From 0f25c0ac859e4996dd4132e4c06f943e890886e7 Mon Sep 17 00:00:00 2001 From: Morten Hjorth-Jensen Date: Wed, 1 May 2024 21:54:10 +0200 Subject: [PATCH] update --- doc/pub/week15/html/week15-bs.html | 211 ++++++- doc/pub/week15/html/week15-reveal.html | 211 ++++++- doc/pub/week15/html/week15-solarized.html | 206 +++++- doc/pub/week15/html/week15.html | 206 +++++- doc/pub/week15/ipynb/ipynb-week15-src.tar.gz | Bin 4281721 -> 4281721 bytes doc/pub/week15/ipynb/week15.ipynb | 626 ++++++++++++++----- doc/pub/week15/pdf/week15.pdf | Bin 5600458 -> 5620908 bytes doc/src/week15/week15.do.txt | 589 ++++------------- 8 files changed, 1425 insertions(+), 624 deletions(-) diff --git a/doc/pub/week15/html/week15-bs.html b/doc/pub/week15/html/week15-bs.html index 1d750038..0b8a907b 100644 --- a/doc/pub/week15/html/week15-bs.html +++ b/doc/pub/week15/html/week15-bs.html @@ -77,6 +77,14 @@ None, 'explicit-expression-for-the-derivative'), ('Final expression', 2, None, 'final-expression'), + ('Kullback-Leibler divergence', + 2, + None, + 'kullback-leibler-divergence'), + ('Jensen-Shannon divergence', + 2, + None, + 'jensen-shannon-divergence'), ('Generative model, basic overview (Borrowed from Rashcka et ' 'al)', 2, @@ -136,6 +144,15 @@ 2, None, 'steps-in-building-a-gan-borrowed-from-rashcka-et-al'), + ('Generative Adversarial Networks', + 2, + None, + 'generative-adversarial-networks'), + ('Optimal value for $D$', 2, None, 'optimal-value-for-d'), + ('What does the Loss Function Represent?', + 2, + None, + 'what-does-the-loss-function-represent'), ('More references', 2, None, 'more-references'), ('Writing Our First Generative Adversarial Network', 2, @@ -218,6 +235,8 @@
  • The derivative of the partition function
  • Explicit expression for the derivative
  • Final expression
    • +
  • Kullback-Leibler divergence
    • +
  • Jensen-Shannon divergence
  • Generative model, basic overview (Borrowed from Rashcka et al)
  • Reminder on VAEs
  • Evidence Lower Bound
    • @@ -249,6 +268,9 @@
  • Deafault choice
  • Design of GANs
  • Steps in building a GAN (Borrowed from Rashcka et al)
    • +
  • Generative Adversarial Networks
    • +
  • Optimal value for \( D \)
    • +
  • What does the Loss Function Represent?
  • More references
  • Writing Our First Generative Adversarial Network
  • Implementing the networks (Borrowed from Rashcka et al)
    • @@ -310,7 +332,7 @@

    Plans for the
    1. Summary of Variational Autoencoders
    2. Generative Adversarial Networks (GANs), see https://lilianweng.github.io/posts/2017-08-20-gan/ for nice overview
      3. -
    3. Start discussion of diffusion models
      4. +
    4. Start discussion of diffusion models, motivation
    5. Video of lecture
    6. Whiteboard notes
    @@ -535,6 +557,48 @@

    Final expression

    sampling as the standard sampling rule.

    + +

    Kullback-Leibler divergence

    + +

    Before we continue, we need to remind ourselves about the +Kullback-Leibler divergence introduced earlier. This will also allow +us to introduce another measure used in connection with the training +of Generative Adversarial Networks, the so-called Jensen-Shannon divergence.. +These metrics are useful for quantifying the similarity between two probability distributions. +

    + +

    The Kullback–Leibler (KL) divergence, labeled \( D_{KL} \), measures how one probability distribution \( p \) diverges from a second expected probability distribution \( q \), +that is +

    +$$ +D_{KL}(p \| q) = \int_x p(x) \log \frac{p(x)}{q(x)} dx. +$$ + +

    The KL-divegrnece \( D_{KL} \) achieves the minimum zero when \( p(x) == q(x) \) everywhere.

    + +

    Note that the KL divergence is asymmetric. In cases where \( p(x) \) is +close to zero, but \( q(x) \) is significantly non-zero, the \( q \)'s effect +is disregarded. It could cause buggy results when we just want to +measure the similarity between two equally important distributions. +

    + + +

    Jensen-Shannon divergence

    + +

    The Jensen–Shannon (JS) divergence is another measure of similarity between +two probability distributions, bounded by \( [0, 1] \). The JS-divergence is +symmetric and more smooth than the KL-divergence. +It is defined as +

    +$$ +D_{JS}(p \| q) = \frac{1}{2} D_{KL}(p \| \frac{p + q}{2}) + \frac{1}{2} D_{KL}(q \| \frac{p + q}{2}) +$$ + +

    Many practitioners believe that one reason behind GANs' big success is +switching the loss function from asymmetric KL-divergence in +traditional maximum-likelihood approach to symmetric JS-divergence. +

    +

    Generative model, basic overview (Borrowed from Rashcka et al)

    @@ -627,14 +691,14 @@

    The derivation from last w $$ \begin{align*} -\log p(\boldsymbol{x}) & = \log p(\boldsymbol{x}) \int q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})dz && \text{(Multiply by $1 = \int q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})d\boldsymbol{h}$)}\\ - & = \int q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})(\log p(\boldsymbol{x}))dz && \text{(Bring evidence into integral)}\\ +\log p(\boldsymbol{x}) & = \log p(\boldsymbol{x}) \int q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})dh && \text{(Multiply by $1 = \int q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})d\boldsymbol{h}$)}\\ + & = \int q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})(\log p(\boldsymbol{x}))dh && \text{(Bring evidence into integral)}\\ & = \mathbb{E}_{q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}\left[\log p(\boldsymbol{x})\right] && \text{(Definition of Expectation)}\\ & = \mathbb{E}_{q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}\left[\log\frac{p(\boldsymbol{x}, \boldsymbol{h})}{p(\boldsymbol{h}|\boldsymbol{x})}\right]&& \\ & = \mathbb{E}_{q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}\left[\log\frac{p(\boldsymbol{x}, \boldsymbol{h})q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}{p(\boldsymbol{h}|\boldsymbol{x})q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}\right]&& \text{(Multiply by $1 = \frac{q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}{q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}$)}\\ & = \mathbb{E}_{q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}\left[\log\frac{p(\boldsymbol{x}, \boldsymbol{h})}{q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}\right] + \mathbb{E}_{q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}\left[\log\frac{q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}{p(\boldsymbol{h}|\boldsymbol{x})}\right] && \text{(Split the Expectation)}\\ & = \mathbb{E}_{q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}\left[\log\frac{p(\boldsymbol{x}, \boldsymbol{h})}{q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}\right] + - KL(q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})\vert\vert p(\boldsymbol{h}|\boldsymbol{x})) && \text{(Definition of KL Divergence)}\\ + D_{KL}(q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})\vert\vert p(\boldsymbol{h}|\boldsymbol{x})) && \text{(Definition of KL Divergence)}\\ & \geq \mathbb{E}_{q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}\left[\log\frac{p(\boldsymbol{x}, \boldsymbol{h})}{q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}\right] && \text{(KL Divergence always $\geq 0$)} \end{align*} $$ @@ -675,7 +739,7 @@

    Dissecting the equations

    {\mathbb{E}_{q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}\left[\log\frac{p(\boldsymbol{x}, \boldsymbol{h})}{q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}\right]} &= {\mathbb{E}_{q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}\left[\log\frac{p_{\boldsymbol{\theta}}(\boldsymbol{x}|\boldsymbol{h})p(\boldsymbol{h})}{q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}\right]} && {\text{(Chain Rule of Probability)}}\\ &= {\mathbb{E}_{q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}\left[\log p_{\boldsymbol{\theta}}(\boldsymbol{x}|\boldsymbol{h})\right] + \mathbb{E}_{q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}\left[\log\frac{p(\boldsymbol{h})}{q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}\right]} && {\text{(Split the Expectation)}}\\ -&= \underbrace{{\mathbb{E}_{q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}\left[\log p_{\boldsymbol{\theta}}(\boldsymbol{x}|\boldsymbol{h})\right]}}_\text{reconstruction term} - \underbrace{{KL(q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}\vert\vert{p(\boldsymbol{h}))}}_\text{prior matching term} && {\text{(Definition of KL Divergence)}} +&= \underbrace{{\mathbb{E}_{q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}\left[\log p_{\boldsymbol{\theta}}(\boldsymbol{x}|\boldsymbol{h})\right]}}_\text{reconstruction term} - \underbrace{{D_{KL}(q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}\vert\vert{p(\boldsymbol{h}))}}_\text{prior matching term} && {\text{(Definition of KL Divergence)}} \end{align*} $$ @@ -724,7 +788,7 @@

    Analytical evaluation

    Then, the KL divergence term of the ELBO can be computed analytically, and the reconstruction term can be approximated using a Monte Carlo estimate. Our objective can then be rewritten as:

    $$ \begin{align*} - \mathrm{argmax}_{\boldsymbol{\phi}, \boldsymbol{\theta}} \mathbb{E}_{q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}\left[\log p_{\boldsymbol{\theta}}(\boldsymbol{x}|\boldsymbol{h})\right] - KL(q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})\vert\vert p(\boldsymbol{h})) \approx \mathrm{argmax}_{\boldsymbol{\phi}, \boldsymbol{\theta}} \sum_{l=1}^{L}\log p_{\boldsymbol{\theta}}(\boldsymbol{x}|\boldsymbol{h}^{(l)}) - KL(q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})\vert\vert p(\boldsymbol{h})) + \mathrm{argmax}_{\boldsymbol{\phi}, \boldsymbol{\theta}} \mathbb{E}_{q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}\left[\log p_{\boldsymbol{\theta}}(\boldsymbol{x}|\boldsymbol{h})\right] - D_{KL}(q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})\vert\vert p(\boldsymbol{h})) \approx \mathrm{argmax}_{\boldsymbol{\phi}, \boldsymbol{\theta}} \sum_{l=1}^{L}\log p_{\boldsymbol{\theta}}(\boldsymbol{x}|\boldsymbol{h}^{(l)}) - D_{KL}(q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})\vert\vert p(\boldsymbol{h})) \end{align*} $$ @@ -982,6 +1046,140 @@

    Step

    + +

    Generative Adversarial Networks

    +

    Generative adversarial networks (GANs) have shown great results in +many generative tasks to replicate the real-world rich content such as +images, human language, and music. It is inspired by game theory: two +models, a generator and a discriminator, are competing with each other while +making each other stronger at the same time. However, it is rather +challenging to train a GANs model, +training instability or failure to converge. +

    + +

    Generative adversarial networks consist of two models (in their simplest form as two opposing feed forward neural networks)

    +
      +
    1. A discriminator \( D \) estimates the probability of a given sample coming from the real dataset. It works as a critic and is optimized to tell the fake samples from the realo ones
      2. +
    2. A generator \( G \) outputs synthetic samples given a noise variable input \( z \) (\( z \) brings in potential output diversity). It is trained to capture the real data distribution in order to generate samples that can be as real as possible, or in other words, can trick the discriminator to offer a high probability.
      3. +
    +

    At the end of the training, the generator can be used to generate for +example new images. In this sense we have trained a model which can +produce new samples. We say that we have implicitely defined a +probability. +

    + +

    These two models compete against each other during the training +process: the generator \( G \) is trying hard to trick the discriminator, +while the critic model \( D \) is trying hard not to be cheated. This +interesting zero-sum game between two models motivates both to improve +their functionalities. +

    + + + + + + + +

    On one hand, we want to make sure the discriminator \( D \)'s decisions +over real data are accurate by maximizing \( \mathbb{E}_{x \sim +p_{r}(x)} [\log D(x)] \). Meanwhile, given a fake sample \( G(z), z \sim +p_z(z) \), the discriminator is expected to output a probability, +\( D(G(z)) \), close to zero by maximizing \( \mathbb{E}_{z \sim p_{z}(z)} +[\log (1 - D(G(z)))] \). +

    + +

    On the other hand, the generator is trained to increase the chances of +\( D \) producing a high probability for a fake example, thus to minimize +\( \mathbb{E}_{z \sim p_{z}(z)} [\log (1 - D(G(z)))] \). +

    + +

    When combining both aspects together, \( D \) and \( G \) are playing a \textit{minimax game} in which we should optimize the following loss function:

    + +$$ +\begin{aligned} +\min_G \max_D L(D, G) +& = \mathbb{E}_{x \sim p_{r}(x)} [\log D(x)] + \mathbb{E}_{z \sim p_z(z)} [\log(1 - D(G(z)))] \\ +& = \mathbb{E}_{x \sim p_{r}(x)} [\log D(x)] + \mathbb{E}_{x \sim p_g(x)} [\log(1 - D(x)] +\end{aligned} +$$ + +

    where \( \mathbb{E}_{x \sim p_{r}(x)} [\log D(x)] \) has no impact on \( G \) during gradient descent updates.

