +$$ +
© 1999-2024, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license diff --git a/doc/pub/week13/html/week13.html b/doc/pub/week13/html/week13.html index 5b2af4f6..dc941593 100644 --- a/doc/pub/week13/html/week13.html +++ b/doc/pub/week13/html/week13.html @@ -144,6 +144,7 @@ 2, None, 'plans-for-the-week-april-15-19-2024'), + ('Readings', 2, None, 'readings'), ('Reminder from last week and layout of lecture this week', 2, None, @@ -160,6 +161,7 @@ 2, None, 'energy-based-models-and-langevin-sampling'), + ('Langevin sampling', 2, None, 'langevin-sampling'), ('Theory of Variational Autoencoders', 2, None, @@ -169,10 +171,25 @@ 2, None, 'schematic-image-of-an-autoencoder'), - ('Kullback-Leibler relative entropy', + ('Mathematics of Variational Autoencoders', 2, None, - 'kullback-leibler-relative-entropy'), + 'mathematics-of-variational-autoencoders'), + ('Using the conditional probability', + 2, + None, + 'using-the-conditional-probability'), + ('VAEs versus autoencoders', 2, None, 'vaes-versus-autoencoders'), + ('Gradient descent', 2, None, 'gradient-descent'), + ('Are VAEs just modified autoencoders?', + 2, + None, + 'are-vaes-just-modified-autoencoders'), + ('Training VAEs', 2, None, 'training-vaes'), + ('Kullback-Leibler relative entropy (notation to be updated)', + 2, + None, + 'kullback-leibler-relative-entropy-notation-to-be-updated'), ('Kullback-Leibler divergence', 2, None, @@ -192,6 +209,7 @@ ('Difference of moments', 2, None, 'difference-of-moments'), ('More observations', 2, None, 'more-observations'), ('Adding hyperparameters', 2, None, 'adding-hyperparameters'), + ('Back to VAEs', 2, None, 'back-to-vaes'), ('Code examples using Keras', 2, None, @@ -225,7 +243,8 @@ ('Exploring the Latent Space', 2, None, - 'exploring-the-latent-space')]} + 'exploring-the-latent-space'), + ('Using the KL divergence', 2, None, 'using-the-kl-divergence')]} end of tocinfo --> @@ -283,11 +302,14 @@

Plans for the week April 15-19, 202

+ +









+

Readings

-Readings +

    -
  1. Reading recommendation: Goodfellow et al chapter 20.10-20-14
    2. +
  2. Reading recommendation: Goodfellow et al, for VAEs and GANs see sections 20.10-20.11
  3. To create Boltzmann machine using Keras, see Babcock and Bali chapter 4, see https://github.com/PacktPublishing/Hands-On-Generative-AI-with-Python-and-TensorFlow-2/blob/master/Chapter_4/models/rbm.py
  4. See Foster, chapter 7 on energy-based models at https://github.com/davidADSP/Generative_Deep_Learning_2nd_Edition/tree/main/notebooks/07_ebm/01_ebm
@@ -448,6 +470,11 @@

Energy-based models and Lange

That notebook is based on a recent article by Du and Mordatch, Implicit generation and modeling with energy-based models, see https://arxiv.org/pdf/1903.08689.pdf.

+









+

Langevin sampling

+ +

Note: Notes to be added

+









Theory of Variational Autoencoders

@@ -474,15 +501,113 @@

The Autoencoder again

\tilde{\boldsymbol{x}} = g(\boldsymbol{h},\boldsymbol{V})). $$ -

The goal is to minimize the construction error.

+

The goal is to minimize the construction error, often done by optimizing the means squared error.











Schematic image of an Autoencoder

- +

+
+

+
+

+ +









+

Mathematics of Variational Autoencoders

+ +

We have defined earlier a probability (marginal) distribution with hidden variables \( \boldsymbol{h} \) and parameters \( \boldsymbol{\Theta} \) as

+$$ +p(\boldsymbol{x};\boldsymbol{\Theta}) = \int d\boldsymbol{h}p(\boldsymbol{x},\boldsymbol{h};\boldsymbol{\Theta}), +$$ + +

for continuous variables \( \boldsymbol{h} \) and

+$$ +p(\boldsymbol{x};\boldsymbol{\Theta}) = \sum_{\boldsymbol{h}}p(\boldsymbol{x},\boldsymbol{h};\boldsymbol{\Theta}), +$$ + +

for discrete stochastic events \( \boldsymbol{h} \). The variables \( \boldsymbol{h} \) are normally called the latent variables in the theory of autoencoders. We will also call then for that here.

+ +









+

Using the conditional probability

+ +

Using the the definition of the conditional probabilities \( p(\boldsymbol{x}\vert\boldsymbol{h};\boldsymbol{\Theta}) \), \( p(\boldsymbol{h}\vert\boldsymbol{x};\boldsymbol{\Theta}) \) and +and the prior \( p(\boldsymbol{h}) \), we can rewrite the above equation as +

+$$ +p(\boldsymbol{x};\boldsymbol{\Theta}) = \sum_{\boldsymbol{h}}p(\boldsymbol{x}\vert\boldsymbol{h};\boldsymbol{\Theta})p(\boldsymbol{h}, +$$ + +

which allows us to make the dependence of \( \boldsymbol{x} \) on \( \boldsymbol{h} \) +explicit by using the law of total probability. The intuition behind +this approach for finding the marginal probability for \( \boldsymbol{x} \) is to +optimize the above equations with respect to the parameters +\( \boldsymbol{\Theta} \). This is done normally by maximizing the probability, +the so-called maximum-likelihood approach discussed earlier. +

+ +









+

VAEs versus autoencoders

+ +

This trained probability is assumed to be able to produce similar +samples as the input. In VAEs it is then common to compare via for +example the mean-squared error or the cross-entropy the predicted +values with the input values. Compared with autoencoders, we are now +producing a probability instead of a functions which mimicks the +input. +

+ +

In VAEs, the choice of this output distribution is often Gaussian, +meaning that the conditional probability is +

+$$ +p(\boldsymbol{x}\vert\boldsymbol{h};\boldsymbol{\Theta})=N(\boldsymbol{x}\vert f(\boldsymbol{h};\boldsymbol{\Theta}), \sigma^2\times \boldsymbol{I}), +$$ + +

with mean value given by the function \( f(\boldsymbol{h};\boldsymbol{\Theta}) \) and a +diagonal covariance matrix multiplied by a parameter \( \sigma^2 \) which +is treated as a hyperparameter. +

+ +









+

Gradient descent

+ +

By having a Gaussian distribution, we can use gradient descent (or any +other optimization technique) to increase \( p(\boldsymbol{x};\boldsymbol{\Theta}) \) by +making \( f(\boldsymbol{h};\boldsymbol{\Theta}) \) approach \( \boldsymbol{x} \) for some \( \boldsymbol{h} \), +gradually making the training data more likely under the generative +model. The important property is simply that the marginal probability +can be computed, and it is continuous in \( \boldsymbol{\Theta} \).. +

+ +









+

Are VAEs just modified autoencoders?

+ +

The mathematical basis of VAEs actually has relatively little to do +with classical autoencoders, for example the sparse autoencoders or +denoising autoencoders discussed earlier. +

+ +

VAEs approximately maximize the probability equation discussed +above. They are called autoencoders only because the final training +objective that derives from this setup does have an encoder and a +decoder, and resembles a traditional autoencoder. Unlike sparse +autoencoders, there are generally no tuning parameters analogous to +the sparsity penalties. And unlike sparse and denoising autoencoders, +we can sample directly from \( p(\boldsymbol{x}) \) without performing Markov +Chain Monte Carlo. +

+ +









+

Training VAEs

+ +

To solve the integral or sum for \( p(\boldsymbol{x}) \), there are two problems +that VAEs must deal with: how to define the latent variables \( \boldsymbol{h} \), +that is decide what information they represent, and how to deal with +the integral over \( \boldsymbol{h} \). VAEs give a definite answer to both. +











-

Kullback-Leibler relative entropy

+

Kullback-Leibler relative entropy (notation to be updated)

When the goal of the training is to approximate a probability distribution, as it is in generative modeling, another relevant @@ -608,9 +733,12 @@

Adding hyperparameters

it is limited to the training data. Hence, in unsupervised training as well it is important to prevent overfitting to the training data. Thus it is common to add regularizers to the cost function in the same -manner as we discussed for say linear regression. +manner as discussed for say linear regression.

+









+

Back to VAEs

+









Code examples using Keras

@@ -836,10 +964,7 @@

Zero-sum game

$$ -\begin{equation} -v(\theta^{(g)}, \theta^{(d)}) -\label{_auto1} -\end{equation} $$ @@ -1752,6 +1877,164 @@

Exploring the Latent Space

A pretty cool result! We see that our generator indeed has learned a distribution which qualitatively looks a whole lot like the MNIST dataset.

+ +









+

Using the KL divergence

+ +

In practice, for most \( \boldsymbol{h} \), \( p(\boldsymbol{x}\vert \boldsymbol{h}; \boldsymbol{\Theta}) \) +will be nearly zero, and hence contribute almost nothing to our +estimate of \( p(\boldsymbol{x}) \). +

+ +

The key idea behind the variational autoencoder is to attempt to +sample values of \( \boldsymbol{h} \) that are likely to have produced \( \boldsymbol{x} \), +and compute \( p(\boldsymbol{x}) \) just from those. +

+ +

This means that we need a new function \( Q(\boldsymbol{h}|\boldsymbol{x}) \) which can +take a value of \( \boldsymbol{x} \) and give us a distribution over \( \boldsymbol{h} \) +values that are likely to produce \( \boldsymbol{x} \). Hopefully the space of +\( \boldsymbol{h} \) values that are likely under \( Q \) will be much smaller than +the space of all \( \boldsymbol{h} \)'s that are likely under the prior +\( p(\boldsymbol{h}) \). This lets us, for example, compute \( E_{\boldsymbol{h}\sim +Q}p(\boldsymbol{x}\vert \boldsymbol{h}) \) relatively easily. Note that we drop +\( \boldsymbol{\Theta} \) from here and for notational simplicity. +

+ +

However, if \( \boldsymbol{h} \) is sampled from an arbitrary distribution with +PDF \( Q(\boldsymbol{h}) \), which is not \( \mathcal{N}(0,I) \), then how does that +help us optimize \( p(\boldsymbol{x}) \)? The first thing we need to do is relate +\( E_{\boldsymbol{h}\sim Q}P(\boldsymbol{x}\vert \boldsymbol{h}) \) and \( p(\boldsymbol{x}) \). We will see where \( Q \) comes from later. +

+ +

The relationship between \( E_{\boldsymbol{h}\sim Q}p(\boldsymbol{x}\vert \boldsymbol{h}) \) and \( p(\boldsymbol{x}) \) is one of the cornerstones of variational Bayesian methods. +We begin with the definition of Kullback-Leibler divergence (KL divergence or \( \mathcal{D} \)) between \( p(\boldsymbol{h}\vert \boldsymbol{x}) \) and \( Q(\boldsymbol{h}) \), for some arbitrary \( Q \) (which may or may not depend on \( \boldsymbol{x} \)): +