    + + +

    Optimal value for \( D \)

    + +

    Now we have a well-defined loss function. Let's first examine what is the best value for \( D \).

    + +$$ +L(G, D) = \int_x \bigg( p_{r}(x) \log(D(x)) + p_g (x) \log(1 - D(x)) \bigg) dx +$$ + +

    Since we are interested in what is the best value of \( D(x) \) to maximize \( L(G, D) \), let us label

    + +$$ +\tilde{x} = D(x), +A=p_{r}(x), +B=p_g(x) +$$ + +

    And then what is inside the integral (we can safely ignore the integral because \( x \) is sampled over all the possible values) is:

    + +$$ +\begin{align*} +f(\tilde{x}) +& = A log\tilde{x} + B log(1-\tilde{x}) \\ +\frac{d f(\tilde{x})}{d \tilde{x}} +& = A \frac{1}{ln10} \frac{1}{\tilde{x}} - B \frac{1}{ln10} \frac{1}{1 - \tilde{x}} \\ +& = \frac{1}{ln10} (\frac{A}{\tilde{x}} - \frac{B}{1-\tilde{x}}) \\ +& = \frac{1}{ln10} \frac{A - (A + B)\tilde{x}}{\tilde{x} (1 - \tilde{x})} \\ +\end{align*} +$$ + +

    Thus, set \( \frac{d f(\tilde{x})}{d \tilde{x}} = 0 \), we get the best value of the discriminator: \( D^*(x) = \tilde{x}^* = \frac{A}{A + B} = \frac{p_{r}(x)}{p_{r}(x) + p_g(x)} \in [0, 1] \). +Once the generator is trained to its optimal, \( p_g \) gets very close to \( p_{r} \). When \( p_g = p_{r} \), \( D^*(x) \) becomes \( 1/2 \). +

    + +

    When both \( G \) and \( D \) are at their optimal values, we have \( p_g = p_{r} \) and \( D^*(x) = 1/2 \) and the loss function becomes:

    + +$$ +\begin{align*} +L(G, D^*) +&= \int_x \bigg( p_{r}(x) \log(D^*(x)) + p_g (x) \log(1 - D^*(x)) \bigg) dx \\ +&= \log \frac{1}{2} \int_x p_{r}(x) dx + \log \frac{1}{2} \int_x p_g(x) dx \\ +&= -2\log2 +\end{align*} +$$ + + + +

    What does the Loss Function Represent?

    + +

    The JS divergence between \( p_{r} \) and \( p_g \) can be computed as:

    + +$$ +\begin{align*} +D_{JS}(p_{r} \| p_g) +=& \frac{1}{2} D_{KL}(p_{r} || \frac{p_{r} + p_g}{2}) + \frac{1}{2} D_{KL}(p_{g} || \frac{p_{r} + p_g}{2}) \\ +=& \frac{1}{2} \bigg( \log2 + \int_x p_{r}(x) \log \frac{p_{r}(x)}{p_{r} + p_g(x)} dx \bigg) + \\& \frac{1}{2} \bigg( \log2 + \int_x p_g(x) \log \frac{p_g(x)}{p_{r} + p_g(x)} dx \bigg) \\ +=& \frac{1}{2} \bigg( \log4 + L(G, D^*) \bigg) +\end{align*} +$$ + +

    Thus,

    + +$$ +L(G, D^*) = 2D_{JS}(p_{r} \| p_g) - 2\log2 +$$ + +

    Essentially the loss function of GAN quantifies the similarity between +the generative data distribution \( p_g \) and the real sample +distribution \( p_{r} \) by JS divergence when the discriminator is +optimal. The best \( G^* \) that replicates the real data distribution +leads to the minimum \( L(G^*, D^*) = -2\log2 \) which is aligned with +equations above. +

    +

    More references

    @@ -1697,6 +1895,7 @@

    Diffusion learning

    smooth target distribution, this method can capture data distributions of arbitrary form.

    + diff --git a/doc/pub/week15/html/week15-reveal.html b/doc/pub/week15/html/week15-reveal.html index c496b5a9..fdaad2bb 100644 --- a/doc/pub/week15/html/week15-reveal.html +++ b/doc/pub/week15/html/week15-reveal.html @@ -203,7 +203,7 @@

    Plans for the week of April 2

    1. Summary of Variational Autoencoders

    2. Generative Adversarial Networks (GANs), see https://lilianweng.github.io/posts/2017-08-20-gan/ for nice overview
      3. -

    3. Start discussion of diffusion models
      4. +

    4. Start discussion of diffusion models, motivation

    5. Video of lecture

    6. Whiteboard notes
    @@ -477,6 +477,54 @@

    Final expression

    +
    +

    Kullback-Leibler divergence

    + +

    Before we continue, we need to remind ourselves about the +Kullback-Leibler divergence introduced earlier. This will also allow +us to introduce another measure used in connection with the training +of Generative Adversarial Networks, the so-called Jensen-Shannon divergence.. +These metrics are useful for quantifying the similarity between two probability distributions. +

    + +

    The Kullback–Leibler (KL) divergence, labeled \( D_{KL} \), measures how one probability distribution \( p \) diverges from a second expected probability distribution \( q \), +that is +

    +

     
    +$$ +D_{KL}(p \| q) = \int_x p(x) \log \frac{p(x)}{q(x)} dx. +$$ +

     
    + +

    The KL-divegrnece \( D_{KL} \) achieves the minimum zero when \( p(x) == q(x) \) everywhere.

    + +

    Note that the KL divergence is asymmetric. In cases where \( p(x) \) is +close to zero, but \( q(x) \) is significantly non-zero, the \( q \)'s effect +is disregarded. It could cause buggy results when we just want to +measure the similarity between two equally important distributions. +

    +
    + +
    +

    Jensen-Shannon divergence

    + +

    The Jensen–Shannon (JS) divergence is another measure of similarity between +two probability distributions, bounded by \( [0, 1] \). The JS-divergence is +symmetric and more smooth than the KL-divergence. +It is defined as +

    +

     
    +$$ +D_{JS}(p \| q) = \frac{1}{2} D_{KL}(p \| \frac{p + q}{2}) + \frac{1}{2} D_{KL}(q \| \frac{p + q}{2}) +$$ +

     
    + +

    Many practitioners believe that one reason behind GANs' big success is +switching the loss function from asymmetric KL-divergence in +traditional maximum-likelihood approach to symmetric JS-divergence. +

    +
    +

    Generative model, basic overview (Borrowed from Rashcka et al)

    @@ -581,14 +629,14 @@

    The derivation from last week

    $$ \begin{align*} -\log p(\boldsymbol{x}) & = \log p(\boldsymbol{x}) \int q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})dz && \text{(Multiply by $1 = \int q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})d\boldsymbol{h}$)}\\ - & = \int q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})(\log p(\boldsymbol{x}))dz && \text{(Bring evidence into integral)}\\ +\log p(\boldsymbol{x}) & = \log p(\boldsymbol{x}) \int q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})dh && \text{(Multiply by $1 = \int q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})d\boldsymbol{h}$)}\\ + & = \int q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})(\log p(\boldsymbol{x}))dh && \text{(Bring evidence into integral)}\\ & = \mathbb{E}_{q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}\left[\log p(\boldsymbol{x})\right] && \text{(Definition of Expectation)}\\ & = \mathbb{E}_{q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}\left[\log\frac{p(\boldsymbol{x}, \boldsymbol{h})}{p(\boldsymbol{h}|\boldsymbol{x})}\right]&& \\ & = \mathbb{E}_{q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}\left[\log\frac{p(\boldsymbol{x}, \boldsymbol{h})q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}{p(\boldsymbol{h}|\boldsymbol{x})q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}\right]&& \text{(Multiply by $1 = \frac{q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}{q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}$)}\\ & = \mathbb{E}_{q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}\left[\log\frac{p(\boldsymbol{x}, \boldsymbol{h})}{q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}\right] + \mathbb{E}_{q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}\left[\log\frac{q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}{p(\boldsymbol{h}|\boldsymbol{x})}\right] && \text{(Split the Expectation)}\\ & = \mathbb{E}_{q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}\left[\log\frac{p(\boldsymbol{x}, \boldsymbol{h})}{q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}\right] + - KL(q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})\vert\vert p(\boldsymbol{h}|\boldsymbol{x})) && \text{(Definition of KL Divergence)}\\ + D_{KL}(q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})\vert\vert p(\boldsymbol{h}|\boldsymbol{x})) && \text{(Definition of KL Divergence)}\\ & \geq \mathbb{E}_{q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}\left[\log\frac{p(\boldsymbol{x}, \boldsymbol{h})}{q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}\right] && \text{(KL Divergence always $\geq 0$)} \end{align*} $$ @@ -632,7 +680,7 @@

    Dissecting the equations

    {\mathbb{E}_{q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}\left[\log\frac{p(\boldsymbol{x}, \boldsymbol{h})}{q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}\right]} &= {\mathbb{E}_{q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}\left[\log\frac{p_{\boldsymbol{\theta}}(\boldsymbol{x}|\boldsymbol{h})p(\boldsymbol{h})}{q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}\right]} && {\text{(Chain Rule of Probability)}}\\ &= {\mathbb{E}_{q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}\left[\log p_{\boldsymbol{\theta}}(\boldsymbol{x}|\boldsymbol{h})\right] + \mathbb{E}_{q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}\left[\log\frac{p(\boldsymbol{h})}{q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}\right]} && {\text{(Split the Expectation)}}\\ -&= \underbrace{{\mathbb{E}_{q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}\left[\log p_{\boldsymbol{\theta}}(\boldsymbol{x}|\boldsymbol{h})\right]}}_\text{reconstruction term} - \underbrace{{KL(q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}\vert\vert{p(\boldsymbol{h}))}}_\text{prior matching term} && {\text{(Definition of KL Divergence)}} +&= \underbrace{{\mathbb{E}_{q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}\left[\log p_{\boldsymbol{\theta}}(\boldsymbol{x}|\boldsymbol{h})\right]}}_\text{reconstruction term} - \underbrace{{D_{KL}(q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}\vert\vert{p(\boldsymbol{h}))}}_\text{prior matching term} && {\text{(Definition of KL Divergence)}} \end{align*} $$

     
    @@ -687,7 +735,7 @@

    Analytical evaluation

     
    $$ \begin{align*} - \mathrm{argmax}_{\boldsymbol{\phi}, \boldsymbol{\theta}} \mathbb{E}_{q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}\left[\log p_{\boldsymbol{\theta}}(\boldsymbol{x}|\boldsymbol{h})\right] - KL(q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})\vert\vert p(\boldsymbol{h})) \approx \mathrm{argmax}_{\boldsymbol{\phi}, \boldsymbol{\theta}} \sum_{l=1}^{L}\log p_{\boldsymbol{\theta}}(\boldsymbol{x}|\boldsymbol{h}^{(l)}) - KL(q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})\vert\vert p(\boldsymbol{h})) + \mathrm{argmax}_{\boldsymbol{\phi}, \boldsymbol{\theta}} \mathbb{E}_{q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}\left[\log p_{\boldsymbol{\theta}}(\boldsymbol{x}|\boldsymbol{h})\right] - D_{KL}(q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})\vert\vert p(\boldsymbol{h})) \approx \mathrm{argmax}_{\boldsymbol{\phi}, \boldsymbol{\theta}} \sum_{l=1}^{L}\log p_{\boldsymbol{\theta}}(\boldsymbol{x}|\boldsymbol{h}^{(l)}) - D_{KL}(q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})\vert\vert p(\boldsymbol{h})) \end{align*} $$

     
    @@ -980,6 +1028,157 @@

    Steps in building a

    +
    +

    Generative Adversarial Networks

    +

    Generative adversarial networks (GANs) have shown great results in +many generative tasks to replicate the real-world rich content such as +images, human language, and music. It is inspired by game theory: two +models, a generator and a discriminator, are competing with each other while +making each other stronger at the same time. However, it is rather +challenging to train a GANs model, +training instability or failure to converge. +

    + +

    Generative adversarial networks consist of two models (in their simplest form as two opposing feed forward neural networks)

    +
      +

    1. A discriminator \( D \) estimates the probability of a given sample coming from the real dataset. It works as a critic and is optimized to tell the fake samples from the realo ones
      2. +

    2. A generator \( G \) outputs synthetic samples given a noise variable input \( z \) (\( z \) brings in potential output diversity). It is trained to capture the real data distribution in order to generate samples that can be as real as possible, or in other words, can trick the discriminator to offer a high probability.
      3. +
    +

    +

    At the end of the training, the generator can be used to generate for +example new images. In this sense we have trained a model which can +produce new samples. We say that we have implicitely defined a +probability. +

    + +

    These two models compete against each other during the training +process: the generator \( G \) is trying hard to trick the discriminator, +while the critic model \( D \) is trying hard not to be cheated. This +interesting zero-sum game between two models motivates both to improve +their functionalities. +

    + + + + + + + +

    On one hand, we want to make sure the discriminator \( D \)'s decisions +over real data are accurate by maximizing \( \mathbb{E}_{x \sim +p_{r}(x)} [\log D(x)] \). Meanwhile, given a fake sample \( G(z), z \sim +p_z(z) \), the discriminator is expected to output a probability, +\( D(G(z)) \), close to zero by maximizing \( \mathbb{E}_{z \sim p_{z}(z)} +[\log (1 - D(G(z)))] \). +

    + +

    On the other hand, the generator is trained to increase the chances of +\( D \) producing a high probability for a fake example, thus to minimize +\( \mathbb{E}_{z \sim p_{z}(z)} [\log (1 - D(G(z)))] \). +

    + +

    When combining both aspects together, \( D \) and \( G \) are playing a \textit{minimax game} in which we should optimize the following loss function:

    + +

     
    +$$ +\begin{aligned} +\min_G \max_D L(D, G) +& = \mathbb{E}_{x \sim p_{r}(x)} [\log D(x)] + \mathbb{E}_{z \sim p_z(z)} [\log(1 - D(G(z)))] \\ +& = \mathbb{E}_{x \sim p_{r}(x)} [\log D(x)] + \mathbb{E}_{x \sim p_g(x)} [\log(1 - D(x)] +\end{aligned} +$$ +

     
    + +

    where \( \mathbb{E}_{x \sim p_{r}(x)} [\log D(x)] \) has no impact on \( G \) during gradient descent updates.

    +
    + +
    +

    Optimal value for \( D \)

    + +

    Now we have a well-defined loss function. Let's first examine what is the best value for \( D \).

    + +

     
    +$$ +L(G, D) = \int_x \bigg( p_{r}(x) \log(D(x)) + p_g (x) \log(1 - D(x)) \bigg) dx +$$ +

     
    + +

    Since we are interested in what is the best value of \( D(x) \) to maximize \( L(G, D) \), let us label

    + +

     
    +$$ +\tilde{x} = D(x), +A=p_{r}(x), +B=p_g(x) +$$ +

     
    + +

    And then what is inside the integral (we can safely ignore the integral because \( x \) is sampled over all the possible values) is:

    + +

     
    +$$ +\begin{align*} +f(\tilde{x}) +& = A log\tilde{x} + B log(1-\tilde{x}) \\ +\frac{d f(\tilde{x})}{d \tilde{x}} +& = A \frac{1}{ln10} \frac{1}{\tilde{x}} - B \frac{1}{ln10} \frac{1}{1 - \tilde{x}} \\ +& = \frac{1}{ln10} (\frac{A}{\tilde{x}} - \frac{B}{1-\tilde{x}}) \\ +& = \frac{1}{ln10} \frac{A - (A + B)\tilde{x}}{\tilde{x} (1 - \tilde{x})} \\ +\end{align*} +$$ +

     
    + +

    Thus, set \( \frac{d f(\tilde{x})}{d \tilde{x}} = 0 \), we get the best value of the discriminator: \( D^*(x) = \tilde{x}^* = \frac{A}{A + B} = \frac{p_{r}(x)}{p_{r}(x) + p_g(x)} \in [0, 1] \). +Once the generator is trained to its optimal, \( p_g \) gets very close to \( p_{r} \). When \( p_g = p_{r} \), \( D^*(x) \) becomes \( 1/2 \). +

    + +

    When both \( G \) and \( D \) are at their optimal values, we have \( p_g = p_{r} \) and \( D^*(x) = 1/2 \) and the loss function becomes:

    + +

     
    +$$ +\begin{align*} +L(G, D^*) +&= \int_x \bigg( p_{r}(x) \log(D^*(x)) + p_g (x) \log(1 - D^*(x)) \bigg) dx \\ +&= \log \frac{1}{2} \int_x p_{r}(x) dx + \log \frac{1}{2} \int_x p_g(x) dx \\ +&= -2\log2 +\end{align*} +$$ +

     
    +

    + +
    +

    What does the Loss Function Represent?

    + +

    The JS divergence between \( p_{r} \) and \( p_g \) can be computed as:

    + +

     
    +$$ +\begin{align*} +D_{JS}(p_{r} \| p_g) +=& \frac{1}{2} D_{KL}(p_{r} || \frac{p_{r} + p_g}{2}) + \frac{1}{2} D_{KL}(p_{g} || \frac{p_{r} + p_g}{2}) \\ +=& \frac{1}{2} \bigg( \log2 + \int_x p_{r}(x) \log \frac{p_{r}(x)}{p_{r} + p_g(x)} dx \bigg) + \\& \frac{1}{2} \bigg( \log2 + \int_x p_g(x) \log \frac{p_g(x)}{p_{r} + p_g(x)} dx \bigg) \\ +=& \frac{1}{2} \bigg( \log4 + L(G, D^*) \bigg) +\end{align*} +$$ +

     
    + +

    Thus,

    + +

     
    +$$ +L(G, D^*) = 2D_{JS}(p_{r} \| p_g) - 2\log2 +$$ +

     
    + +

    Essentially the loss function of GAN quantifies the similarity between +the generative data distribution \( p_g \) and the real sample +distribution \( p_{r} \) by JS divergence when the discriminator is +optimal. The best \( G^* \) that replicates the real data distribution +leads to the minimum \( L(G^*, D^*) = -2\log2 \) which is aligned with +equations above. +

    +
    +

    More references

    diff --git a/doc/pub/week15/html/week15-solarized.html b/doc/pub/week15/html/week15-solarized.html index 8c5fd574..37e3193f 100644 --- a/doc/pub/week15/html/week15-solarized.html +++ b/doc/pub/week15/html/week15-solarized.html @@ -104,6 +104,14 @@ None, 'explicit-expression-for-the-derivative'), ('Final expression', 2, None, 'final-expression'), + ('Kullback-Leibler divergence', + 2, + None, + 'kullback-leibler-divergence'), + ('Jensen-Shannon divergence', + 2, + None, + 'jensen-shannon-divergence'), ('Generative model, basic overview (Borrowed from Rashcka et ' 'al)', 2, @@ -163,6 +171,15 @@ 2, None, 'steps-in-building-a-gan-borrowed-from-rashcka-et-al'), + ('Generative Adversarial Networks', + 2, + None, + 'generative-adversarial-networks'), + ('Optimal value for $D$', 2, None, 'optimal-value-for-d'), + ('What does the Loss Function Represent?', + 2, + None, + 'what-does-the-loss-function-represent'), ('More references', 2, None, 'more-references'), ('Writing Our First Generative Adversarial Network', 2, @@ -245,7 +262,7 @@

    Plans for the week of April 2
    1. Summary of Variational Autoencoders
    2. Generative Adversarial Networks (GANs), see https://lilianweng.github.io/posts/2017-08-20-gan/ for nice overview
      3. -
    3. Start discussion of diffusion models
      4. +
    4. Start discussion of diffusion models, motivation
    5. Video of lecture
    6. Whiteboard notes
    @@ -468,6 +485,48 @@

    Final expression

    sampling as the standard sampling rule.

    +









    +

    Kullback-Leibler divergence

    + +

    Before we continue, we need to remind ourselves about the +Kullback-Leibler divergence introduced earlier. This will also allow +us to introduce another measure used in connection with the training +of Generative Adversarial Networks, the so-called Jensen-Shannon divergence.. +These metrics are useful for quantifying the similarity between two probability distributions. +

    + +

    The Kullback–Leibler (KL) divergence, labeled \( D_{KL} \), measures how one probability distribution \( p \) diverges from a second expected probability distribution \( q \), +that is +

    +$$ +D_{KL}(p \| q) = \int_x p(x) \log \frac{p(x)}{q(x)} dx. +$$ + +

    The KL-divegrnece \( D_{KL} \) achieves the minimum zero when \( p(x) == q(x) \) everywhere.

    + +

    Note that the KL divergence is asymmetric. In cases where \( p(x) \) is +close to zero, but \( q(x) \) is significantly non-zero, the \( q \)'s effect +is disregarded. It could cause buggy results when we just want to +measure the similarity between two equally important distributions. +

    + +









    +

    Jensen-Shannon divergence

    + +

    The Jensen–Shannon (JS) divergence is another measure of similarity between +two probability distributions, bounded by \( [0, 1] \). The JS-divergence is +symmetric and more smooth than the KL-divergence. +It is defined as +

    +$$ +D_{JS}(p \| q) = \frac{1}{2} D_{KL}(p \| \frac{p + q}{2}) + \frac{1}{2} D_{KL}(q \| \frac{p + q}{2}) +$$ + +

    Many practitioners believe that one reason behind GANs' big success is +switching the loss function from asymmetric KL-divergence in +traditional maximum-likelihood approach to symmetric JS-divergence. +

    +









    Generative model, basic overview (Borrowed from Rashcka et al)

    @@ -560,14 +619,14 @@

    The derivation from last week

    $$ \begin{align*} -\log p(\boldsymbol{x}) & = \log p(\boldsymbol{x}) \int q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})dz && \text{(Multiply by $1 = \int q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})d\boldsymbol{h}$)}\\ - & = \int q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})(\log p(\boldsymbol{x}))dz && \text{(Bring evidence into integral)}\\ +\log p(\boldsymbol{x}) & = \log p(\boldsymbol{x}) \int q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})dh && \text{(Multiply by $1 = \int q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})d\boldsymbol{h}$)}\\ + & = \int q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})(\log p(\boldsymbol{x}))dh && \text{(Bring evidence into integral)}\\ & = \mathbb{E}_{q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}\left[\log p(\boldsymbol{x})\right] && \text{(Definition of Expectation)}\\ & = \mathbb{E}_{q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}\left[\log\frac{p(\boldsymbol{x}, \boldsymbol{h})}{p(\boldsymbol{h}|\boldsymbol{x})}\right]&& \\ & = \mathbb{E}_{q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}\left[\log\frac{p(\boldsymbol{x}, \boldsymbol{h})q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}{p(\boldsymbol{h}|\boldsymbol{x})q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}\right]&& \text{(Multiply by $1 = \frac{q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}{q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}$)}\\ & = \mathbb{E}_{q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}\left[\log\frac{p(\boldsymbol{x}, \boldsymbol{h})}{q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}\right] + \mathbb{E}_{q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}\left[\log\frac{q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}{p(\boldsymbol{h}|\boldsymbol{x})}\right] && \text{(Split the Expectation)}\\ & = \mathbb{E}_{q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}\left[\log\frac{p(\boldsymbol{x}, \boldsymbol{h})}{q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}\right] + - KL(q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})\vert\vert p(\boldsymbol{h}|\boldsymbol{x})) && \text{(Definition of KL Divergence)}\\ + D_{KL}(q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})\vert\vert p(\boldsymbol{h}|\boldsymbol{x})) && \text{(Definition of KL Divergence)}\\ & \geq \mathbb{E}_{q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}\left[\log\frac{p(\boldsymbol{x}, \boldsymbol{h})}{q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}\right] && \text{(KL Divergence always $\geq 0$)} \end{align*} $$ @@ -608,7 +667,7 @@