+$$ + \mathcal{D}\left[Q(z)\|P(z|X)\right]=E_{z\sim Q}\left[\log Q(z) - \log P(z|X) \right]. +$$ + +

We can get both \( p(\boldsymbol{x}) \) and \( p(\boldsymbol{x}\vert \boldsymbol{h}) \) into this equation by applying Bayes rule to \( p(\boldsymbol{h}|\boldsymbol{x}) \)

+$$ + \mathcal{D}\left[Q(z)\|P(z|X)\right]=E_{z\sim Q}\left[\log Q(z) - \log P(X|z) - \log P(z) \right] + \log P(X). +$$ + +

Here, \( \log P(X) \) comes out of the expectation because it does not depend on \( z \). +Negating both sides, rearranging, and contracting part of \( E_{z\sim Q} \) into a KL-divergence terms yields: +

+$$ +\log P(X) - \mathcal{D}\left[Q(z)\|P(z|X)\right]=E_{z\sim Q}\left[\log P(X|z) \right] - \mathcal{D}\left[Q(z)\|P(z)\right]. +$$ + +

By Bayes rule, we have:

+$$ +E_{z\sim Q}\left[\log P(Y_i|z,X_i)\right]=E_{z\sim Q}\left[\log P(z|Y_i,X_i) - \log P(z|X_i) + \log P(Y_i|X_i) \right] +$$ + +

Rearranging the terms and subtracting \( E_{z\sim Q}\log Q(z) \) from both sides:

+$$ +\begin{array}{c} +\log P(Y_i|X_i) - E_{z\sim Q}\left[\log Q(z)-\log P(z|X_i,Y_i)\right]=\hspace{10em}\\ +\hspace{10em}E_{z\sim Q}\left[\log P(Y_i|z,X_i)+\log P(z|X_i)-\log Q(z)\right] +\end{array} +$$ + +

Note that \( X \) is fixed, and \( Q \) can be \textit{any} distribution, not +just a distribution which does a good job mapping \( X \) to the \( z \)'s +that can produce \( X \). +

+ +

Since we are interested in inferring \( P(X) \), it makes sense to +construct a \( Q \) which \textit{does} depend on \( X \), and in particular, +one which makes \( \mathcal{D}\left[Q(z)\|P(z|X)\right] \) small: +

+$$ +\log P(X) - \mathcal{D}\left[Q(z|X)\|P(z|X)\right]=E_{z\sim Q}\left[\log P(X|z) \right] - \mathcal{D}\left[Q(z|X)\|P(z)\right]. +$$ + +

Hence, during training, it makes sense to choose a \( Q \) which will make +\( E_{z\sim Q}[\log Q(z)- \) $\log P(z|X_i,Y_i)]$ (a +\( \mathcal{D} \)-divergence) small, such that the right hand side is a +close approximation to \( \log P(Y_i|X_i) \). +

+ +

This equation serves as the core of the variational autoencoder, and +it's worth spending some time thinking about what it says In two +sentences, the left hand side has the quantity we want to maximize: +\( \log P(X) \) (plus an error term, which makes \( Q \) produce \( z \)'s that +can reproduce a given \( X \); this term will become small if \( Q \) is +high-capacity). The right hand side is something we can optimize via +stochastic gradient descent given the right choice of \( Q \) (although it +may not be obvious yet how). +

+ +

So how can we perform stochastic gradient descent?

+ +

First we need to be a bit more specific about the form that \( Q(z|X) \) +will take. The usual choice is to say that +\( Q(z|X)=\mathcal{N}(z|\mu(X;\vartheta),\Sigma(X;\vartheta)) \), where +\( \mu \) and \( \Sigma \) are arbitrary deterministic functions with +parameters \( \vartheta \) that can be learned from data (we will omit +\( \vartheta \) in later equations). In practice, \( \mu \) and \( \Sigma \) are +again implemented via neural networks, and \( \Sigma \) is constrained to +be a diagonal matrix. The name variational "autoencoder" comes from +the fact that \( \mu \) and \( \Sigma \) are "encoding" \( X \) into the latent +space \( z \). The advantages of this choice are computational, as they +make it clear how to compute the right hand side. The last +term---\( \mathcal{D}\left[Q(z|X)\|P(z)\right] \)---is now a KL-divergence +between two multivariate Gaussian distributions, which can be computed +in closed form as: +

+$$ +\begin{array}{c} + \mathcal{D}[\mathcal{N}(\mu_0,\Sigma_0) \| \mathcal{N}(\mu_1,\Sigma_1)] = \hspace{20em}\\ + \hspace{5em}\frac{ 1 }{ 2 } \left( \mathrm{tr} \left( \Sigma_1^{-1} \Sigma_0 \right) + \left( \mu_1 - \mu_0\right)^\top \Sigma_1^{-1} ( \mu_1 - \mu_0 ) - k + \log \left( \frac{ \det \Sigma_1 }{ \det \Sigma_0 } \right) \right) +\end{array} +$$ + +

where \( k \) is the dimensionality of the distribution. In our case, this simplifies to:

+$$ +\begin{array}{c} + \mathcal{D}[\mathcal{N}(\mu(X),\Sigma(X)) \| \mathcal{N}(0,I)] = \hspace{20em}\\ +\hspace{6em}\frac{ 1 }{ 2 } \left( \mathrm{tr} \left( \Sigma(X) \right) + \left( \mu(X)\right)^\top ( \mu(X) ) - k - \log\det\left( \Sigma(X) \right) \right). +\end{array} +$$ + +

The first term on the right hand side is a bit more tricky. +We could use sampling to estimate \( E_{z\sim Q}\left[\log P(X|z) \right] \), but getting a good estimate would require passing many samples of \( z \) through \( f \), which would be expensive. +Hence, as is standard in stochastic gradient descent, we take one sample of \( z \) and treat \( \log P(X|z) \) for that \( z \) as an approximation of \( E_{z\sim Q}\left[\log P(X|z) \right] \). +After all, we are already doing stochastic gradient descent over different values of \( X \) sampled from a dataset \( D \). +The full equation we want to optimize is: +

+ +$$ +\begin{array}{c} + E_{X\sim D}\left[\log P(X) - \mathcal{D}\left[Q(z|X)\|P(z|X)\right]\right]=\hspace{16em}\\ +\hspace{10em}E_{X\sim D}\left[E_{z\sim Q}\left[\log P(X|z) \right] - \mathcal{D}\left[Q(z|X)\|P(z)\right]\right]. +\end{array} +$$ + +

If we take the gradient of this equation, the gradient symbol can be moved into the expectations. +Therefore, we can sample a single value of \( X \) and a single value of \( z \) from the distribution \( Q(z|X) \), and compute the gradient of: +

+$$ +\begin{equation} + \log P(X|z)-\mathcal{D}\left[Q(z|X)\|P(z)\right]. +\label{_auto1} +\end{equation} +$$ + +

We can then average the gradient of this function over arbitrarily many samples of \( X \) and \( z \), and the result converges to the gradient.

+ +

There is, however, a significant problem +\( E_{z\sim Q}\left[\log P(X|z) \right] \) depends not just on the parameters of \( P \), but also on the parameters of \( Q \). +

+ +

In order to make VAEs work, it is essential to drive \( Q \) to produce codes for \( X \) that \( P \) can reliably decode.

+$$ + E_{X\sim D}\left[E_{\epsilon\sim\mathcal{N}(0,I)}[\log P(X|z=\mu(X)+\Sigma^{1/2}(X)*\epsilon)]-\mathcal{D}\left[Q(z|X)\|P(z)\right]\right]. +$$ +
© 1999-2024, Morten Hjorth-Jensen. 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zLj2CFG{13QHp+9Ruv5+)OxU^_f^*pcN8A7uDleKwu&S*TlJ_>F8eC=bGNNRfDe7\n", - "\n", - "\n", - "**Readings.**\n", - "\n", - "1. Reading recommendation: Goodfellow et al chapter 20.10-20-14\n", + "" + ] + }, + { + "cell_type": "markdown", + "id": "07b48a42", + "metadata": { + "editable": true + }, + "source": [ + "## Readings\n", + "1. Reading recommendation: Goodfellow et al, for VAEs and GANs see sections 20.10-20.11\n", "\n", "2. To create Boltzmann machine using Keras, see Babcock and Bali chapter 4, see \n", "\n", @@ -62,7 +69,7 @@ }, { "cell_type": "markdown", - "id": "6f3b1a0e", + "id": "4a593d0c", "metadata": { "editable": true }, @@ -78,7 +85,7 @@ }, { "cell_type": "markdown", - "id": "4c4518b7", + "id": "8c41077c", "metadata": { "editable": true }, @@ -89,7 +96,7 @@ { "cell_type": "code", "execution_count": 1, - "id": "a38cb8bb", + "id": "26f3c67f", "metadata": { "collapsed": false, "editable": true @@ -204,7 +211,7 @@ }, { "cell_type": "markdown", - "id": "6a863f37", + "id": "761e5203", "metadata": { "editable": true }, @@ -216,7 +223,7 @@ }, { "cell_type": "markdown", - "id": "ca225537", + "id": "9204df6f", "metadata": { "editable": true }, @@ -230,7 +237,19 @@ }, { "cell_type": "markdown", - "id": "ff575c58", + "id": "7c2000bc", + "metadata": { + "editable": true + }, + "source": [ + "## Langevin sampling\n", + "\n", + "**Note**: Notes to be added" + ] + }, + { + "cell_type": "markdown", + "id": "b57de47c", "metadata": { "editable": true }, @@ -242,7 +261,7 @@ }, { "cell_type": "markdown", - "id": "13c1450c", + "id": "90cbcc91", "metadata": { "editable": true }, @@ -258,7 +277,7 @@ }, { "cell_type": "markdown", - "id": "5cdb38eb", + "id": "231c7c76", "metadata": { "editable": true }, @@ -270,7 +289,7 @@ }, { "cell_type": "markdown", - "id": "e75d2c06", + "id": "e4039926", "metadata": { "editable": true }, @@ -281,7 +300,7 @@ }, { "cell_type": "markdown", - "id": "04094e6d", + "id": "132111d9", "metadata": { "editable": true }, @@ -293,34 +312,233 @@ }, { "cell_type": "markdown", - "id": "69a127f0", + "id": "f9996a71", "metadata": { "editable": true }, "source": [ - "The goal is to minimize the construction error." + "The goal is to minimize the construction error, often done by optimizing the means squared error." ] }, { "cell_type": "markdown", - "id": "67c0a90b", + "id": "c4327625", "metadata": { "editable": true }, "source": [ "## Schematic image of an Autoencoder\n", "\n", - "" + "\n", + "\n", + "\n", + "

Figure 1:

\n", + "" ] }, { "cell_type": "markdown", - "id": "789ace31", + "id": "6d32b3e5", "metadata": { "editable": true }, "source": [ - "## Kullback-Leibler relative entropy\n", + "## Mathematics of Variational Autoencoders\n", + "\n", + "We have defined earlier a probability (marginal) distribution with hidden variables $\\boldsymbol{h}$ and parameters $\\boldsymbol{\\Theta}$ as" + ] + }, + { + "cell_type": "markdown", + "id": "a09c0733", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "p(\\boldsymbol{x};\\boldsymbol{\\Theta}) = \\int d\\boldsymbol{h}p(\\boldsymbol{x},\\boldsymbol{h};\\boldsymbol{\\Theta}),\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "ef6b1e7c", + "metadata": { + "editable": true + }, + "source": [ + "for continuous variables $\\boldsymbol{h}$ and" + ] + }, + { + "cell_type": "markdown", + "id": "c0ca4ce9", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "p(\\boldsymbol{x};\\boldsymbol{\\Theta}) = \\sum_{\\boldsymbol{h}}p(\\boldsymbol{x},\\boldsymbol{h};\\boldsymbol{\\Theta}),\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "abc0a344", + "metadata": { + "editable": true + }, + "source": [ + "for discrete stochastic events $\\boldsymbol{h}$. The variables $\\boldsymbol{h}$ are normally called the **latent variables** in the theory of autoencoders. We will also call then for that here." + ] + }, + { + "cell_type": "markdown", + "id": "95f0590f", + "metadata": { + "editable": true + }, + "source": [ + "## Using the conditional probability\n", + "\n", + "Using the the definition of the conditional probabilities $p(\\boldsymbol{x}\\vert\\boldsymbol{h};\\boldsymbol{\\Theta})$, $p(\\boldsymbol{h}\\vert\\boldsymbol{x};\\boldsymbol{\\Theta})$ and \n", + "and the prior $p(\\boldsymbol{h})$, we can rewrite the above equation as" + ] + }, + { + "cell_type": "markdown", + "id": "0418643f", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "p(\\boldsymbol{x};\\boldsymbol{\\Theta}) = \\sum_{\\boldsymbol{h}}p(\\boldsymbol{x}\\vert\\boldsymbol{h};\\boldsymbol{\\Theta})p(\\boldsymbol{h},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "4a9128c2", + "metadata": { + "editable": true + }, + "source": [ + "which allows us to make the dependence of $\\boldsymbol{x}$ on $\\boldsymbol{h}$\n", + "explicit by using the law of total probability. The intuition behind\n", + "this approach for finding the marginal probability for $\\boldsymbol{x}$ is to\n", + "optimize the above equations with respect to the parameters\n", + "$\\boldsymbol{\\Theta}$. This is done normally by maximizing the probability,\n", + "the so-called maximum-likelihood approach discussed earlier." + ] + }, + { + "cell_type": "markdown", + "id": "2bcab5be", + "metadata": { + "editable": true + }, + "source": [ + "## VAEs versus autoencoders\n", + "\n", + "This trained probability is assumed to be able to produce similar\n", + "samples as the input. In VAEs it is then common to compare via for\n", + "example the mean-squared error or the cross-entropy the predicted\n", + "values with the input values. Compared with autoencoders, we are now\n", + "producing a probability instead of a functions which mimicks the\n", + "input.\n", + "\n", + "In VAEs, the choice of this output distribution is often Gaussian,\n", + "meaning that the conditional probability is" + ] + }, + { + "cell_type": "markdown", + "id": "120515d6", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "p(\\boldsymbol{x}\\vert\\boldsymbol{h};\\boldsymbol{\\Theta})=N(\\boldsymbol{x}\\vert f(\\boldsymbol{h};\\boldsymbol{\\Theta}), \\sigma^2\\times \\boldsymbol{I}),\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "e7937800", + "metadata": { + "editable": true + }, + "source": [ + "with mean value given by the function $f(\\boldsymbol{h};\\boldsymbol{\\Theta})$ and a\n", + "diagonal covariance matrix multiplied by a parameter $\\sigma^2$ which\n", + "is treated as a hyperparameter." + ] + }, + { + "cell_type": "markdown", + "id": "837a0123", + "metadata": { + "editable": true + }, + "source": [ + "## Gradient descent\n", + "\n", + "By having a Gaussian distribution, we can use gradient descent (or any\n", + "other optimization technique) to increase $p(\\boldsymbol{x};\\boldsymbol{\\Theta})$ by\n", + "making $f(\\boldsymbol{h};\\boldsymbol{\\Theta})$ approach $\\boldsymbol{x}$ for some $\\boldsymbol{h}$,\n", + "gradually making the training data more likely under the generative\n", + "model. The important property is simply that the marginal probability\n", + "can be computed, and it is continuous in $\\boldsymbol{\\Theta}$.." + ] + }, + { + "cell_type": "markdown", + "id": "a8471ff4", + "metadata": { + "editable": true + }, + "source": [ + "## Are VAEs just modified autoencoders?\n", + "\n", + "The mathematical basis of VAEs actually has relatively little to do\n", + "with classical autoencoders, for example the sparse autoencoders or\n", + "denoising autoencoders discussed earlier.\n", + "\n", + "VAEs approximately maximize the probability equation discussed\n", + "above. They are called autoencoders only because the final training\n", + "objective that derives from this setup does have an encoder and a\n", + "decoder, and resembles a traditional autoencoder. Unlike sparse\n", + "autoencoders, there are generally no tuning parameters analogous to\n", + "the sparsity penalties. And unlike sparse and denoising autoencoders,\n", + "we can sample directly from $p(\\boldsymbol{x})$ without performing Markov\n", + "Chain Monte Carlo." + ] + }, + { + "cell_type": "markdown", + "id": "34436c42", + "metadata": { + "editable": true + }, + "source": [ + "## Training VAEs\n", + "\n", + "To solve the integral or sum for $p(\\boldsymbol{x})$, there are two problems\n", + "that VAEs must deal with: how to define the latent variables $\\boldsymbol{h}$,\n", + "that is decide what information they represent, and how to deal with\n", + "the integral over $\\boldsymbol{h}$. VAEs give a definite answer to both." + ] + }, + { + "cell_type": "markdown", + "id": "dcb0681c", + "metadata": { + "editable": true + }, + "source": [ + "## Kullback-Leibler relative entropy (notation to be updated)\n", "\n", "When the goal of the training is to approximate a probability\n", "distribution, as it is in generative modeling, another relevant\n", @@ -333,7 +551,7 @@ }, { "cell_type": "markdown", - "id": "37323bfa", + "id": "8b98f3ed", "metadata": { "editable": true }, @@ -347,7 +565,7 @@ }, { "cell_type": "markdown", - "id": "9d74c9a3", + "id": "f8923141", "metadata": { "editable": true }, @@ -361,7 +579,7 @@ }, { "cell_type": "markdown", - "id": "fb48fde1", + "id": "0db3a2b6", "metadata": { "editable": true }, @@ -382,7 +600,7 @@ }, { "cell_type": "markdown", - "id": "77696018", + "id": "9e92fc4c", "metadata": { "editable": true }, @@ -403,7 +621,7 @@ }, { "cell_type": "markdown", - "id": "feb5a374", + "id": "c079c8d3", "metadata": { "editable": true }, @@ -418,7 +636,7 @@ }, { "cell_type": "markdown", - "id": "d82efd30", + "id": "3be06ae7", "metadata": { "editable": true }, @@ -434,7 +652,7 @@ }, { "cell_type": "markdown", - "id": "a4ac64cb", + "id": "e1548b0c", "metadata": { "editable": true }, @@ -444,7 +662,7 @@ }, { "cell_type": "markdown", - "id": "8e54f20c", + "id": "51264cb8", "metadata": { "editable": true }, @@ -456,7 +674,7 @@ }, { "cell_type": "markdown", - "id": "e00dde69", + "id": "484a967b", "metadata": { "editable": true }, @@ -473,7 +691,7 @@ }, { "cell_type": "markdown", - "id": "5d2aebb2", + "id": "c4b01a24", "metadata": { "editable": true }, @@ -494,7 +712,7 @@ }, { "cell_type": "markdown", - "id": "cae0300f", + "id": "de4af2c7", "metadata": { "editable": true }, @@ -513,7 +731,7 @@ }, { "cell_type": "markdown", - "id": "4e47497b", + "id": "73d9684d", "metadata": { "editable": true }, @@ -527,12 +745,22 @@ "it is limited to the training data. Hence, in unsupervised training as\n", "well it is important to prevent overfitting to the training data. Thus\n", "it is common to add regularizers to the cost function in the same\n", - "manner as we discussed for say linear regression." + "manner as discussed for say linear regression." + ] + }, + { + "cell_type": "markdown", + "id": "2c02bb33", + "metadata": { + "editable": true + }, + "source": [ + "## Back to VAEs" ] }, { "cell_type": "markdown", - "id": "94de7899", + "id": "9299e540", "metadata": { "editable": true }, @@ -544,7 +772,7 @@ }, { "cell_type": "markdown", - "id": "f241ef68", + "id": "4e53fe60", "metadata": { "editable": true }, @@ -555,7 +783,7 @@ { "cell_type": "code", "execution_count": 2, - "id": "5d5cbed5", + "id": "e8eb3742", "metadata": { "collapsed": false, "editable": true @@ -666,7 +894,7 @@ }, { "cell_type": "markdown", - "id": "a475e2d7", + "id": "47597299", "metadata": { "editable": true }, @@ -682,7 +910,7 @@ }, { "cell_type": "markdown", - "id": "c1cd08d5", + "id": "7aeb947e", "metadata": { "editable": true }, @@ -703,7 +931,7 @@ }, { "cell_type": "markdown", - "id": "9cb9dde2", + "id": "927b10d6", "metadata": { "editable": true }, @@ -715,7 +943,7 @@ }, { "cell_type": "markdown", - "id": "94bd4d04", + "id": "73e1f1a4", "metadata": { "editable": true }, @@ -740,7 +968,7 @@ }, { "cell_type": "markdown", - "id": "41a6d884", + "id": "f62ef66f", "metadata": { "editable": true }, @@ -758,7 +986,7 @@ }, { "cell_type": "markdown", - "id": "4c4ced8e", + "id": "268689d6", "metadata": { "editable": true }, @@ -770,7 +998,7 @@ }, { "cell_type": "markdown", - "id": "24c0fbe4", + "id": "f60b6a3c", "metadata": { "editable": true }, @@ -786,7 +1014,7 @@ }, { "cell_type": "markdown", - "id": "8c16d11f", + "id": "9085e160", "metadata": { "editable": true }, @@ -798,7 +1026,7 @@ }, { "cell_type": "markdown", - "id": "f883edc4", + "id": "8f3a0ea3", "metadata": { "editable": true }, @@ -809,7 +1037,7 @@ }, { "cell_type": "markdown", - "id": "49277a25", + "id": "46652aaa", "metadata": { "editable": true }, @@ -823,7 +1051,7 @@ }, { "cell_type": "markdown", - "id": "bc17df16", + "id": "c8fd98ab", "metadata": { "editable": true }, @@ -835,7 +1063,7 @@ }, { "cell_type": "markdown", - "id": "56d256d9", + "id": "9f086d9d", "metadata": { "editable": true }, @@ -846,25 +1074,19 @@ }, { "cell_type": "markdown", - "id": "b5668a30", + "id": "f1f44e87", "metadata": { "editable": true }, "source": [ - "\n", - "