    Dissecting the equations

    {\mathbb{E}_{q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}\left[\log\frac{p(\boldsymbol{x}, \boldsymbol{h})}{q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}\right]} &= {\mathbb{E}_{q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}\left[\log\frac{p_{\boldsymbol{\theta}}(\boldsymbol{x}|\boldsymbol{h})p(\boldsymbol{h})}{q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}\right]} && {\text{(Chain Rule of Probability)}}\\ &= {\mathbb{E}_{q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}\left[\log p_{\boldsymbol{\theta}}(\boldsymbol{x}|\boldsymbol{h})\right] + \mathbb{E}_{q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}\left[\log\frac{p(\boldsymbol{h})}{q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}\right]} && {\text{(Split the Expectation)}}\\ -&= \underbrace{{\mathbb{E}_{q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}\left[\log p_{\boldsymbol{\theta}}(\boldsymbol{x}|\boldsymbol{h})\right]}}_\text{reconstruction term} - \underbrace{{KL(q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}\vert\vert{p(\boldsymbol{h}))}}_\text{prior matching term} && {\text{(Definition of KL Divergence)}} +&= \underbrace{{\mathbb{E}_{q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}\left[\log p_{\boldsymbol{\theta}}(\boldsymbol{x}|\boldsymbol{h})\right]}}_\text{reconstruction term} - \underbrace{{D_{KL}(q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}\vert\vert{p(\boldsymbol{h}))}}_\text{prior matching term} && {\text{(Definition of KL Divergence)}} \end{align*} $$ @@ -657,7 +716,7 @@

    Analytical evaluation

    Then, the KL divergence term of the ELBO can be computed analytically, and the reconstruction term can be approximated using a Monte Carlo estimate. Our objective can then be rewritten as:

    $$ \begin{align*} - \mathrm{argmax}_{\boldsymbol{\phi}, \boldsymbol{\theta}} \mathbb{E}_{q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}\left[\log p_{\boldsymbol{\theta}}(\boldsymbol{x}|\boldsymbol{h})\right] - KL(q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})\vert\vert p(\boldsymbol{h})) \approx \mathrm{argmax}_{\boldsymbol{\phi}, \boldsymbol{\theta}} \sum_{l=1}^{L}\log p_{\boldsymbol{\theta}}(\boldsymbol{x}|\boldsymbol{h}^{(l)}) - KL(q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})\vert\vert p(\boldsymbol{h})) + \mathrm{argmax}_{\boldsymbol{\phi}, \boldsymbol{\theta}} \mathbb{E}_{q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}\left[\log p_{\boldsymbol{\theta}}(\boldsymbol{x}|\boldsymbol{h})\right] - D_{KL}(q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})\vert\vert p(\boldsymbol{h})) \approx \mathrm{argmax}_{\boldsymbol{\phi}, \boldsymbol{\theta}} \sum_{l=1}^{L}\log p_{\boldsymbol{\theta}}(\boldsymbol{x}|\boldsymbol{h}^{(l)}) - D_{KL}(q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})\vert\vert p(\boldsymbol{h})) \end{align*} $$ @@ -915,6 +974,140 @@

    Steps in building a

    +









    +

    Generative Adversarial Networks

    +

    Generative adversarial networks (GANs) have shown great results in +many generative tasks to replicate the real-world rich content such as +images, human language, and music. It is inspired by game theory: two +models, a generator and a discriminator, are competing with each other while +making each other stronger at the same time. However, it is rather +challenging to train a GANs model, +training instability or failure to converge. +

    + +

    Generative adversarial networks consist of two models (in their simplest form as two opposing feed forward neural networks)

    +
      +
    1. A discriminator \( D \) estimates the probability of a given sample coming from the real dataset. It works as a critic and is optimized to tell the fake samples from the realo ones
      2. +
    2. A generator \( G \) outputs synthetic samples given a noise variable input \( z \) (\( z \) brings in potential output diversity). It is trained to capture the real data distribution in order to generate samples that can be as real as possible, or in other words, can trick the discriminator to offer a high probability.
      3. +
    +

    At the end of the training, the generator can be used to generate for +example new images. In this sense we have trained a model which can +produce new samples. We say that we have implicitely defined a +probability. +

    + +

    These two models compete against each other during the training +process: the generator \( G \) is trying hard to trick the discriminator, +while the critic model \( D \) is trying hard not to be cheated. This +interesting zero-sum game between two models motivates both to improve +their functionalities. +

    + + + + + + + +

    On one hand, we want to make sure the discriminator \( D \)'s decisions +over real data are accurate by maximizing \( \mathbb{E}_{x \sim +p_{r}(x)} [\log D(x)] \). Meanwhile, given a fake sample \( G(z), z \sim +p_z(z) \), the discriminator is expected to output a probability, +\( D(G(z)) \), close to zero by maximizing \( \mathbb{E}_{z \sim p_{z}(z)} +[\log (1 - D(G(z)))] \). +

    + +

    On the other hand, the generator is trained to increase the chances of +\( D \) producing a high probability for a fake example, thus to minimize +\( \mathbb{E}_{z \sim p_{z}(z)} [\log (1 - D(G(z)))] \). +

    + +

    When combining both aspects together, \( D \) and \( G \) are playing a \textit{minimax game} in which we should optimize the following loss function:

    + +$$ +\begin{aligned} +\min_G \max_D L(D, G) +& = \mathbb{E}_{x \sim p_{r}(x)} [\log D(x)] + \mathbb{E}_{z \sim p_z(z)} [\log(1 - D(G(z)))] \\ +& = \mathbb{E}_{x \sim p_{r}(x)} [\log D(x)] + \mathbb{E}_{x \sim p_g(x)} [\log(1 - D(x)] +\end{aligned} +$$ + +

    where \( \mathbb{E}_{x \sim p_{r}(x)} [\log D(x)] \) has no impact on \( G \) during gradient descent updates.

    + +









    +

    Optimal value for \( D \)

    + +

    Now we have a well-defined loss function. Let's first examine what is the best value for \( D \).

    + +$$ +L(G, D) = \int_x \bigg( p_{r}(x) \log(D(x)) + p_g (x) \log(1 - D(x)) \bigg) dx +$$ + +

    Since we are interested in what is the best value of \( D(x) \) to maximize \( L(G, D) \), let us label

    + +$$ +\tilde{x} = D(x), +A=p_{r}(x), +B=p_g(x) +$$ + +

    And then what is inside the integral (we can safely ignore the integral because \( x \) is sampled over all the possible values) is:

    + +$$ +\begin{align*} +f(\tilde{x}) +& = A log\tilde{x} + B log(1-\tilde{x}) \\ +\frac{d f(\tilde{x})}{d \tilde{x}} +& = A \frac{1}{ln10} \frac{1}{\tilde{x}} - B \frac{1}{ln10} \frac{1}{1 - \tilde{x}} \\ +& = \frac{1}{ln10} (\frac{A}{\tilde{x}} - \frac{B}{1-\tilde{x}}) \\ +& = \frac{1}{ln10} \frac{A - (A + B)\tilde{x}}{\tilde{x} (1 - \tilde{x})} \\ +\end{align*} +$$ + +

    Thus, set \( \frac{d f(\tilde{x})}{d \tilde{x}} = 0 \), we get the best value of the discriminator: \( D^*(x) = \tilde{x}^* = \frac{A}{A + B} = \frac{p_{r}(x)}{p_{r}(x) + p_g(x)} \in [0, 1] \). +Once the generator is trained to its optimal, \( p_g \) gets very close to \( p_{r} \). When \( p_g = p_{r} \), \( D^*(x) \) becomes \( 1/2 \). +

    + +

    When both \( G \) and \( D \) are at their optimal values, we have \( p_g = p_{r} \) and \( D^*(x) = 1/2 \) and the loss function becomes:

    + +$$ +\begin{align*} +L(G, D^*) +&= \int_x \bigg( p_{r}(x) \log(D^*(x)) + p_g (x) \log(1 - D^*(x)) \bigg) dx \\ +&= \log \frac{1}{2} \int_x p_{r}(x) dx + \log \frac{1}{2} \int_x p_g(x) dx \\ +&= -2\log2 +\end{align*} +$$ + + +









    +

    What does the Loss Function Represent?

    + +

    The JS divergence between \( p_{r} \) and \( p_g \) can be computed as:

    + +$$ +\begin{align*} +D_{JS}(p_{r} \| p_g) +=& \frac{1}{2} D_{KL}(p_{r} || \frac{p_{r} + p_g}{2}) + \frac{1}{2} D_{KL}(p_{g} || \frac{p_{r} + p_g}{2}) \\ +=& \frac{1}{2} \bigg( \log2 + \int_x p_{r}(x) \log \frac{p_{r}(x)}{p_{r} + p_g(x)} dx \bigg) + \\& \frac{1}{2} \bigg( \log2 + \int_x p_g(x) \log \frac{p_g(x)}{p_{r} + p_g(x)} dx \bigg) \\ +=& \frac{1}{2} \bigg( \log4 + L(G, D^*) \bigg) +\end{align*} +$$ + +

    Thus,

    + +$$ +L(G, D^*) = 2D_{JS}(p_{r} \| p_g) - 2\log2 +$$ + +

    Essentially the loss function of GAN quantifies the similarity between +the generative data distribution \( p_g \) and the real sample +distribution \( p_{r} \) by JS divergence when the discriminator is +optimal. The best \( G^* \) that replicates the real data distribution +leads to the minimum \( L(G^*, D^*) = -2\log2 \) which is aligned with +equations above. +

    +









    More references

    @@ -1630,6 +1823,7 @@

    Diffusion learning

    smooth target distribution, this method can capture data distributions of arbitrary form.

    +
    © 1999-2024, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license diff --git a/doc/pub/week15/html/week15.html b/doc/pub/week15/html/week15.html index 9bb01f5f..39303bd7 100644 --- a/doc/pub/week15/html/week15.html +++ b/doc/pub/week15/html/week15.html @@ -181,6 +181,14 @@ None, 'explicit-expression-for-the-derivative'), ('Final expression', 2, None, 'final-expression'), + ('Kullback-Leibler divergence', + 2, + None, + 'kullback-leibler-divergence'), + ('Jensen-Shannon divergence', + 2, + None, + 'jensen-shannon-divergence'), ('Generative model, basic overview (Borrowed from Rashcka et ' 'al)', 2, @@ -240,6 +248,15 @@ 2, None, 'steps-in-building-a-gan-borrowed-from-rashcka-et-al'), + ('Generative Adversarial Networks', + 2, + None, + 'generative-adversarial-networks'), + ('Optimal value for $D$', 2, None, 'optimal-value-for-d'), + ('What does the Loss Function Represent?', + 2, + None, + 'what-does-the-loss-function-represent'), ('More references', 2, None, 'more-references'), ('Writing Our First Generative Adversarial Network', 2, @@ -322,7 +339,7 @@

    Plans for the week of April 2
    1. Summary of Variational Autoencoders
    2. Generative Adversarial Networks (GANs), see https://lilianweng.github.io/posts/2017-08-20-gan/ for nice overview
      3. -
    3. Start discussion of diffusion models
      4. +
    4. Start discussion of diffusion models, motivation
    5. Video of lecture
    6. Whiteboard notes
    @@ -545,6 +562,48 @@

    Final expression

    sampling as the standard sampling rule.

    +









    +

    Kullback-Leibler divergence

    + +

    Before we continue, we need to remind ourselves about the +Kullback-Leibler divergence introduced earlier. This will also allow +us to introduce another measure used in connection with the training +of Generative Adversarial Networks, the so-called Jensen-Shannon divergence.. +These metrics are useful for quantifying the similarity between two probability distributions. +

    + +

    The Kullback–Leibler (KL) divergence, labeled \( D_{KL} \), measures how one probability distribution \( p \) diverges from a second expected probability distribution \( q \), +that is +

    +$$ +D_{KL}(p \| q) = \int_x p(x) \log \frac{p(x)}{q(x)} dx. +$$ + +

    The KL-divegrnece \( D_{KL} \) achieves the minimum zero when \( p(x) == q(x) \) everywhere.