\n", - "\n", "$$\n", - "\\begin{equation}\n", - " -v(\\theta^{(g)}, \\theta^{(d)})\n", - "\\label{_auto1} \\tag{1}\n", - "\\end{equation}\n", + "-v(\\theta^{(g)}, \\theta^{(d)})\n", "$$" ] }, { "cell_type": "markdown", - "id": "541f6691", + "id": "101e72f5", "metadata": { "editable": true }, @@ -887,7 +1109,7 @@ }, { "cell_type": "markdown", - "id": "d983ee38", + "id": "e2179e45", "metadata": { "editable": true }, @@ -904,7 +1126,7 @@ }, { "cell_type": "markdown", - "id": "6e8303e5", + "id": "87cd2006", "metadata": { "editable": true }, @@ -917,7 +1139,7 @@ }, { "cell_type": "markdown", - "id": "b99b0716", + "id": "a5250c3e", "metadata": { "editable": true }, @@ -928,7 +1150,7 @@ }, { "cell_type": "markdown", - "id": "ddcb25d6", + "id": "2d8a85a5", "metadata": { "editable": true }, @@ -942,7 +1164,7 @@ }, { "cell_type": "markdown", - "id": "847060e5", + "id": "7564be45", "metadata": { "editable": true }, @@ -955,7 +1177,7 @@ }, { "cell_type": "markdown", - "id": "ea2d547c", + "id": "3b1ac15c", "metadata": { "editable": true }, @@ -967,7 +1189,7 @@ }, { "cell_type": "markdown", - "id": "fb2b961f", + "id": "5b15fc11", "metadata": { "editable": true }, @@ -982,7 +1204,7 @@ }, { "cell_type": "markdown", - "id": "e5360cad", + "id": "d7b0cedf", "metadata": { "editable": true }, @@ -1001,7 +1223,7 @@ }, { "cell_type": "markdown", - "id": "c26e4e35", + "id": "31681017", "metadata": { "editable": true }, @@ -1020,7 +1242,7 @@ { "cell_type": "code", "execution_count": 3, - "id": "1cc26f49", + "id": "6e9d1942", "metadata": { "collapsed": false, "editable": true @@ -1038,7 +1260,7 @@ }, { "cell_type": "markdown", - "id": "16ab04cc", + "id": "b09615cf", "metadata": { "editable": true }, @@ -1049,7 +1271,7 @@ { "cell_type": "code", "execution_count": 4, - "id": "acdfbd97", + "id": "dd57f39d", "metadata": { "collapsed": false, "editable": true @@ -1075,7 +1297,7 @@ }, { "cell_type": "markdown", - "id": "c2e1e358", + "id": "94610e51", "metadata": { "editable": true }, @@ -1086,7 +1308,7 @@ { "cell_type": "code", "execution_count": 5, - "id": "b630975c", + "id": "f7929086", "metadata": { "collapsed": false, "editable": true @@ -1099,7 +1321,7 @@ }, { "cell_type": "markdown", - "id": "40b02fdd", + "id": "1e2b9d59", "metadata": { "editable": true }, @@ -1118,7 +1340,7 @@ { "cell_type": "code", "execution_count": 6, - "id": "34b786bb", + "id": "e9cbaa1c", "metadata": { "collapsed": false, "editable": true @@ -1186,7 +1408,7 @@ }, { "cell_type": "markdown", - "id": "6df7239a", + "id": "12c54220", "metadata": { "editable": true }, @@ -1199,7 +1421,7 @@ { "cell_type": "code", "execution_count": 7, - "id": "0bb7f9f0", + "id": "37a40bd1", "metadata": { "collapsed": false, "editable": true @@ -1239,7 +1461,7 @@ }, { "cell_type": "markdown", - "id": "19cfaf8f", + "id": "b8fef51e", "metadata": { "editable": true }, @@ -1250,7 +1472,7 @@ { "cell_type": "code", "execution_count": 8, - "id": "7fcd2d95", + "id": "636acf90", "metadata": { "collapsed": false, "editable": true @@ -1264,7 +1486,7 @@ { "cell_type": "code", "execution_count": 9, - "id": "e7565186", + "id": "dd901d8c", "metadata": { "collapsed": false, "editable": true @@ -1277,7 +1499,7 @@ }, { "cell_type": "markdown", - "id": "f5895e30", + "id": "a959802a", "metadata": { "editable": true }, @@ -1288,7 +1510,7 @@ { "cell_type": "code", "execution_count": 10, - "id": "9464bb23", + "id": "567812ff", "metadata": { "collapsed": false, "editable": true @@ -1302,7 +1524,7 @@ }, { "cell_type": "markdown", - "id": "734d99c1", + "id": "99185dc3", "metadata": { "editable": true }, @@ -1317,7 +1539,7 @@ { "cell_type": "code", "execution_count": 11, - "id": "283c03f2", + "id": "9868ba30", "metadata": { "collapsed": false, "editable": true @@ -1333,7 +1555,7 @@ { "cell_type": "code", "execution_count": 12, - "id": "4456797c", + "id": "b0ae1928", "metadata": { "collapsed": false, "editable": true @@ -1350,7 +1572,7 @@ }, { "cell_type": "markdown", - "id": "c2a781f0", + "id": "c876db65", "metadata": { "editable": true }, @@ -1362,7 +1584,7 @@ { "cell_type": "code", "execution_count": 13, - "id": "3eb6008b", + "id": "66f5b26c", "metadata": { "collapsed": false, "editable": true @@ -1376,7 +1598,7 @@ }, { "cell_type": "markdown", - "id": "81de819d", + "id": "b75521a4", "metadata": { "editable": true }, @@ -1390,7 +1612,7 @@ { "cell_type": "code", "execution_count": 14, - "id": "764073c3", + "id": "4c125b46", "metadata": { "collapsed": false, "editable": true @@ -1424,7 +1646,7 @@ }, { "cell_type": "markdown", - "id": "3aff1e1d", + "id": "6396a3c9", "metadata": { "editable": true }, @@ -1437,7 +1659,7 @@ { "cell_type": "code", "execution_count": 15, - "id": "2cf78d69", + "id": "2c290385", "metadata": { "collapsed": false, "editable": true @@ -1462,7 +1684,7 @@ }, { "cell_type": "markdown", - "id": "882f9616", + "id": "547d46ef", "metadata": { "editable": true }, @@ -1475,7 +1697,7 @@ { "cell_type": "code", "execution_count": 16, - "id": "75b2124c", + "id": "0dc78b2d", "metadata": { "collapsed": false, "editable": true @@ -1493,7 +1715,7 @@ }, { "cell_type": "markdown", - "id": "a76891d9", + "id": "426496e9", "metadata": { "editable": true }, @@ -1504,7 +1726,7 @@ { "cell_type": "code", "execution_count": 17, - "id": "0dd65bc0", + "id": "89f60891", "metadata": { "collapsed": false, "editable": true @@ -1544,7 +1766,7 @@ }, { "cell_type": "markdown", - "id": "8bf59554", + "id": "bbb9acb4", "metadata": { "editable": true }, @@ -1556,7 +1778,7 @@ { "cell_type": "code", "execution_count": 18, - "id": "6c26eac8", + "id": "cc029b41", "metadata": { "collapsed": false, "editable": true @@ -1568,7 +1790,7 @@ }, { "cell_type": "markdown", - "id": "2603b4e9", + "id": "a27008d5", "metadata": { "editable": true }, @@ -1582,7 +1804,7 @@ { "cell_type": "code", "execution_count": 19, - "id": "92c94d6b", + "id": "043375de", "metadata": { "collapsed": false, "editable": true @@ -1599,7 +1821,7 @@ }, { "cell_type": "markdown", - "id": "a8a5fe0b", + "id": "9b6d7239", "metadata": { "editable": true }, @@ -1613,7 +1835,7 @@ { "cell_type": "code", "execution_count": 20, - "id": "92d63386", + "id": "27096756", "metadata": { "collapsed": false, "editable": true @@ -1630,7 +1852,7 @@ }, { "cell_type": "markdown", - "id": "cb10725a", + "id": "d16b971c", "metadata": { "editable": true }, @@ -1646,7 +1868,7 @@ { "cell_type": "code", "execution_count": 21, - "id": "73ad155d", + "id": "2235ab2b", "metadata": { "collapsed": false, "editable": true @@ -1675,7 +1897,7 @@ { "cell_type": "code", "execution_count": 22, - "id": "35f60297", + "id": "5c2128f7", "metadata": { "collapsed": false, "editable": true @@ -1698,7 +1920,7 @@ { "cell_type": "code", "execution_count": 23, - "id": "7241e7aa", + "id": "68a53f68", "metadata": { "collapsed": false, "editable": true @@ -1711,7 +1933,7 @@ }, { "cell_type": "markdown", - "id": "2b4e56f0", + "id": "e1542e1a", "metadata": { "editable": true }, @@ -1726,7 +1948,7 @@ { "cell_type": "code", "execution_count": 24, - "id": "a13ac5ce", + "id": "5c6bf1f1", "metadata": { "collapsed": false, "editable": true @@ -1753,7 +1975,7 @@ }, { "cell_type": "markdown", - "id": "5c98bf5c", + "id": "fb91389b", "metadata": { "editable": true }, @@ -1768,7 +1990,7 @@ { "cell_type": "code", "execution_count": 25, - "id": "3db9513c", + "id": "1b67c811", "metadata": { "collapsed": false, "editable": true @@ -1783,7 +2005,7 @@ }, { "cell_type": "markdown", - "id": "9b6f10fd", + "id": "cfe9d355", "metadata": { "editable": true }, @@ -1791,6 +2013,337 @@ "A pretty cool result! We see that our generator indeed has learned a\n", "distribution which qualitatively looks a whole lot like the MNIST dataset." ] + }, + { + "cell_type": "markdown", + "id": "dbd85047", + "metadata": { + "editable": true + }, + "source": [ + "## Using the KL divergence\n", + "\n", + "In practice, for most $\\boldsymbol{h}$, $p(\\boldsymbol{x}\\vert \\boldsymbol{h}; \\boldsymbol{\\Theta})$\n", + "will be nearly zero, and hence contribute almost nothing to our\n", + "estimate of $p(\\boldsymbol{x})$.\n", + "\n", + "The key idea behind the variational autoencoder is to attempt to\n", + "sample values of $\\boldsymbol{h}$ that are likely to have produced $\\boldsymbol{x}$,\n", + "and compute $p(\\boldsymbol{x})$ just from those.\n", + "\n", + "This means that we need a new function $Q(\\boldsymbol{h}|\\boldsymbol{x})$ which can\n", + "take a value of $\\boldsymbol{x}$ and give us a distribution over $\\boldsymbol{h}$\n", + "values that are likely to produce $\\boldsymbol{x}$. Hopefully the space of\n", + "$\\boldsymbol{h}$ values that are likely under $Q$ will be much smaller than\n", + "the space of all $\\boldsymbol{h}$'s that are likely under the prior\n", + "$p(\\boldsymbol{h})$. This lets us, for example, compute $E_{\\boldsymbol{h}\\sim\n", + "Q}p(\\boldsymbol{x}\\vert \\boldsymbol{h})$ relatively easily. Note that we drop\n", + "$\\boldsymbol{\\Theta}$ from here and for notational simplicity.\n", + "\n", + "However, if $\\boldsymbol{h}$ is sampled from an arbitrary distribution with\n", + "PDF $Q(\\boldsymbol{h})$, which is not $\\mathcal{N}(0,I)$, then how does that\n", + "help us optimize $p(\\boldsymbol{x})$? The first thing we need to do is relate\n", + "$E_{\\boldsymbol{h}\\sim Q}P(\\boldsymbol{x}\\vert \\boldsymbol{h})$ and $p(\\boldsymbol{x})$. We will see where $Q$ comes from later.\n", + "\n", + "The relationship between $E_{\\boldsymbol{h}\\sim Q}p(\\boldsymbol{x}\\vert \\boldsymbol{h})$ and $p(\\boldsymbol{x})$ is one of the cornerstones of variational Bayesian methods.\n", + "We begin with the definition of Kullback-Leibler divergence (KL divergence or $\\mathcal{D}$) between $p(\\boldsymbol{h}\\vert \\boldsymbol{x})$ and $Q(\\boldsymbol{h})$, for some arbitrary $Q$ (which may or may not depend on $\\boldsymbol{x}$):" + ] + }, + { + "cell_type": "markdown", + "id": "2477d441", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\mathcal{D}\\left[Q(z)\\|P(z|X)\\right]=E_{z\\sim Q}\\left[\\log Q(z) - \\log P(z|X) \\right].\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "46ca3298", + "metadata": { + "editable": true + }, + "source": [ + "We can get both $p(\\boldsymbol{x})$ and $p(\\boldsymbol{x}\\vert \\boldsymbol{h})$ into this equation by applying Bayes rule to $p(\\boldsymbol{h}|\\boldsymbol{x})$" + ] + }, + { + "cell_type": "markdown", + "id": "b024079a", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\mathcal{D}\\left[Q(z)\\|P(z|X)\\right]=E_{z\\sim Q}\\left[\\log Q(z) - \\log P(X|z) - \\log P(z) \\right] + \\log P(X).\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "57cf294b", + "metadata": { + "editable": true + }, + "source": [ + "Here, $\\log P(X)$ comes out of the expectation because it does not depend on $z$.\n", + "Negating both sides, rearranging, and contracting part of $E_{z\\sim Q}$ into a KL-divergence terms yields:" + ] + }, + { + "cell_type": "markdown", + "id": "6d8bf875", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\log P(X) - \\mathcal{D}\\left[Q(z)\\|P(z|X)\\right]=E_{z\\sim Q}\\left[\\log P(X|z) \\right] - \\mathcal{D}\\left[Q(z)\\|P(z)\\right].\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "76456ab4", + "metadata": { + "editable": true + }, + "source": [ + "By Bayes rule, we have:" + ] + }, + { + "cell_type": "markdown", + "id": "1924f38e", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "E_{z\\sim Q}\\left[\\log P(Y_i|z,X_i)\\right]=E_{z\\sim Q}\\left[\\log P(z|Y_i,X_i) - \\log P(z|X_i) + \\log P(Y_i|X_i) \\right]\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "8fe3c445", + "metadata": { + "editable": true + }, + "source": [ + "Rearranging the terms and subtracting $E_{z\\sim Q}\\log Q(z)$ from both sides:" + ] + }, + { + "cell_type": "markdown", + "id": "c5ef147e", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\begin{array}{c}\n", + "\\log P(Y_i|X_i) - E_{z\\sim Q}\\left[\\log Q(z)-\\log P(z|X_i,Y_i)\\right]=\\hspace{10em}\\\\\n", + "\\hspace{10em}E_{z\\sim Q}\\left[\\log P(Y_i|z,X_i)+\\log P(z|X_i)-\\log Q(z)\\right]\n", + "\\end{array}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "cd28fa2e", + "metadata": { + "editable": true + }, + "source": [ + "Note that $X$ is fixed, and $Q$ can be \\textit{any} distribution, not\n", + "just a distribution which does a good job mapping $X$ to the $z$'s\n", + "that can produce $X$.\n", + "\n", + "Since we are interested in inferring $P(X)$, it makes sense to\n", + "construct a $Q$ which \\textit{does} depend on $X$, and in particular,\n", + "one which makes $\\mathcal{D}\\left[Q(z)\\|P(z|X)\\right]$ small:" + ] + }, + { + "cell_type": "markdown", + "id": "0f8ac050", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\log P(X) - \\mathcal{D}\\left[Q(z|X)\\|P(z|X)\\right]=E_{z\\sim Q}\\left[\\log P(X|z) \\right] - \\mathcal{D}\\left[Q(z|X)\\|P(z)\\right].\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "966ded1b", + "metadata": { + "editable": true + }, + "source": [ + "Hence, during training, it makes sense to choose a $Q$ which will make\n", + "$E_{z\\sim Q}[\\log Q(z)-$ $\\log P(z|X_i,Y_i)]$ (a\n", + "$\\mathcal{D}$-divergence) small, such that the right hand side is a\n", + "close approximation to $\\log P(Y_i|X_i)$.\n", + "\n", + "This equation serves as the core of the variational autoencoder, and\n", + "it's worth spending some time thinking about what it says In two\n", + "sentences, the left hand side has the quantity we want to maximize:\n", + "$\\log P(X)$ (plus an error term, which makes $Q$ produce $z$'s that\n", + "can reproduce a given $X$; this term will become small if $Q$ is\n", + "high-capacity). The right hand side is something we can optimize via\n", + "stochastic gradient descent given the right choice of $Q$ (although it\n", + "may not be obvious yet how).\n", + "\n", + "So how can we perform stochastic gradient descent?\n", + "\n", + "First we need to be a bit more specific about the form that $Q(z|X)$\n", + "will take. The usual choice is to say that\n", + "$Q(z|X)=\\mathcal{N}(z|\\mu(X;\\vartheta),\\Sigma(X;\\vartheta))$, where\n", + "$\\mu$ and $\\Sigma$ are arbitrary deterministic functions with\n", + "parameters $\\vartheta$ that can be learned from data (we will omit\n", + "$\\vartheta$ in later equations). In practice, $\\mu$ and $\\Sigma$ are\n", + "again implemented via neural networks, and $\\Sigma$ is constrained to\n", + "be a diagonal matrix. The name variational \"autoencoder\" comes from\n", + "the fact that $\\mu$ and $\\Sigma$ are \"encoding\" $X$ into the latent\n", + "space $z$. The advantages of this choice are computational, as they\n", + "make it clear how to compute the right hand side. The last\n", + "term---$\\mathcal{D}\\left[Q(z|X)\\|P(z)\\right]$---is now a KL-divergence\n", + "between two multivariate Gaussian distributions, which can be computed\n", + "in closed form as:" + ] + }, + { + "cell_type": "markdown", + "id": "d4e8ede5", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\begin{array}{c}\n", + " \\mathcal{D}[\\mathcal{N}(\\mu_0,\\Sigma_0) \\| \\mathcal{N}(\\mu_1,\\Sigma_1)] = \\hspace{20em}\\\\\n", + " \\hspace{5em}\\frac{ 1 }{ 2 } \\left( \\mathrm{tr} \\left( \\Sigma_1^{-1} \\Sigma_0 \\right) + \\left( \\mu_1 - \\mu_0\\right)^\\top \\Sigma_1^{-1} ( \\mu_1 - \\mu_0 ) - k + \\log \\left( \\frac{ \\det \\Sigma_1 }{ \\det \\Sigma_0 } \\right) \\right)\n", + "\\end{array}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "4d1d0d07", + "metadata": { + "editable": true + }, + "source": [ + "where $k$ is the dimensionality of the distribution. In our case, this simplifies to:" + ] + }, + { + "cell_type": "markdown", + "id": "c9edc7c3", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\begin{array}{c}\n", + " \\mathcal{D}[\\mathcal{N}(\\mu(X),\\Sigma(X)) \\| \\mathcal{N}(0,I)] = \\hspace{20em}\\\\\n", + "\\hspace{6em}\\frac{ 1 }{ 2 } \\left( \\mathrm{tr} \\left( \\Sigma(X) \\right) + \\left( \\mu(X)\\right)^\\top ( \\mu(X) ) - k - \\log\\det\\left( \\Sigma(X) \\right) \\right).\n", + "\\end{array}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "1eb68e03", + "metadata": { + "editable": true + }, + "source": [ + "The first term on the right hand side is a bit more tricky.\n", + "We could use sampling to estimate $E_{z\\sim Q}\\left[\\log P(X|z) \\right]$, but getting a good estimate would require passing many samples of $z$ through $f$, which would be expensive.\n", + "Hence, as is standard in stochastic gradient descent, we take one sample of $z$ and treat $\\log P(X|z)$ for that $z$ as an approximation of $E_{z\\sim Q}\\left[\\log P(X|z) \\right]$.\n", + "After all, we are already doing stochastic gradient descent over different values of $X$ sampled from a dataset $D$.\n", + "The full equation we want to optimize is:" + ] + }, + { + "cell_type": "markdown", + "id": "a5b7d69d", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\begin{array}{c}\n", + " E_{X\\sim D}\\left[\\log P(X) - \\mathcal{D}\\left[Q(z|X)\\|P(z|X)\\right]\\right]=\\hspace{16em}\\\\\n", + "\\hspace{10em}E_{X\\sim D}\\left[E_{z\\sim Q}\\left[\\log P(X|z) \\right] - \\mathcal{D}\\left[Q(z|X)\\|P(z)\\right]\\right].\n", + "\\end{array}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "5f06b6bd", + "metadata": { + "editable": true + }, + "source": [ + "If we take the gradient of this equation, the gradient symbol can be moved into the expectations.\n", + "Therefore, we can sample a single value of $X$ and a single value of $z$ from the distribution $Q(z|X)$, and compute the gradient of:" + ] + }, + { + "cell_type": "markdown", + "id": "6bb2eda1", + "metadata": { + "editable": true + }, + "source": [ + "\n", + "