    + +

    Note that the KL divergence is asymmetric. In cases where \( p(x) \) is +close to zero, but \( q(x) \) is significantly non-zero, the \( q \)'s effect +is disregarded. It could cause buggy results when we just want to +measure the similarity between two equally important distributions. +

    + +









    +

    Jensen-Shannon divergence

    + +

    The Jensen–Shannon (JS) divergence is another measure of similarity between +two probability distributions, bounded by \( [0, 1] \). The JS-divergence is +symmetric and more smooth than the KL-divergence. +It is defined as +

    +$$ +D_{JS}(p \| q) = \frac{1}{2} D_{KL}(p \| \frac{p + q}{2}) + \frac{1}{2} D_{KL}(q \| \frac{p + q}{2}) +$$ + +

    Many practitioners believe that one reason behind GANs' big success is +switching the loss function from asymmetric KL-divergence in +traditional maximum-likelihood approach to symmetric JS-divergence. +

    +









    Generative model, basic overview (Borrowed from Rashcka et al)

    @@ -637,14 +696,14 @@

    The derivation from last week

    $$ \begin{align*} -\log p(\boldsymbol{x}) & = \log p(\boldsymbol{x}) \int q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})dz && \text{(Multiply by $1 = \int q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})d\boldsymbol{h}$)}\\ - & = \int q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})(\log p(\boldsymbol{x}))dz && \text{(Bring evidence into integral)}\\ +\log p(\boldsymbol{x}) & = \log p(\boldsymbol{x}) \int q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})dh && \text{(Multiply by $1 = \int q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})d\boldsymbol{h}$)}\\ + & = \int q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})(\log p(\boldsymbol{x}))dh && \text{(Bring evidence into integral)}\\ & = \mathbb{E}_{q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}\left[\log p(\boldsymbol{x})\right] && \text{(Definition of Expectation)}\\ & = \mathbb{E}_{q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}\left[\log\frac{p(\boldsymbol{x}, \boldsymbol{h})}{p(\boldsymbol{h}|\boldsymbol{x})}\right]&& \\ & = \mathbb{E}_{q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}\left[\log\frac{p(\boldsymbol{x}, \boldsymbol{h})q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}{p(\boldsymbol{h}|\boldsymbol{x})q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}\right]&& \text{(Multiply by $1 = \frac{q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}{q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}$)}\\ & = \mathbb{E}_{q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}\left[\log\frac{p(\boldsymbol{x}, \boldsymbol{h})}{q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}\right] + \mathbb{E}_{q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}\left[\log\frac{q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}{p(\boldsymbol{h}|\boldsymbol{x})}\right] && \text{(Split the Expectation)}\\ & = \mathbb{E}_{q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}\left[\log\frac{p(\boldsymbol{x}, \boldsymbol{h})}{q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}\right] + - KL(q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})\vert\vert p(\boldsymbol{h}|\boldsymbol{x})) && \text{(Definition of KL Divergence)}\\ + D_{KL}(q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})\vert\vert p(\boldsymbol{h}|\boldsymbol{x})) && \text{(Definition of KL Divergence)}\\ & \geq \mathbb{E}_{q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}\left[\log\frac{p(\boldsymbol{x}, \boldsymbol{h})}{q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}\right] && \text{(KL Divergence always $\geq 0$)} \end{align*} $$ @@ -685,7 +744,7 @@

    Dissecting the equations

    {\mathbb{E}_{q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}\left[\log\frac{p(\boldsymbol{x}, \boldsymbol{h})}{q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}\right]} &= {\mathbb{E}_{q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}\left[\log\frac{p_{\boldsymbol{\theta}}(\boldsymbol{x}|\boldsymbol{h})p(\boldsymbol{h})}{q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}\right]} && {\text{(Chain Rule of Probability)}}\\ &= {\mathbb{E}_{q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}\left[\log p_{\boldsymbol{\theta}}(\boldsymbol{x}|\boldsymbol{h})\right] + \mathbb{E}_{q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}\left[\log\frac{p(\boldsymbol{h})}{q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}\right]} && {\text{(Split the Expectation)}}\\ -&= \underbrace{{\mathbb{E}_{q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}\left[\log p_{\boldsymbol{\theta}}(\boldsymbol{x}|\boldsymbol{h})\right]}}_\text{reconstruction term} - \underbrace{{KL(q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}\vert\vert{p(\boldsymbol{h}))}}_\text{prior matching term} && {\text{(Definition of KL Divergence)}} +&= \underbrace{{\mathbb{E}_{q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}\left[\log p_{\boldsymbol{\theta}}(\boldsymbol{x}|\boldsymbol{h})\right]}}_\text{reconstruction term} - \underbrace{{D_{KL}(q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}\vert\vert{p(\boldsymbol{h}))}}_\text{prior matching term} && {\text{(Definition of KL Divergence)}} \end{align*} $$ @@ -734,7 +793,7 @@

    Analytical evaluation

    Then, the KL divergence term of the ELBO can be computed analytically, and the reconstruction term can be approximated using a Monte Carlo estimate. Our objective can then be rewritten as:

    $$ \begin{align*} - \mathrm{argmax}_{\boldsymbol{\phi}, \boldsymbol{\theta}} \mathbb{E}_{q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}\left[\log p_{\boldsymbol{\theta}}(\boldsymbol{x}|\boldsymbol{h})\right] - KL(q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})\vert\vert p(\boldsymbol{h})) \approx \mathrm{argmax}_{\boldsymbol{\phi}, \boldsymbol{\theta}} \sum_{l=1}^{L}\log p_{\boldsymbol{\theta}}(\boldsymbol{x}|\boldsymbol{h}^{(l)}) - KL(q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})\vert\vert p(\boldsymbol{h})) + \mathrm{argmax}_{\boldsymbol{\phi}, \boldsymbol{\theta}} \mathbb{E}_{q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})}\left[\log p_{\boldsymbol{\theta}}(\boldsymbol{x}|\boldsymbol{h})\right] - D_{KL}(q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})\vert\vert p(\boldsymbol{h})) \approx \mathrm{argmax}_{\boldsymbol{\phi}, \boldsymbol{\theta}} \sum_{l=1}^{L}\log p_{\boldsymbol{\theta}}(\boldsymbol{x}|\boldsymbol{h}^{(l)}) - D_{KL}(q_{\boldsymbol{\phi}}(\boldsymbol{h}|\boldsymbol{x})\vert\vert p(\boldsymbol{h})) \end{align*} $$ @@ -992,6 +1051,140 @@

    Steps in building a



    +









    +

    Generative Adversarial Networks

    +

    Generative adversarial networks (GANs) have shown great results in +many generative tasks to replicate the real-world rich content such as +images, human language, and music. It is inspired by game theory: two +models, a generator and a discriminator, are competing with each other while +making each other stronger at the same time. However, it is rather +challenging to train a GANs model, +training instability or failure to converge. +

    + +

    Generative adversarial networks consist of two models (in their simplest form as two opposing feed forward neural networks)

    +
      +
    1. A discriminator \( D \) estimates the probability of a given sample coming from the real dataset. It works as a critic and is optimized to tell the fake samples from the realo ones
      2. +
    2. A generator \( G \) outputs synthetic samples given a noise variable input \( z \) (\( z \) brings in potential output diversity). It is trained to capture the real data distribution in order to generate samples that can be as real as possible, or in other words, can trick the discriminator to offer a high probability.
      3. +
    +

    At the end of the training, the generator can be used to generate for +example new images. In this sense we have trained a model which can +produce new samples. We say that we have implicitely defined a +probability. +

    + +

    These two models compete against each other during the training +process: the generator \( G \) is trying hard to trick the discriminator, +while the critic model \( D \) is trying hard not to be cheated. This +interesting zero-sum game between two models motivates both to improve +their functionalities. +

    + + + + + + + +

    On one hand, we want to make sure the discriminator \( D \)'s decisions +over real data are accurate by maximizing \( \mathbb{E}_{x \sim +p_{r}(x)} [\log D(x)] \). Meanwhile, given a fake sample \( G(z), z \sim +p_z(z) \), the discriminator is expected to output a probability, +\( D(G(z)) \), close to zero by maximizing \( \mathbb{E}_{z \sim p_{z}(z)} +[\log (1 - D(G(z)))] \). +

    + +

    On the other hand, the generator is trained to increase the chances of +\( D \) producing a high probability for a fake example, thus to minimize +\( \mathbb{E}_{z \sim p_{z}(z)} [\log (1 - D(G(z)))] \). +

    + +

    When combining both aspects together, \( D \) and \( G \) are playing a \textit{minimax game} in which we should optimize the following loss function:

    + +$$ +\begin{aligned} +\min_G \max_D L(D, G) +& = \mathbb{E}_{x \sim p_{r}(x)} [\log D(x)] + \mathbb{E}_{z \sim p_z(z)} [\log(1 - D(G(z)))] \\ +& = \mathbb{E}_{x \sim p_{r}(x)} [\log D(x)] + \mathbb{E}_{x \sim p_g(x)} [\log(1 - D(x)] +\end{aligned} +$$ + +

    where \( \mathbb{E}_{x \sim p_{r}(x)} [\log D(x)] \) has no impact on \( G \) during gradient descent updates.

    + +









    +

    Optimal value for \( D \)

    + +

    Now we have a well-defined loss function. Let's first examine what is the best value for \( D \).

    + +$$ +L(G, D) = \int_x \bigg( p_{r}(x) \log(D(x)) + p_g (x) \log(1 - D(x)) \bigg) dx +$$ + +

    Since we are interested in what is the best value of \( D(x) \) to maximize \( L(G, D) \), let us label

    + +$$ +\tilde{x} = D(x), +A=p_{r}(x), +B=p_g(x) +$$ + +

    And then what is inside the integral (we can safely ignore the integral because \( x \) is sampled over all the possible values) is:

    + +$$ +\begin{align*} +f(\tilde{x}) +& = A log\tilde{x} + B log(1-\tilde{x}) \\ +\frac{d f(\tilde{x})}{d \tilde{x}} +& = A \frac{1}{ln10} \frac{1}{\tilde{x}} - B \frac{1}{ln10} \frac{1}{1 - \tilde{x}} \\ +& = \frac{1}{ln10} (\frac{A}{\tilde{x}} - \frac{B}{1-\tilde{x}}) \\ +& = \frac{1}{ln10} \frac{A - (A + B)\tilde{x}}{\tilde{x} (1 - \tilde{x})} \\ +\end{align*} +$$ + +

    Thus, set \( \frac{d f(\tilde{x})}{d \tilde{x}} = 0 \), we get the best value of the discriminator: \( D^*(x) = \tilde{x}^* = \frac{A}{A + B} = \frac{p_{r}(x)}{p_{r}(x) + p_g(x)} \in [0, 1] \). +Once the generator is trained to its optimal, \( p_g \) gets very close to \( p_{r} \). When \( p_g = p_{r} \), \( D^*(x) \) becomes \( 1/2 \). +

    + +

    When both \( G \) and \( D \) are at their optimal values, we have \( p_g = p_{r} \) and \( D^*(x) = 1/2 \) and the loss function becomes:

    + +$$ +\begin{align*} +L(G, D^*) +&= \int_x \bigg( p_{r}(x) \log(D^*(x)) + p_g (x) \log(1 - D^*(x)) \bigg) dx \\ +&= \log \frac{1}{2} \int_x p_{r}(x) dx + \log \frac{1}{2} \int_x p_g(x) dx \\ +&= -2\log2 +\end{align*} +$$ + + +









    +

    What does the Loss Function Represent?

    + +

    The JS divergence between \( p_{r} \) and \( p_g \) can be computed as:

    + +$$ +\begin{align*} +D_{JS}(p_{r} \| p_g) +=& \frac{1}{2} D_{KL}(p_{r} || \frac{p_{r} + p_g}{2}) + \frac{1}{2} D_{KL}(p_{g} || \frac{p_{r} + p_g}{2}) \\ +=& \frac{1}{2} \bigg( \log2 + \int_x p_{r}(x) \log \frac{p_{r}(x)}{p_{r} + p_g(x)} dx \bigg) + \\& \frac{1}{2} \bigg( \log2 + \int_x p_g(x) \log \frac{p_g(x)}{p_{r} + p_g(x)} dx \bigg) \\ +=& \frac{1}{2} \bigg( \log4 + L(G, D^*) \bigg) +\end{align*} +$$ + +

    Thus,

    + +$$ +L(G, D^*) = 2D_{JS}(p_{r} \| p_g) - 2\log2 +$$ + +

    Essentially the loss function of GAN quantifies the similarity between +the generative data distribution \( p_g \) and the real sample +distribution \( p_{r} \) by JS divergence when the discriminator is +optimal. The best \( G^* \) that replicates the real data distribution +leads to the minimum \( L(G^*, D^*) = -2\log2 \) which is aligned with +equations above. +

    +









    More references

    @@ -1707,6 +1900,7 @@

    Diffusion learning

    smooth target distribution, this method can capture data distributions of arbitrary form.