\n", + "\n", + "$$\n", + "\\begin{equation}\n", + " \\log P(X|z)-\\mathcal{D}\\left[Q(z|X)\\|P(z)\\right].\n", + "\\label{_auto1} \\tag{1}\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "db3d86d0", + "metadata": { + "editable": true + }, + "source": [ + "We can then average the gradient of this function over arbitrarily many samples of $X$ and $z$, and the result converges to the gradient.\n", + "\n", + "There is, however, a significant problem\n", + "$E_{z\\sim Q}\\left[\\log P(X|z) \\right]$ depends not just on the parameters of $P$, but also on the parameters of $Q$.\n", + "\n", + "In order to make VAEs work, it is essential to drive $Q$ to produce codes for $X$ that $P$ can reliably decode." + ] + }, + { + "cell_type": "markdown", + "id": "fabec023", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "E_{X\\sim D}\\left[E_{\\epsilon\\sim\\mathcal{N}(0,I)}[\\log P(X|z=\\mu(X)+\\Sigma^{1/2}(X)*\\epsilon)]-\\mathcal{D}\\left[Q(z|X)\\|P(z)\\right]\\right].\n", + "$$" + ] } ], "metadata": {}, diff --git a/doc/pub/week13/pdf/week13.pdf b/doc/pub/week13/pdf/week13.pdf index 040ac073e52ae1bb1ddf96e9d030f96a2464c879..a9a9429e7c477238307c8c86c74a6592f030a993 100644 GIT binary patch delta 227164 zcmb5VWn5KJ*DtCdiXcdB8Yv|3QWAHd&-0#h?zvyCAN(n?*IILpImW-{b}wyWJ3}QMerm`90VST0Xks5H52CN> zHp+jlaocdg0l)g``R@2f+#{!GBl372nAbB2ny5V(@w5};9NV?Rhr;m&16x#9BcvWK_vxpWU#l^ir-UD1V-@^vfs-h_AeFV}K8;Ls{j27ek zAKajJJRoaW9C;NqekwI4Y;OMBDR$o2H`L79-=uylmz>~Wx<@6jc$Y2BUeq2hc|{|F zEc7g8;=cQh8izORVo{tbciuHOWRMmn-W$o1?j$c1rD++c#pj~4%OJqJQf;oV@fPX8 zFpQVxPW+gvH9NPhob2j6H8W0s?*X1JS*+++YKKs-EIgy=Yj?j}>FjsYFG!PzDZU&j zej5C&ha$&=UcZ`8pdm1b{HuF#i9`A9+Hd*sbHtB5 zEK)OPraTMgN#8Y`m(94_T{M&aqMk^t;@d@sg<393{2;3XwI_uWU+j_yOYaK2Z=dvT z*S_sDAElQyKGL0+b~mt|u4Ug*?`PVAOGglE=vaG#lAgm{SI`z&E)tW_@Gdp0O+-d% zpNo7Rc4#yLy65*XkzY0VfUf3Y3gfozBFYu7%IVHV>G 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z&;lu!rW`TdF@E2=UgDv^grUx(0r;8aEhGErV7Kw$a(O_6dF}*`9rbVfm z1rLS6$NdJ0bF{Gsqrp&7WmZDQG0UPW8YEgTLhc8DB@ULipPcWFrvHzQ1$ z2?Aw7`N~UPkdK3;nJ(vVB5~5YBtU}j5|+zlpy(zp54bDKIhVC9LC8LXxH6DPsV&Y( z&Nt zmb@}c=vYeDs=Ci*sFHCbjT>oS>z31A2dGKVyjYfO!Nc3IEKt*q*lUM61;$c=+8vjy z0o`t0c%qdh!}dL;O#KLlM!ZY3uODVSin5L*Nw{g#b@sfIcg?4%5E@>IH)$KVA5jy=R><8c?^6LZQ01A!b>m=eJ%tfdn3!IrzYTyZ`_I diff --git a/doc/src/week13/week13.do.txt b/doc/src/week13/week13.do.txt index b3eaafe4..17d7d0f3 100644 --- a/doc/src/week13/week13.do.txt +++ b/doc/src/week13/week13.do.txt @@ -14,13 +14,15 @@ o Generative Adversarial Networks (GANs) #o "Video of lecture":"https://youtu.be/0VBmdP_iCzA" #o "Whiteboard notes":"https://github.com/CompPhysics/AdvancedMachineLearning/blob/main/doc/HandwrittenNotes/NotesApr262023.pdf" !eblock -!bblock Readings -o Reading recommendation: Goodfellow et al chapter 20.10-20-14 + +!split +===== Readings ===== +!bblock +o Reading recommendation: Goodfellow et al, for VAEs and GANs see sections 20.10-20.11 o To create Boltzmann machine using Keras, see Babcock and Bali chapter 4, see URL:"https://github.com/PacktPublishing/Hands-On-Generative-AI-with-Python-and-TensorFlow-2/blob/master/Chapter_4/models/rbm.py" o See Foster, chapter 7 on energy-based models at URL:"https://github.com/davidADSP/Generative_Deep_Learning_2nd_Edition/tree/main/notebooks/07_ebm/01_ebm" !eblock - #todo: add about Langevin sampling, see https://www.lyndonduong.com/sgmcmc/ # add material about VAEs # add code on RBMs @@ -157,7 +159,10 @@ See discussions in Foster, chapter 7 on energy-based models at URL:"https://gith That notebook is based on a recent article by Du and Mordatch, _Implicit generation and modeling with energy-based models_, see URL:"https://arxiv.org/pdf/1903.08689.pdf." +!split +===== Langevin sampling ===== +_Note_: Notes to be added !split ===== Theory of Variational Autoencoders ===== @@ -188,7 +193,7 @@ the final ouput \] !et -The goal is to minimize the construction error. +The goal is to minimize the construction error, often done by optimizing the means squared error. !split ===== Schematic image of an Autoencoder ===== @@ -197,12 +202,100 @@ FIGURE: [figures/ae1.png, width=700 frac=1.0] !split -===== Mathematics of VAEs ===== +===== Mathematics of Variational Autoencoders ===== + +We have defined earlier a probability (marginal) distribution with hidden variables $\bm{h}$ and parameters $\bm{\Theta}$ as +!bt +\[ +p(\bm{x};\bm{\Theta}) = \int d\bm{h}p(\bm{x},\bm{h};\bm{\Theta}), +\] +!et +for continuous variables $\bm{h}$ and +!bt +\[ +p(\bm{x};\bm{\Theta}) = \sum_{\bm{h}}p(\bm{x},\bm{h};\bm{\Theta}), +\] +!et +for discrete stochastic events $\bm{h}$. The variables $\bm{h}$ are normally called the _latent variables_ in the theory of autoencoders. We will also call then for that here. + +!split +===== Using the conditional probability ===== + +Using the the definition of the conditional probabilities $p(\bm{x}\vert\bm{h};\bm{\Theta})$, $p(\bm{h}\vert\bm{x};\bm{\Theta})$ and +and the prior $p(\bm{h})$, we can rewrite the above equation as +!bt +\[ +p(\bm{x};\bm{\Theta}) = \sum_{\bm{h}}p(\bm{x}\vert\bm{h};\bm{\Theta})p(\bm{h}, +\] +!et -More material will be added here +which allows us to make the dependence of $\bm{x}$ on $\bm{h}$ +explicit by using the law of total probability. The intuition behind +this approach for finding the marginal probability for $\bm{x}$ is to +optimize the above equations with respect to the parameters +$\bm{\Theta}$. This is done normally by maximizing the probability, +the so-called maximum-likelihood approach discussed earlier. !split -===== Kullback-Leibler relative entropy (to be corrected) ===== +===== VAEs versus autoencoders ===== + +This trained probability is assumed to be able to produce similar +samples as the input. In VAEs it is then common to compare via for +example the mean-squared error or the cross-entropy the predicted +values with the input values. Compared with autoencoders, we are now +producing a probability instead of a functions which mimicks the +input. + +In VAEs, the choice of this output distribution is often Gaussian, +meaning that the conditional probability is +!bt +\[ +p(\bm{x}\vert\bm{h};\bm{\Theta})=N(\bm{x}\vert f(\bm{h};\bm{\Theta}), \sigma^2\times \bm{I}), +\] +!et +with mean value given by the function $f(\bm{h};\bm{\Theta})$ and a +diagonal covariance matrix multiplied by a parameter $\sigma^2$ which +is treated as a hyperparameter. + +!split +===== Gradient descent ===== + +By having a Gaussian distribution, we can use gradient descent (or any +other optimization technique) to increase $p(\bm{x};\bm{\Theta})$ by +making $f(\bm{h};\bm{\Theta})$ approach $\bm{x}$ for some $\bm{h}$, +gradually making the training data more likely under the generative +model. The important property is simply that the marginal probability +can be computed, and it is continuous in $\bm{\Theta}$.. + +!split +===== Are VAEs just modified autoencoders? ===== + +The mathematical basis of VAEs actually has relatively little to do +with classical autoencoders, for example the sparse autoencoders or +denoising autoencoders discussed earlier. + +VAEs approximately maximize the probability equation discussed +above. They are called autoencoders only because the final training +objective that derives from this setup does have an encoder and a +decoder, and resembles a traditional autoencoder. Unlike sparse +autoencoders, there are generally no tuning parameters analogous to +the sparsity penalties. And unlike sparse and denoising autoencoders, +we can sample directly from $p(\bm{x})$ without performing Markov +Chain Monte Carlo. + + +!split +===== Training VAEs ===== + +To solve the integral or sum for $p(\bm{x})$, there are two problems +that VAEs must deal with: how to define the latent variables $\bm{h}$, +that is decide what information they represent, and how to deal with +the integral over $\bm{h}$. VAEs give a definite answer to both. + + + +!split +===== Kullback-Leibler relative entropy (notation to be updated) ===== When the goal of the training is to approximate a probability distribution, as it is in generative modeling, another relevant @@ -320,7 +413,12 @@ approximate is not the _true_ distribution we wish to estimate, it is limited to the training data. Hence, in unsupervised training as well it is important to prevent overfitting to the training data. Thus it is common to add regularizers to the cost function in the same -manner as we discussed for say linear regression. +manner as discussed for say linear regression. + + +!split +===== Back to VAEs ===== + !split @@ -530,9 +628,9 @@ determines the reward for the discriminator, while the generator gets the conjugate reward !bt -\begin{equation} +\[ -v(\theta^{(g)}, \theta^{(d)}) -\end{equation} +\] !et @@ -1034,222 +1132,168 @@ distribution which qualitatively looks a whole lot like the MNIST dataset. !split -===== Marginal distribution again ===== +===== Using the KL divergence ===== -We have defined earlier a probability (marginal) distribution with hidden variables $\bm{h}$ and parameters $\bm{\Theta}$ as +In practice, for most $\bm{h}$, $p(\bm{x}\vert \bm{h}; \bm{\Theta})$ +will be nearly zero, and hence contribute almost nothing to our +estimate of $p(\bm{x})$. + +The key idea behind the variational autoencoder is to attempt to +sample values of $\bm{h}$ that are likely to have produced $\bm{x}$, +and compute $p(\bm{x})$ just from those. + +This means that we need a new function $Q(\bm{h}|\bm{x})$ which can +take a value of $\bm{x}$ and give us a distribution over $\bm{h}$ +values that are likely to produce $\bm{x}$. Hopefully the space of +$\bm{h}$ values that are likely under $Q$ will be much smaller than +the space of all $\bm{h}$'s that are likely under the prior +$p(\bm{h})$. This lets us, for example, compute $E_{\bm{h}\sim +Q}p(\bm{x}\vert \bm{h})$ relatively easily. Note that we drop +$\bm{\Theta}$ from here and for notational simplicity. + + + +However, if $\bm{h}$ is sampled from an arbitrary distribution with +PDF $Q(\bm{h})$, which is not $\mathcal{N}(0,I)$, then how does that +help us optimize $p(\bm{x})$? The first thing we need to do is relate +$E_{\bm{h}\sim Q}P(\bm{x}\vert \bm{h})$ and $p(\bm{x})$. We will see where $Q$ comes from later. + +The relationship between $E_{\bm{h}\sim Q}p(\bm{x}\vert \bm{h})$ and $p(\bm{x})$ is one of the cornerstones of variational Bayesian methods. +We begin with the definition of Kullback-Leibler divergence (KL divergence or $\mathcal{D}$) between $p(\bm{h}\vert \bm{x})$ and $Q(\bm{h})$, for some arbitrary $Q$ (which may or may not depend on $\bm{x}$): !bt \[ -p(\bm{x};\bm{\Theta}) = \int d\bm{h}p(\bm{x},\bm{h};\bm{\Theta}), + \mathcal{D}\left[Q(z)\|P(z|X)\right]=E_{z\sim Q}\left[\log Q(z) - \log P(z|X) \right]. \] !et -for continuous variables $\bm{h}$ and + + +We can get both $p(\bm{x})$ and $p(\bm{x}\vert \bm{h})$ into this equation by applying Bayes rule to $p(\bm{h}|\bm{x})$ !bt \[ -p(\bm{x};\bm{\Theta}) = \sum_{\bm{h}}p(\bm{x},\bm{h};\bm{\Theta}), + \mathcal{D}\left[Q(z)\|P(z|X)\right]=E_{z\sim Q}\left[\log Q(z) - \log P(X|z) - \log P(z) \right] + \log P(X). \] !et -for discrete stochastic events $\bm{h}$. The variables $\bm{h}$ are normally called the _latent variables_ in the theory of autoencoders. We will also call then for that here. - -!split -===== Using the conditional probability ===== -Using the the definition of the conditional probabilities $p(\bm{x}\vert\bm{h};\bm{\Theta})$, $p(\bm{h}\vert\bm{x};\bm{\Theta})$ and -and the prior $p{\bm{h})$, we can rewrite the above equation as +Here, $\log P(X)$ comes out of the expectation because it does not depend on $z$. +Negating both sides, rearranging, and contracting part of $E_{z\sim Q}$ into a KL-divergence terms yields: !bt \[ -p(\bm{x};\bm{\Theta}) = \sum_{\bm{h}}p(\bm{x}\vert\bm{h};\bm{\Theta})p(\bm{h}, +\log P(X) - \mathcal{D}\left[Q(z)\|P(z|X)\right]=E_{z\sim Q}\left[\log P(X|z) \right] - \mathcal{D}\left[Q(z)\|P(z)\right]. \] !et - -which allows us to make the dependence of $\bm{x}$ on $\bm{h}$ -explicit by using the law of total probability. The intuition behind -this approach for finding the marginal probability for $\bm{x}$ is to -optimize the above equations with respect to the parameters -$\bm{\Theta}$. This is done normally by maximizing the probability, -the so-called maximum-likelihood approach discussed earlier. - -!split -===== VAEs versus autoencoders ===== - -This trained probability is assumed to be able to produce similar -samples as the input. In VAEs it is then common to compare via for -example the mean-squared error or the cross-entropy the predicted -values with the input values. Compared with autoencoders, we are now -producing a probability instead of a functions which mimicks the -input. - -In VAEs, the choice of this output distribution is often Gaussian, -meaning that the conditional probability is +By Bayes rule, we have: !bt \[ -p(\bm{x}\vert\bm{h};\bm{\Theta})=N(\bm{x}\vert f(\bm{h};\bm{\Theta}), \sigma^2\times \bm{I}), +E_{z\sim Q}\left[\log P(Y_i|z,X_i)\right]=E_{z\sim Q}\left[\log P(z|Y_i,X_i) - \log P(z|X_i) + \log P(Y_i|X_i) \right] \] !et -with mean value given by the function $f(\bm{h};\bm{\Theta})$ and a -diagonal covariance matrix multiplied by a parameter $\sigma^2$ which -is treated as a hyperparameter. - -!split -====== Gradient descent ===== -By having a Gaussian distribution, we can use gradient descent (or any -other optimization technique) to increase $p(\bm{x};\bm{\Theta})) by -making $f(\bm{h}; \bm{\Theta})$ approach $bm{x}$ for some $\bm{h}$, -gradually making the training data more likely under the generative -model. The important property is simply that the marginal probability -can be computed, and is continuous in $\bm{\Theta}$.. +Rearranging the terms and subtracting $E_{z\sim Q}\log Q(z)$ from both sides: +!bt +\[ +\begin{array}{c} +\log P(Y_i|X_i) - E_{z\sim Q}\left[\log Q(z)-\log P(z|X_i,Y_i)\right]=\hspace{10em}\\ +\hspace{10em}E_{z\sim Q}\left[\log P(Y_i|z,X_i)+\log P(z|X_i)-\log Q(z)\right] +\end{array} +\] +!et +Note that $X$ is fixed, and $Q$ can be \textit{any} distribution, not +just a distribution which does a good job mapping $X$ to the $z$'s +that can produce $X$. -!split -===== Are VAEs just modified autoencoders? ===== -The mathematical basis of VAEs actually has relatively little to do -with classical autoencoders, for example the sparse autoencoders or -denoising autoencoders discussed earlier. +Since we are interested in inferring $P(X)$, it makes sense to +construct a $Q$ which \textit{does} depend on $X$, and in particular, +one which makes $\mathcal{D}\left[Q(z)\|P(z|X)\right]$ small: +!bt +\[ +\log P(X) - \mathcal{D}\left[Q(z|X)\|P(z|X)\right]=E_{z\sim Q}\left[\log P(X|z) \right] - \mathcal{D}\left[Q(z|X)\|P(z)\right]. +\] +!et -VAEs approximately maximize -the probability equation discussed above. They are called -autoencoders only because the final training objective that -derives from this setup does have an encoder and a decoder, and -resembles a traditional autoencoder. Unlike sparse autoencoders, there -are generally no tuning parameters analogous to the sparsity -penalties. And unlike sparse and denoising autoencoders, we can sample directly from $p(\bm{x})$ -without performing Markov Chain Monte Carlo. - -To solve the integral or sum for $p(\bm{x})$, there are two problems that VAEs must deal with: how to define the latent variables $\bm{h}$, that is decide what information they represent, and how to deal with the integral over $\bm{h}$. -VAEs give a definite answer to both. - - -In practice, for most $z$, $P(X|z)$ will be nearly zero, and hence contribute almost nothing to our estimate of $P(X)$. - -The key idea behind the variational autoencoder is to attempt to sample values of $z$ that are likely to have produced $X$, and compute $P(X)$ just from those. -This means that we need a new function $Q(z|X)$ which can take a value of $X$ and give us a distribution over $z$ values that are likely to produce $X$. -Hopefully the space of $z$ values that are likely under $Q$ will be much smaller than the space of all $z$'s that are likely under the prior $P(z)$. -This lets us, for example, compute $E_{z\sim Q}P(X|z)$ relatively easily. -%Given that we are sampling $z$ from some distribution other than $\mathcal{N}(0,I)$, however, the math becomes a bit less straightforward. -%Hence, the variational ``autoencoder'' framework first samples $z$ from some distribution different from $\mathcal{N}(0,1)$ (specifically, a distribution of $z$ values which are likely to give rise to $Y_i$ given $X_i$), and uses that sample to approximate $P(Y|X)$ in the following way. -However, if $z$ is sampled from an arbitrary distribution with PDF $Q(z)$, which is not $\mathcal{N}(0,I)$, then how does that help us optimize $P(X)$? -The first thing we need to do is relate $E_{z\sim Q}P(X|z)$ and $P(X)$. -We'll see where $Q$ comes from later. - -The relationship between $E_{z\sim Q}P(X|z)$ and $P(X)$ is one of the cornerstones of variational Bayesian methods. -We begin with the definition of Kullback-Leibler divergence (KL divergence or $\mathcal{D}$) between $P(z|X)$ and $Q(z)$, for some arbitrary $Q$ (which may or may not depend on $X$): -\begin{equation} - \mathcal{D}\left[Q(z)\|P(z|X)\right]=E_{z\sim Q}\left[\log Q(z) - \log P(z|X) \right]. -\label{eq:kl} -\end{equation} -\noindent We can get both $P(X)$ and $P(X|z)$ into this equation by applying Bayes rule to $P(z|X)$: -\begin{equation} - \mathcal{D}\left[Q(z)\|P(z|X)\right]=E_{z\sim Q}\left[\log Q(z) - \log P(X|z) - \log P(z) \right] + \log P(X). -\end{equation} -\noindent Here, $\log P(X)$ comes out of the expectation because it does not depend on $z$. Negating both sides, rearranging, and contracting part of $E_{z\sim Q}$ into a KL-divergence terms yields: -\begin{equation} - \log P(X) - \mathcal{D}\left[Q(z)\|P(z|X)\right]=E_{z\sim Q}\left[\log P(X|z) \right] - \mathcal{D}\left[Q(z)\|P(z)\right]. -\end{equation} -%By Bayes rule, we have: -%\vspace{-0.05in} -%\begin{equation} -% E_{z\sim Q}\left[\log P(Y_i|z,X_i)\right]=E_{z\sim Q}\left[\log P(z|Y_i,X_i) - \log P(z|X_i) + \log P(Y_i|X_i) \right] -%\end{equation} -%\vspace{-0.05in} -%\noindent Rearranging the terms and subtracting $E_{z\sim Q}\log Q(z)$ from both sides: -%\vspace{-0.05in} -%\begin{equation} -%\begin{array}{c} -% \log P(Y_i|X_i) - E_{z\sim Q}\left[\log Q(z)-\log P(z|X_i,Y_i)\right]=\hspace{10em}\\ -% \hspace{10em}E_{z\sim Q}\left[\log P(Y_i|z,X_i)+\log P(z|X_i)-\log Q(z)\right] -%\end{array} -%\end{equation} -%\vspace{-0.05in} -\noindent Note that $X$ is fixed, and $Q$ can be \textit{any} distribution, not just a distribution which does a good job mapping $X$ to the $z$'s that can produce $X$. -Since we're interested in inferring $P(X)$, it makes sense to construct a $Q$ which \textit{does} depend on $X$, and in particular, one which makes $\mathcal{D}\left[Q(z)\|P(z|X)\right]$ small: -\begin{equation} - \log P(X) - \mathcal{D}\left[Q(z|X)\|P(z|X)\right]=E_{z\sim Q}\left[\log P(X|z) \right] - \mathcal{D}\left[Q(z|X)\|P(z)\right]. - \label{eq:variational} -\end{equation} -%Hence, during training, it makes sense to choose a $Q$ which will make $E_{z\sim Q}[\log Q(z)-$ -%$\log P(z|X_i,Y_i)]$ (a $\mathcal{D}$-divergence) small, such that the right hand side is a close approximation to $\log P(Y_i|X_i)$. -\noindent This equation serves as the core of the variational autoencoder, and it's worth spending some time thinking about what it says\footnote{ -Historically, this math (particularly Equation~\ref{eq:variational}) was known long before VAEs. -For example, Helmholtz Machines~\cite{dayan1995helmholtz} (see Equation 5) use nearly identical mathematics, with one crucial difference. -The integral in our expectations is replaced with a sum in Dayan et al.