    +
    © 1999-2024, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license diff --git a/doc/pub/week15/ipynb/ipynb-week15-src.tar.gz b/doc/pub/week15/ipynb/ipynb-week15-src.tar.gz index 831ba04f66c975a0871cad853a484246e98f7d7c..77b6090aaecb816883e3580c0587585fcc1d163d 100644 GIT binary patch delta 215 zcmWN_OF}^b06@`teeRP`Ns+(ui=u~$A}ykgv;%`XvW7O$+(InJ%p07^+5H>rVpP}R z!S*j&A%_ykksQm3oJyRbB;-tzaxN)JOGYl_Qm!N`*OHSPxs|*W^4UxPisL>L&Is)(H}2s4}j diff --git a/doc/pub/week15/ipynb/week15.ipynb b/doc/pub/week15/ipynb/week15.ipynb index 46204ccc..b91d3d41 100644 --- a/doc/pub/week15/ipynb/week15.ipynb +++ b/doc/pub/week15/ipynb/week15.ipynb @@ -2,7 +2,7 @@ "cells": [ { "cell_type": "markdown", - "id": "bbc251d3", + "id": "c7cf0fce", "metadata": { "editable": true }, @@ -14,7 +14,7 @@ }, { "cell_type": "markdown", - "id": "8154bff2", + "id": "e66f7980", "metadata": { "editable": true }, @@ -27,7 +27,7 @@ }, { "cell_type": "markdown", - "id": "30bdfd80", + "id": "b25a54f4", "metadata": { "editable": true }, @@ -40,7 +40,7 @@ "\n", "2. Generative Adversarial Networks (GANs), see for nice overview\n", "\n", - "3. Start discussion of diffusion models\n", + "3. Start discussion of diffusion models, motivation\n", "\n", "4. [Video of lecture](https://youtu.be/Cg8n9aWwHuU)\n", "\n", @@ -49,7 +49,7 @@ }, { "cell_type": "markdown", - "id": "b0ea3794", + "id": "7304c3a4", "metadata": { "editable": true }, @@ -65,7 +65,7 @@ }, { "cell_type": "markdown", - "id": "32939787", + "id": "1f29a278", "metadata": { "editable": true }, @@ -81,7 +81,7 @@ }, { "cell_type": "markdown", - "id": "89bf1911", + "id": "ad62142e", "metadata": { "editable": true }, @@ -91,7 +91,7 @@ }, { "cell_type": "markdown", - "id": "f04c1132", + "id": "3c906e0c", "metadata": { "editable": true }, @@ -103,7 +103,7 @@ }, { "cell_type": "markdown", - "id": "306c3a9e", + "id": "9e92737f", "metadata": { "editable": true }, @@ -115,7 +115,7 @@ }, { "cell_type": "markdown", - "id": "38d85424", + "id": "ea1bf8fd", "metadata": { "editable": true }, @@ -125,7 +125,7 @@ }, { "cell_type": "markdown", - "id": "72d0e5a4", + "id": "1d00646e", "metadata": { "editable": true }, @@ -137,7 +137,7 @@ }, { "cell_type": "markdown", - "id": "edc35f6e", + "id": "a3f6475d", "metadata": { "editable": true }, @@ -149,7 +149,7 @@ }, { "cell_type": "markdown", - "id": "eab97d57", + "id": "8f95be62", "metadata": { "editable": true }, @@ -163,7 +163,7 @@ }, { "cell_type": "markdown", - "id": "e36868b7", + "id": "184db513", "metadata": { "editable": true }, @@ -175,7 +175,7 @@ }, { "cell_type": "markdown", - "id": "05ccd2ab", + "id": "a9270d8e", "metadata": { "editable": true }, @@ -187,7 +187,7 @@ }, { "cell_type": "markdown", - "id": "e5f7b578", + "id": "e58357fd", "metadata": { "editable": true }, @@ -199,7 +199,7 @@ }, { "cell_type": "markdown", - "id": "fdc48043", + "id": "6b570b28", "metadata": { "editable": true }, @@ -209,7 +209,7 @@ }, { "cell_type": "markdown", - "id": "32dd3ff0", + "id": "c938f39d", "metadata": { "editable": true }, @@ -221,7 +221,7 @@ }, { "cell_type": "markdown", - "id": "5e333dc2", + "id": "13461367", "metadata": { "editable": true }, @@ -236,7 +236,7 @@ }, { "cell_type": "markdown", - "id": "9d4996e0", + "id": "1050da2f", "metadata": { "editable": true }, @@ -248,7 +248,7 @@ }, { "cell_type": "markdown", - "id": "bfbcdfd5", + "id": "094c2485", "metadata": { "editable": true }, @@ -258,7 +258,7 @@ }, { "cell_type": "markdown", - "id": "2bac65fd", + "id": "7a888b05", "metadata": { "editable": true }, @@ -270,7 +270,7 @@ }, { "cell_type": "markdown", - "id": "f513d787", + "id": "27b7e939", "metadata": { "editable": true }, @@ -283,7 +283,7 @@ }, { "cell_type": "markdown", - "id": "7344a35e", + "id": "0a367b8d", "metadata": { "editable": true }, @@ -295,7 +295,7 @@ }, { "cell_type": "markdown", - "id": "135b8bb8", + "id": "67e2aac1", "metadata": { "editable": true }, @@ -307,7 +307,7 @@ }, { "cell_type": "markdown", - "id": "aea87bfd", + "id": "552dd312", "metadata": { "editable": true }, @@ -317,7 +317,7 @@ }, { "cell_type": "markdown", - "id": "cce8712d", + "id": "3bc3b94c", "metadata": { "editable": true }, @@ -329,7 +329,7 @@ }, { "cell_type": "markdown", - "id": "5bcfef5b", + "id": "008ec2d0", "metadata": { "editable": true }, @@ -341,7 +341,7 @@ }, { "cell_type": "markdown", - "id": "665445d0", + "id": "90a0b8a3", "metadata": { "editable": true }, @@ -353,7 +353,7 @@ }, { "cell_type": "markdown", - "id": "bdc994e9", + "id": "fce8a947", "metadata": { "editable": true }, @@ -364,7 +364,7 @@ }, { "cell_type": "markdown", - "id": "97db2763", + "id": "aed7c7b0", "metadata": { "editable": true }, @@ -376,7 +376,7 @@ }, { "cell_type": "markdown", - "id": "6274f6f9", + "id": "f59709c5", "metadata": { "editable": true }, @@ -390,7 +390,7 @@ }, { "cell_type": "markdown", - "id": "a8c268f3", + "id": "b6a46f04", "metadata": { "editable": true }, @@ -402,7 +402,7 @@ }, { "cell_type": "markdown", - "id": "5fb324c3", + "id": "d309349e", "metadata": { "editable": true }, @@ -412,7 +412,7 @@ }, { "cell_type": "markdown", - "id": "3ab2b400", + "id": "a29d8581", "metadata": { "editable": true }, @@ -424,7 +424,7 @@ }, { "cell_type": "markdown", - "id": "ad53bab5", + "id": "d844c4dd", "metadata": { "editable": true }, @@ -436,7 +436,7 @@ }, { "cell_type": "markdown", - "id": "80022c49", + "id": "684424fc", "metadata": { "editable": true }, @@ -448,7 +448,7 @@ }, { "cell_type": "markdown", - "id": "ab06bb70", + "id": "540c038b", "metadata": { "editable": true }, @@ -459,7 +459,7 @@ }, { "cell_type": "markdown", - "id": "7f086749", + "id": "1dd6f171", "metadata": { "editable": true }, @@ -474,7 +474,7 @@ }, { "cell_type": "markdown", - "id": "65f4974d", + "id": "99fdc241", "metadata": { "editable": true }, @@ -486,7 +486,7 @@ }, { "cell_type": "markdown", - "id": "01b6074d", + "id": "0366d1a3", "metadata": { "editable": true }, @@ -498,7 +498,7 @@ }, { "cell_type": "markdown", - "id": "abc68191", + "id": "0ad245f3", "metadata": { "editable": true }, @@ -508,7 +508,7 @@ }, { "cell_type": "markdown", - "id": "42d55ede", + "id": "62a879d0", "metadata": { "editable": true }, @@ -519,7 +519,7 @@ }, { "cell_type": "markdown", - "id": "2713bc10", + "id": "2dd7dc01", "metadata": { "editable": true }, @@ -531,7 +531,7 @@ }, { "cell_type": "markdown", - "id": "bcbc944d", + "id": "afd4999e", "metadata": { "editable": true }, @@ -541,7 +541,7 @@ }, { "cell_type": "markdown", - "id": "f470f0b7", + "id": "610cb88e", "metadata": { "editable": true }, @@ -553,7 +553,7 @@ }, { "cell_type": "markdown", - "id": "6cc4dc33", + "id": "3093f6b2", "metadata": { "editable": true }, @@ -565,7 +565,7 @@ }, { "cell_type": "markdown", - "id": "b8d79f98", + "id": "ffac2f38", "metadata": { "editable": true }, @@ -577,7 +577,7 @@ }, { "cell_type": "markdown", - "id": "133d8b3f", + "id": "c3d13ab4", "metadata": { "editable": true }, @@ -589,7 +589,7 @@ }, { "cell_type": "markdown", - "id": "9fc2ebb7", + "id": "28f81e5e", "metadata": { "editable": true }, @@ -601,7 +601,7 @@ }, { "cell_type": "markdown", - "id": "431a67c4", + "id": "62fc9e76", "metadata": { "editable": true }, @@ -611,7 +611,7 @@ }, { "cell_type": "markdown", - "id": "3df67560", + "id": "f4c94f15", "metadata": { "editable": true }, @@ -623,7 +623,7 @@ }, { "cell_type": "markdown", - "id": "0838e65c", + "id": "8d9acf3f", "metadata": { "editable": true }, @@ -633,7 +633,7 @@ }, { "cell_type": "markdown", - "id": "822bf5ac", + "id": "b15612ff", "metadata": { "editable": true }, @@ -645,7 +645,7 @@ }, { "cell_type": "markdown", - "id": "a93722a2", + "id": "e1390374", "metadata": { "editable": true }, @@ -656,7 +656,92 @@ }, { "cell_type": "markdown", - "id": "d6d8c42e", + "id": "f391425b", + "metadata": { + "editable": true + }, + "source": [ + "## Kullback-Leibler divergence\n", + "\n", + "Before we continue, we need to remind ourselves about the\n", + "Kullback-Leibler divergence introduced earlier. This will also allow\n", + "us to introduce another measure used in connection with the training\n", + "of Generative Adversarial Networks, the so-called Jensen-Shannon divergence..\n", + "These metrics are useful for quantifying the similarity between two probability distributions.\n", + "\n", + "The Kullback–Leibler (KL) divergence, labeled $D_{KL}$, measures how one probability distribution $p$ diverges from a second expected probability distribution $q$,\n", + "that is" + ] + }, + { + "cell_type": "markdown", + "id": "71586dc3", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "D_{KL}(p \\| q) = \\int_x p(x) \\log \\frac{p(x)}{q(x)} dx.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "6042b046", + "metadata": { + "editable": true + }, + "source": [ + "The KL-divegrnece $D_{KL}$ achieves the minimum zero when $p(x) == q(x)$ everywhere.\n", + "\n", + "Note that the KL divergence is asymmetric. In cases where $p(x)$ is\n", + "close to zero, but $q(x)$ is significantly non-zero, the $q$'s effect\n", + "is disregarded. It could cause buggy results when we just want to\n", + "measure the similarity between two equally important distributions." + ] + }, + { + "cell_type": "markdown", + "id": "d20cf644", + "metadata": { + "editable": true + }, + "source": [ + "## Jensen-Shannon divergence\n", + "\n", + "The Jensen–Shannon (JS) divergence is another measure of similarity between\n", + "two probability distributions, bounded by $[0, 1]$. The JS-divergence is\n", + "symmetric and more smooth than the KL-divergence.