~\cite{dayan1995helmholtz}, because Helmholtz Machines assume a discrete distribution for the latent variables. -This choice prevents the transformations that make gradient descent tractable in VAEs. -}. -In two sentences, the left hand side has the quantity we want to maximize: $\log P(X)$ (plus an error term, which makes $Q$ produce $z$'s that can reproduce a given $X$; this term will become small if $Q$ is high-capacity). -The right hand side is something we can optimize via stochastic gradient descent given the right choice of $Q$ (although it may not be obvious yet how). -Note that the framework---in particular, the right hand side of Equation~\ref{eq:variational}---has suddenly taken a form which looks like an autoencoder, since $Q$ is ``encoding'' $X$ into $z$, and $P$ is ``decoding'' it to reconstruct $X$. -We'll explore this connection in more detail later. -%That is, we have solved our problem of sampling $z$ by training a distribution $Q$ to predict which values of $z$ are likely to produce $X$, and not considering the rest. %we can optimize $P(X)$ in our model just by optimizing the right hand side of this equation! - -Now for a bit more detail on Equatinon~\ref{eq:variational}. -Starting with the left hand side, we are maximizing $\log P(X)$ while simultaneously minimizing $\mathcal{D}\left[Q(z|X)\|P(z|X)\right]$. -$P(z|X)$ is not something we can compute analytically: it describes the values of $z$ that are likely to give rise to a sample like $X$ under our model in Figure~\ref{fig:model}. -However, the second term on the left is pulling $Q(z|x)$ to match $P(z|X)$. -Assuming we use an arbitrarily high-capacity model for $Q(z|x)$, then $Q(z|x)$ will hopefully actually \textit{match} $P(z|X)$, in which case this KL-divergence term will be zero, and we will be directly optimizing $\log P(X)$. -As an added bonus, we have made the intractable $P(z|X)$ tractable: we can just use $Q(z|x)$ to compute it. - -\subsection{Optimizing the objective} - -So how can we perform stochastic gradient descent on the right hand side of Equation~\ref{eq:variational}? -First we need to be a bit more specific about the form that $Q(z|X)$ will take. -The usual choice is to say that $Q(z|X)=\mathcal{N}(z|\mu(X;\vartheta),\Sigma(X;\vartheta))$, where $\mu$ and $\Sigma$ are arbitrary deterministic functions with parameters $\vartheta$ that can be learned from data (we will omit $\vartheta$ in later equations). -In practice, $\mu$ and $\Sigma$ are again implemented via neural networks, and $\Sigma$ is constrained to be a diagonal matrix. -%The name variational ``autoencoder'' comes from the fact that $\mu$ and $\Sigma$ are ``encoding'' $X$ into the latent space $z$. -The advantages of this choice are computational, as they make it clear how to compute the right hand side. -The last term---$\mathcal{D}\left[Q(z|X)\|P(z)\right]$---is now a KL-divergence between two multivariate Gaussian distributions, which can be computed in closed form as: -\begin{equation} +Hence, during training, it makes sense to choose a $Q$ which will make +$E_{z\sim Q}[\log Q(z)-$ $\log P(z|X_i,Y_i)]$ (a +$\mathcal{D}$-divergence) small, such that the right hand side is a +close approximation to $\log P(Y_i|X_i)$. + + +This equation serves as the core of the variational autoencoder, and +it's worth spending some time thinking about what it says In two +sentences, the left hand side has the quantity we want to maximize: +$\log P(X)$ (plus an error term, which makes $Q$ produce $z$'s that +can reproduce a given $X$; this term will become small if $Q$ is +high-capacity). The right hand side is something we can optimize via +stochastic gradient descent given the right choice of $Q$ (although it +may not be obvious yet how). + + + +So how can we perform stochastic gradient descent? + +First we need to be a bit more specific about the form that $Q(z|X)$ +will take. The usual choice is to say that +$Q(z|X)=\mathcal{N}(z|\mu(X;\vartheta),\Sigma(X;\vartheta))$, where +$\mu$ and $\Sigma$ are arbitrary deterministic functions with +parameters $\vartheta$ that can be learned from data (we will omit +$\vartheta$ in later equations). In practice, $\mu$ and $\Sigma$ are +again implemented via neural networks, and $\Sigma$ is constrained to +be a diagonal matrix. The name variational ``autoencoder'' comes from +the fact that $\mu$ and $\Sigma$ are ``encoding'' $X$ into the latent +space $z$. The advantages of this choice are computational, as they +make it clear how to compute the right hand side. The last +term---$\mathcal{D}\left[Q(z|X)\|P(z)\right]$---is now a KL-divergence +between two multivariate Gaussian distributions, which can be computed +in closed form as: +!bt +\[ \begin{array}{c} \mathcal{D}[\mathcal{N}(\mu_0,\Sigma_0) \| \mathcal{N}(\mu_1,\Sigma_1)] = \hspace{20em}\\ \hspace{5em}\frac{ 1 }{ 2 } \left( \mathrm{tr} \left( \Sigma_1^{-1} \Sigma_0 \right) + \left( \mu_1 - \mu_0\right)^\top \Sigma_1^{-1} ( \mu_1 - \mu_0 ) - k + \log \left( \frac{ \det \Sigma_1 }{ \det \Sigma_0 } \right) \right) \end{array} -\end{equation} -\noindent where $k$ is the dimensionality of the distribution. In our case, this simplifies to: -\begin{equation} +\] +!et + +where $k$ is the dimensionality of the distribution. In our case, this simplifies to: +!bt +\[ \begin{array}{c} \mathcal{D}[\mathcal{N}(\mu(X),\Sigma(X)) \| \mathcal{N}(0,I)] = \hspace{20em}\\ \hspace{6em}\frac{ 1 }{ 2 } \left( \mathrm{tr} \left( \Sigma(X) \right) + \left( \mu(X)\right)^\top ( \mu(X) ) - k - \log\det\left( \Sigma(X) \right) \right). \end{array} -\end{equation} +\] +!et -The first term on the right hand side of Equation~\ref{eq:variational} is a bit more tricky. +The first term on the right hand side is a bit more tricky. We could use sampling to estimate $E_{z\sim Q}\left[\log P(X|z) \right]$, but getting a good estimate would require passing many samples of $z$ through $f$, which would be expensive. Hence, as is standard in stochastic gradient descent, we take one sample of $z$ and treat $\log P(X|z)$ for that $z$ as an approximation of $E_{z\sim Q}\left[\log P(X|z) \right]$. After all, we are already doing stochastic gradient descent over different values of $X$ sampled from a dataset $D$. -%, since we need to compute Equation~\ref{eq:variational} on The full equation we want to optimize is: -\begin{equation} + +!bt +\[ \begin{array}{c} E_{X\sim D}\left[\log P(X) - \mathcal{D}\left[Q(z|X)\|P(z|X)\right]\right]=\hspace{16em}\\ \hspace{10em}E_{X\sim D}\left[E_{z\sim Q}\left[\log P(X|z) \right] - \mathcal{D}\left[Q(z|X)\|P(z)\right]\right]. \end{array} - \label{eq:expected} -\end{equation} +\] +!et + If we take the gradient of this equation, the gradient symbol can be moved into the expectations. Therefore, we can sample a single value of $X$ and a single value of $z$ from the distribution $Q(z|X)$, and compute the gradient of: +!bt \begin{equation} \log P(X|z)-\mathcal{D}\left[Q(z|X)\|P(z)\right]. - \label{eq:onesamp} \end{equation} -We can then average the gradient of this function over arbitrarily many samples of $X$ and $z$, and the result converges to the gradient of Equation~\ref{eq:expected}. +!et + +We can then average the gradient of this function over arbitrarily many samples of $X$ and $z$, and the result converges to the gradient. -There is, however, a significant problem with Equation~\ref{eq:onesamp}. +There is, however, a significant problem $E_{z\sim Q}\left[\log P(X|z) \right]$ depends not just on the parameters of $P$, but also on the parameters of $Q$. -However, in Equation~\ref{eq:onesamp}, this dependency has disappeared! -In order to make VAEs work, it's essential to drive $Q$ to produce codes for $X$ that $P$ can reliably decode. -To see the problem a different way, the network described in Equation~\ref{eq:onesamp} is much like the network shown in Figure~\ref{fig:net} (left). -The forward pass of this network works fine and, if the output is averaged over many samples of $X$ and $z$, produces the correct expected value. -However, we need to back-propagate the error through a layer that samples $z$ from $Q(z|X)$, which is a non-continuous operation and has no gradient. -Stochastic gradient descent via backpropagation can handle stochastic inputs, but not stochastic units within the network! -The solution, called the ``reparameterization trick'' in~\cite{Kingma14a}, is to move the sampling to an input layer. -Given $\mu(X)$ and $\Sigma(X)$---the mean and covariance of $Q(z|X)$---we can sample from $\mathcal{N}(\mu(X),\Sigma(X))$ by first sampling $\epsilon \sim \mathcal{N}(0,I)$, then computing $z=\mu(X)+\Sigma^{1/2}(X)*\epsilon$. %, where $\circ$ denotes elementwise product. -Thus, the equation we actually take the gradient of is: -\begin{equation} - E_{X\sim D}\left[E_{\epsilon\sim\mathcal{N}(0,I)}[\log P(X|z=\mu(X)+\Sigma^{1/2}(X)*\epsilon)]-\mathcal{D}\left[Q(z|X)\|P(z)\right]\right]. -\end{equation} -This is shown schematically in Figure~\ref{fig:net} (right). -Note that none of the expectations are with respect to distributions that depend on our model parameters, so we can safely move a gradient symbol into them while maintaning equality. -That is, given a fixed $X$ and $\epsilon$, this function is deterministic and continuous in the parameters of $P$ and $Q$, meaning backpropagation can compute a gradient that will work for stochastic gradient descent. -It's worth pointing out that the ``reparameterization trick'' only works if we can sample from $Q(z|X)$ by evaluating a function $h(\eta,X)$, where $\eta$ is noise from a distribution that is not learned. -Furthermore, $h$ must be \textit{continuous} in $X$ so that we can backprop through it. -This means $Q(z|X)$ (and therefore $P(z)$) can't be a discrete distribution! -If $Q$ is discrete, then for a fixed $\eta$, either $h$ needs to ignore $X$, or there needs to be some point at which $h(\eta,X)$ ``jumps'' from one possible value in $Q$'s sample space to another, i.e., a discontinuity. +In order to make VAEs work, it is essential to drive $Q$ to produce codes for $X$ that $P$ can reliably decode. +!bt +\[ + E_{X\sim D}\left[E_{\epsilon\sim\mathcal{N}(0,I)}[\log P(X|z=\mu(X)+\Sigma^{1/2}(X)*\epsilon)]-\mathcal{D}\left[Q(z|X)\|P(z)\right]\right]. +\] +!et