\n", + "It is defined as" + ] + }, + { + "cell_type": "markdown", + "id": "1d9ded24", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "D_{JS}(p \\| q) = \\frac{1}{2} D_{KL}(p \\| \\frac{p + q}{2}) + \\frac{1}{2} D_{KL}(q \\| \\frac{p + q}{2})\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "e42178b6", + "metadata": { + "editable": true + }, + "source": [ + "Many practitioners believe that one reason behind GANs' big success is\n", + "switching the loss function from asymmetric KL-divergence in\n", + "traditional maximum-likelihood approach to symmetric JS-divergence." + ] + }, + { + "cell_type": "markdown", + "id": "09f95d9a", "metadata": { "editable": true }, @@ -672,7 +757,7 @@ }, { "cell_type": "markdown", - "id": "f17356a3", + "id": "18a2b94d", "metadata": { "editable": true }, @@ -691,7 +776,7 @@ }, { "cell_type": "markdown", - "id": "27fb099f", + "id": "19583482", "metadata": { "editable": true }, @@ -703,7 +788,7 @@ }, { "cell_type": "markdown", - "id": "17548236", + "id": "0836982d", "metadata": { "editable": true }, @@ -713,7 +798,7 @@ }, { "cell_type": "markdown", - "id": "3f05b416", + "id": "ae4be538", "metadata": { "editable": true }, @@ -725,7 +810,7 @@ }, { "cell_type": "markdown", - "id": "2cf43c02", + "id": "7ffb8c95", "metadata": { "editable": true }, @@ -735,7 +820,7 @@ }, { "cell_type": "markdown", - "id": "4c9b8c03", + "id": "62aa7b51", "metadata": { "editable": true }, @@ -757,7 +842,7 @@ }, { "cell_type": "markdown", - "id": "587e829b", + "id": "f8b3736f", "metadata": { "editable": true }, @@ -768,7 +853,7 @@ }, { "cell_type": "markdown", - "id": "6cec0936", + "id": "780e9d0f", "metadata": { "editable": true }, @@ -780,7 +865,7 @@ }, { "cell_type": "markdown", - "id": "008b0d93", + "id": "216c8f3a", "metadata": { "editable": true }, @@ -790,7 +875,7 @@ }, { "cell_type": "markdown", - "id": "03ec09b6", + "id": "05117498", "metadata": { "editable": true }, @@ -802,7 +887,7 @@ }, { "cell_type": "markdown", - "id": "ee878746", + "id": "e095ece6", "metadata": { "editable": true }, @@ -823,7 +908,7 @@ }, { "cell_type": "markdown", - "id": "be988b7e", + "id": "447b2be9", "metadata": { "editable": true }, @@ -835,21 +920,21 @@ }, { "cell_type": "markdown", - "id": "d06829bb", + "id": "709128f2", "metadata": { "editable": true }, "source": [ "$$\n", "\\begin{align*}\n", - "\\log p(\\boldsymbol{x}) & = \\log p(\\boldsymbol{x}) \\int q_{\\boldsymbol{\\phi}}(\\boldsymbol{h}|\\boldsymbol{x})dz && \\text{(Multiply by $1 = \\int q_{\\boldsymbol{\\phi}}(\\boldsymbol{h}|\\boldsymbol{x})d\\boldsymbol{h}$)}\\\\\n", - " & = \\int q_{\\boldsymbol{\\phi}}(\\boldsymbol{h}|\\boldsymbol{x})(\\log p(\\boldsymbol{x}))dz && \\text{(Bring evidence into integral)}\\\\\n", + "\\log p(\\boldsymbol{x}) & = \\log p(\\boldsymbol{x}) \\int q_{\\boldsymbol{\\phi}}(\\boldsymbol{h}|\\boldsymbol{x})dh && \\text{(Multiply by $1 = \\int q_{\\boldsymbol{\\phi}}(\\boldsymbol{h}|\\boldsymbol{x})d\\boldsymbol{h}$)}\\\\\n", + " & = \\int q_{\\boldsymbol{\\phi}}(\\boldsymbol{h}|\\boldsymbol{x})(\\log p(\\boldsymbol{x}))dh && \\text{(Bring evidence into integral)}\\\\\n", " & = \\mathbb{E}_{q_{\\boldsymbol{\\phi}}(\\boldsymbol{h}|\\boldsymbol{x})}\\left[\\log p(\\boldsymbol{x})\\right] && \\text{(Definition of Expectation)}\\\\\n", " & = \\mathbb{E}_{q_{\\boldsymbol{\\phi}}(\\boldsymbol{h}|\\boldsymbol{x})}\\left[\\log\\frac{p(\\boldsymbol{x}, \\boldsymbol{h})}{p(\\boldsymbol{h}|\\boldsymbol{x})}\\right]&& \\\\\n", " & = \\mathbb{E}_{q_{\\boldsymbol{\\phi}}(\\boldsymbol{h}|\\boldsymbol{x})}\\left[\\log\\frac{p(\\boldsymbol{x}, \\boldsymbol{h})q_{\\boldsymbol{\\phi}}(\\boldsymbol{h}|\\boldsymbol{x})}{p(\\boldsymbol{h}|\\boldsymbol{x})q_{\\boldsymbol{\\phi}}(\\boldsymbol{h}|\\boldsymbol{x})}\\right]&& \\text{(Multiply by $1 = \\frac{q_{\\boldsymbol{\\phi}}(\\boldsymbol{h}|\\boldsymbol{x})}{q_{\\boldsymbol{\\phi}}(\\boldsymbol{h}|\\boldsymbol{x})}$)}\\\\\n", " & = \\mathbb{E}_{q_{\\boldsymbol{\\phi}}(\\boldsymbol{h}|\\boldsymbol{x})}\\left[\\log\\frac{p(\\boldsymbol{x}, \\boldsymbol{h})}{q_{\\boldsymbol{\\phi}}(\\boldsymbol{h}|\\boldsymbol{x})}\\right] + \\mathbb{E}_{q_{\\boldsymbol{\\phi}}(\\boldsymbol{h}|\\boldsymbol{x})}\\left[\\log\\frac{q_{\\boldsymbol{\\phi}}(\\boldsymbol{h}|\\boldsymbol{x})}{p(\\boldsymbol{h}|\\boldsymbol{x})}\\right] && \\text{(Split the Expectation)}\\\\\n", " & = \\mathbb{E}_{q_{\\boldsymbol{\\phi}}(\\boldsymbol{h}|\\boldsymbol{x})}\\left[\\log\\frac{p(\\boldsymbol{x}, \\boldsymbol{h})}{q_{\\boldsymbol{\\phi}}(\\boldsymbol{h}|\\boldsymbol{x})}\\right] +\n", - "\t KL(q_{\\boldsymbol{\\phi}}(\\boldsymbol{h}|\\boldsymbol{x})\\vert\\vert p(\\boldsymbol{h}|\\boldsymbol{x})) && \\text{(Definition of KL Divergence)}\\\\\n", + "\t D_{KL}(q_{\\boldsymbol{\\phi}}(\\boldsymbol{h}|\\boldsymbol{x})\\vert\\vert p(\\boldsymbol{h}|\\boldsymbol{x})) && \\text{(Definition of KL Divergence)}\\\\\n", " & \\geq \\mathbb{E}_{q_{\\boldsymbol{\\phi}}(\\boldsymbol{h}|\\boldsymbol{x})}\\left[\\log\\frac{p(\\boldsymbol{x}, \\boldsymbol{h})}{q_{\\boldsymbol{\\phi}}(\\boldsymbol{h}|\\boldsymbol{x})}\\right] && \\text{(KL Divergence always $\\geq 0$)}\n", "\\end{align*}\n", "$$" @@ -857,7 +942,7 @@ }, { "cell_type": "markdown", - "id": "f165c01c", + "id": "2840182f", "metadata": { "editable": true }, @@ -875,7 +960,7 @@ }, { "cell_type": "markdown", - "id": "a5a5ba44", + "id": "a19f85d0", "metadata": { "editable": true }, @@ -893,7 +978,7 @@ }, { "cell_type": "markdown", - "id": "5a0edb20", + "id": "8dd0ce7b", "metadata": { "editable": true }, @@ -905,7 +990,7 @@ }, { "cell_type": "markdown", - "id": "066fb151", + "id": "0e1892c3", "metadata": { "editable": true }, @@ -915,14 +1000,14 @@ "{\\mathbb{E}_{q_{\\boldsymbol{\\phi}}(\\boldsymbol{h}|\\boldsymbol{x})}\\left[\\log\\frac{p(\\boldsymbol{x}, \\boldsymbol{h})}{q_{\\boldsymbol{\\phi}}(\\boldsymbol{h}|\\boldsymbol{x})}\\right]}\n", "&= {\\mathbb{E}_{q_{\\boldsymbol{\\phi}}(\\boldsymbol{h}|\\boldsymbol{x})}\\left[\\log\\frac{p_{\\boldsymbol{\\theta}}(\\boldsymbol{x}|\\boldsymbol{h})p(\\boldsymbol{h})}{q_{\\boldsymbol{\\phi}}(\\boldsymbol{h}|\\boldsymbol{x})}\\right]} && {\\text{(Chain Rule of Probability)}}\\\\\n", "&= {\\mathbb{E}_{q_{\\boldsymbol{\\phi}}(\\boldsymbol{h}|\\boldsymbol{x})}\\left[\\log p_{\\boldsymbol{\\theta}}(\\boldsymbol{x}|\\boldsymbol{h})\\right] + \\mathbb{E}_{q_{\\boldsymbol{\\phi}}(\\boldsymbol{h}|\\boldsymbol{x})}\\left[\\log\\frac{p(\\boldsymbol{h})}{q_{\\boldsymbol{\\phi}}(\\boldsymbol{h}|\\boldsymbol{x})}\\right]} && {\\text{(Split the Expectation)}}\\\\\n", - "&= \\underbrace{{\\mathbb{E}_{q_{\\boldsymbol{\\phi}}(\\boldsymbol{h}|\\boldsymbol{x})}\\left[\\log p_{\\boldsymbol{\\theta}}(\\boldsymbol{x}|\\boldsymbol{h})\\right]}}_\\text{reconstruction term} - \\underbrace{{KL(q_{\\boldsymbol{\\phi}}(\\boldsymbol{h}|\\boldsymbol{x})}\\vert\\vert{p(\\boldsymbol{h}))}}_\\text{prior matching term} && {\\text{(Definition of KL Divergence)}}\n", + "&= \\underbrace{{\\mathbb{E}_{q_{\\boldsymbol{\\phi}}(\\boldsymbol{h}|\\boldsymbol{x})}\\left[\\log p_{\\boldsymbol{\\theta}}(\\boldsymbol{x}|\\boldsymbol{h})\\right]}}_\\text{reconstruction term} - \\underbrace{{D_{KL}(q_{\\boldsymbol{\\phi}}(\\boldsymbol{h}|\\boldsymbol{x})}\\vert\\vert{p(\\boldsymbol{h}))}}_\\text{prior matching term} && {\\text{(Definition of KL Divergence)}}\n", "\\end{align*}\n", "$$" ] }, { "cell_type": "markdown", - "id": "b1c188ac", + "id": "c87201d3", "metadata": { "editable": true }, @@ -940,7 +1025,7 @@ }, { "cell_type": "markdown", - "id": "61b1725e", + "id": "bfab7537", "metadata": { "editable": true }, @@ -960,7 +1045,7 @@ }, { "cell_type": "markdown", - "id": "d134dd70", + "id": "a3919404", "metadata": { "editable": true }, @@ -972,7 +1057,7 @@ }, { "cell_type": "markdown", - "id": "c914ac07", + "id": "4c967472", "metadata": { "editable": true }, @@ -987,7 +1072,7 @@ }, { "cell_type": "markdown", - "id": "182be149", + "id": "52a6f5da", "metadata": { "editable": true }, @@ -999,21 +1084,21 @@ }, { "cell_type": "markdown", - "id": "d26c7329", + "id": "d57e59e7", "metadata": { "editable": true }, "source": [ "$$\n", "\\begin{align*}\n", - " \\mathrm{argmax}_{\\boldsymbol{\\phi}, \\boldsymbol{\\theta}} \\mathbb{E}_{q_{\\boldsymbol{\\phi}}(\\boldsymbol{h}|\\boldsymbol{x})}\\left[\\log p_{\\boldsymbol{\\theta}}(\\boldsymbol{x}|\\boldsymbol{h})\\right] - KL(q_{\\boldsymbol{\\phi}}(\\boldsymbol{h}|\\boldsymbol{x})\\vert\\vert p(\\boldsymbol{h})) \\approx \\mathrm{argmax}_{\\boldsymbol{\\phi}, \\boldsymbol{\\theta}} \\sum_{l=1}^{L}\\log p_{\\boldsymbol{\\theta}}(\\boldsymbol{x}|\\boldsymbol{h}^{(l)}) - KL(q_{\\boldsymbol{\\phi}}(\\boldsymbol{h}|\\boldsymbol{x})\\vert\\vert p(\\boldsymbol{h}))\n", + " \\mathrm{argmax}_{\\boldsymbol{\\phi}, \\boldsymbol{\\theta}} \\mathbb{E}_{q_{\\boldsymbol{\\phi}}(\\boldsymbol{h}|\\boldsymbol{x})}\\left[\\log p_{\\boldsymbol{\\theta}}(\\boldsymbol{x}|\\boldsymbol{h})\\right] - D_{KL}(q_{\\boldsymbol{\\phi}}(\\boldsymbol{h}|\\boldsymbol{x})\\vert\\vert p(\\boldsymbol{h})) \\approx \\mathrm{argmax}_{\\boldsymbol{\\phi}, \\boldsymbol{\\theta}} \\sum_{l=1}^{L}\\log p_{\\boldsymbol{\\theta}}(\\boldsymbol{x}|\\boldsymbol{h}^{(l)}) - D_{KL}(q_{\\boldsymbol{\\phi}}(\\boldsymbol{h}|\\boldsymbol{x})\\vert\\vert p(\\boldsymbol{h}))\n", "\\end{align*}\n", "$$" ] }, { "cell_type": "markdown", - "id": "2a89d0f3", + "id": "921da238", "metadata": { "editable": true }, @@ -1023,7 +1108,7 @@ }, { "cell_type": "markdown", - "id": "7666826e", + "id": "00966db8", "metadata": { "editable": true }, @@ -1040,7 +1125,7 @@ }, { "cell_type": "markdown", - "id": "a6ff50b2", + "id": "3b9fb44a", "metadata": { "editable": true }, @@ -1057,7 +1142,7 @@ }, { "cell_type": "markdown", - "id": "1300c538", + "id": "93627bb2", "metadata": { "editable": true }, @@ -1071,7 +1156,7 @@ }, { "cell_type": "markdown", - "id": "ebd27757", + "id": "82348034", "metadata": { "editable": true }, @@ -1089,7 +1174,7 @@ }, { "cell_type": "markdown", - "id": "624b1963", + "id": "29eb0946", "metadata": { "editable": true }, @@ -1101,7 +1186,7 @@ }, { "cell_type": "markdown", - "id": "eaf66572", + "id": "c96afc97", "metadata": { "editable": true }, @@ -1115,7 +1200,7 @@ }, { "cell_type": "markdown", - "id": "f4d09b90", + "id": "9a77743c", "metadata": { "editable": true }, @@ -1131,7 +1216,7 @@ }, { "cell_type": "markdown", - "id": "df030b4e", + "id": "8059d00f", "metadata": { "editable": true }, @@ -1150,7 +1235,7 @@ }, { "cell_type": "markdown", - "id": "179d5913", + "id": "68e50592", "metadata": { "editable": true }, @@ -1168,7 +1253,7 @@ }, { "cell_type": "markdown", - "id": "e954e4d0", + "id": "9fe155ca", "metadata": { "editable": true }, @@ -1189,7 +1274,7 @@ }, { "cell_type": "markdown", - "id": "9912338d", + "id": "834bb255", "metadata": { "editable": true }, @@ -1201,7 +1286,7 @@ }, { "cell_type": "markdown", - "id": "9fadae1d", + "id": "4f134c60", "metadata": { "editable": true }, @@ -1226,7 +1311,7 @@ }, { "cell_type": "markdown", - "id": "d205f4c0", + "id": "db32ac68", "metadata": { "editable": true }, @@ -1242,7 +1327,7 @@ }, { "cell_type": "markdown", - "id": "8afeef95", + "id": "ba47c9c1", "metadata": { "editable": true }, @@ -1260,7 +1345,7 @@ }, { "cell_type": "markdown", - "id": "6e728830", + "id": "fb895c87", "metadata": { "editable": true }, @@ -1272,7 +1357,7 @@ }, { "cell_type": "markdown", - "id": "5318fb35", + "id": "444ccf63", "metadata": { "editable": true }, @@ -1288,7 +1373,7 @@ }, { "cell_type": "markdown", - "id": "70ed2bca", + "id": "1a3f4662", "metadata": { "editable": true }, @@ -1300,7 +1385,7 @@ }, { "cell_type": "markdown", - "id": "b2f662ca", + "id": "7ddadfca", "metadata": { "editable": true }, @@ -1311,7 +1396,7 @@ }, { "cell_type": "markdown", - "id": "ca69c67d", + "id": "a244b68b", "metadata": { "editable": true }, @@ -1325,7 +1410,7 @@ }, { "cell_type": "markdown", - "id": "4d8a5c1f", + "id": "0dcc133a", "metadata": { "editable": true }, @@ -1337,7 +1422,7 @@ }, { "cell_type": "markdown", - "id": "eaeafa28", + "id": "1fb4338d", "metadata": { "editable": true }, @@ -1348,7 +1433,7 @@ }, { "cell_type": "markdown", - "id": "f35d0439", + "id": "c41ac754", "metadata": { "editable": true }, @@ -1360,7 +1445,7 @@ }, { "cell_type": "markdown", - "id": "e9df164f", + "id": "8f6aa934", "metadata": { "editable": true }, @@ -1383,7 +1468,7 @@ }, { "cell_type": "markdown", - "id": "ac025bad", + "id": "ab5ce09f", "metadata": { "editable": true }, @@ -1400,7 +1485,7 @@ }, { "cell_type": "markdown", - "id": "1548d332", + "id": "30d20331", "metadata": { "editable": true }, @@ -1413,7 +1498,7 @@ }, { "cell_type": "markdown", - "id": "415e7020", + "id": "a4aed469", "metadata": { "editable": true }, @@ -1424,7 +1509,7 @@ }, { "cell_type": "markdown", - "id": "afa15d31", + "id": "7dc442ca", "metadata": { "editable": true }, @@ -1438,7 +1523,7 @@ }, { "cell_type": "markdown", - "id": "db584443", + "id": "00bbdf9f", "metadata": { "editable": true }, @@ -1451,7 +1536,7 @@ }, { "cell_type": "markdown", - "id": "65ecceb4", + "id": "f328f999", "metadata": { "editable": true }, @@ -1463,7 +1548,7 @@ }, { "cell_type": "markdown", - "id": "93110eb4", + "id": "6a2e4da3", "metadata": { "editable": true }, @@ -1478,7 +1563,7 @@ }, { "cell_type": "markdown", - "id": "f5789340", + "id": "bcca6687", "metadata": { "editable": true }, @@ -1494,7 +1579,258 @@ }, { "cell_type": "markdown", - "id": "4bbe2c2b", + "id": "98cb8499", + "metadata": { + "editable": true + }, + "source": [ + "## Generative Adversarial Networks\n", + "Generative adversarial networks (GANs) have shown great results in\n", + "many generative tasks to replicate the real-world rich content such as\n", + "images, human language, and music. It is inspired by game theory: two\n", + "models, a generator and a discriminator, are competing with each other while\n", + "making each other stronger at the same time. However, it is rather\n", + "challenging to train a GANs model, \n", + "training instability or failure to converge.\n", + "\n", + "Generative adversarial networks consist of two models (in their simplest form as two opposing feed forward neural networks)\n", + "1. A discriminator $D$ estimates the probability of a given sample coming from the real dataset. It works as a critic and is optimized to tell the fake samples from the realo ones\n", + "\n", + "2. A generator $G$ outputs synthetic samples given a noise variable input $z$ ($z$ brings in potential output diversity). It is trained to capture the real data distribution in order to generate samples that can be as real as possible, or in other words, can trick the discriminator to offer a high probability.\n", + "\n", + "At the end of the training, the generator can be used to generate for\n", + "example new images. In this sense we have trained a model which can\n", + "produce new samples. We say that we have implicitely defined a\n", + "probability.\n", + "\n", + "These two models compete against each other during the training\n", + "process: the generator $G$ is trying hard to trick the discriminator,\n", + "while the critic model $D$ is trying hard not to be cheated. This\n", + "interesting zero-sum game between two models motivates both to improve\n", + "their functionalities.\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "On one hand, we want to make sure the discriminator $D$'s decisions\n", + "over real data are accurate by maximizing $\\mathbb{E}_{x \\sim\n", + "p_{r}(x)} [\\log D(x)]$. Meanwhile, given a fake sample $G(z), z \\sim\n", + "p_z(z)$, the discriminator is expected to output a probability,\n", + "$D(G(z))$, close to zero by maximizing $\\mathbb{E}_{z \\sim p_{z}(z)}\n", + "[\\log (1 - D(G(z)))]$.\n", + "\n", + "On the other hand, the generator is trained to increase the chances of\n", + "$D$ producing a high probability for a fake example, thus to minimize\n", + "$\\mathbb{E}_{z \\sim p_{z}(z)} [\\log (1 - D(G(z)))]$.\n", + "\n", + "When combining both aspects together, $D$ and $G$ are playing a \\textit{minimax game} in which we should optimize the following loss function:" + ] + }, + { + "cell_type": "markdown", + "id": "e6af787f", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\begin{aligned}\n", + "\\min_G \\max_D L(D, G) \n", + "& = \\mathbb{E}_{x \\sim p_{r}(x)} [\\log D(x)] + \\mathbb{E}_{z \\sim p_z(z)} [\\log(1 - D(G(z)))] \\\\\n", + "& = \\mathbb{E}_{x \\sim p_{r}(x)} [\\log D(x)] + \\mathbb{E}_{x \\sim p_g(x)} [\\log(1 - D(x)]\n", + "\\end{aligned}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "d9a15e9a", + "metadata": { + "editable": true + }, + "source": [ + "where $\\mathbb{E}_{x \\sim p_{r}(x)} [\\log D(x)]$ has no impact on $G$ during gradient descent updates." + ] + }, + { + "cell_type": "markdown", + "id": "ee24c8ff", + "metadata": { + "editable": true + }, + "source": [ + "## Optimal value for $D$\n", + "\n", + "Now we have a well-defined loss function. Let's first examine what is the best value for $D$." + ] + }, + { + "cell_type": "markdown", + "id": "f811ecdf", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "L(G, D) = \\int_x \\bigg( p_{r}(x) \\log(D(x)) + p_g (x) \\log(1 - D(x)) \\bigg) dx\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "27776912", + "metadata": { + "editable": true + }, + "source": [ + "Since we are interested in what is the best value of $D(x)$ to maximize $L(G, D)$, let us label" + ] + }, + { + "cell_type": "markdown", + "id": "6d3b9d73", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\tilde{x} = D(x), \n", + "A=p_{r}(x), \n", + "B=p_g(x)\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "b4911af2", + "metadata": { + "editable": true + }, + "source": [ + "And then what is inside the integral (we can safely ignore the integral because $x$ is sampled over all the possible values) is:" + ] + }, + { + "cell_type": "markdown", + "id": "5162efda", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\begin{align*}\n", + "f(\\tilde{x}) \n", + "& = A log\\tilde{x} + B log(1-\\tilde{x}) \\\\\n", + "\\frac{d f(\\tilde{x})}{d \\tilde{x}}\n", + "& = A \\frac{1}{ln10} \\frac{1}{\\tilde{x}} - B \\frac{1}{ln10} \\frac{1}{1 - \\tilde{x}} \\\\\n", + "& = \\frac{1}{ln10} (\\frac{A}{\\tilde{x}} - \\frac{B}{1-\\tilde{x}}) \\\\\n", + "& = \\frac{1}{ln10} \\frac{A - (A + B)\\tilde{x}}{\\tilde{x} (1 - \\tilde{x})} \\\\\n", + "\\end{align*}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "bee57995", + "metadata": { + "editable": true + }, + "source": [ + "Thus, set $\\frac{d f(\\tilde{x})}{d \\tilde{x}} = 0$, we get the best value of the discriminator: $D^*(x) = \\tilde{x}^* = \\frac{A}{A + B} = \\frac{p_{r}(x)}{p_{r}(x) + p_g(x)} \\in [0, 1]$.\n", + "Once the generator is trained to its optimal, $p_g$ gets very close to $p_{r}$. When $p_g = p_{r}$, $D^*(x)$ becomes $1/2$.\n", + "\n", + "When both $G$ and $D$ are at their optimal values, we have $p_g = p_{r}$ and $D^*(x) = 1/2$ and the loss function becomes:" + ] + }, + { + "cell_type": "markdown", + "id": "27ffb307", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\begin{align*}\n", + "L(G, D^*) \n", + "&= \\int_x \\bigg( p_{r}(x) \\log(D^*(x)) + p_g (x) \\log(1 - D^*(x)) \\bigg) dx \\\\\n", + "&= \\log \\frac{1}{2} \\int_x p_{r}(x) dx + \\log \\frac{1}{2} \\int_x p_g(x) dx \\\\\n", + "&= -2\\log2\n", + "\\end{align*}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "97400366", + "metadata": { + "editable": true + }, + "source": [ + "## What does the Loss Function Represent?\n", + "\n", + "The JS divergence between $p_{r}$ and $p_g$ can be computed as:" + ] + }, + { + "cell_type": "markdown", + "id": "f12302f0", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\begin{align*}\n", + "D_{JS}(p_{r} \\| p_g) \n", + "=& \\frac{1}{2} D_{KL}(p_{r} || \\frac{p_{r} + p_g}{2}) + \\frac{1}{2} D_{KL}(p_{g} || \\frac{p_{r} + p_g}{2}) \\\\\n", + "=& \\frac{1}{2} \\bigg( \\log2 + \\int_x p_{r}(x) \\log \\frac{p_{r}(x)}{p_{r} + p_g(x)} dx \\bigg) + \\\\& \\frac{1}{2} \\bigg( \\log2 + \\int_x p_g(x) \\log \\frac{p_g(x)}{p_{r} + p_g(x)} dx \\bigg) \\\\\n", + "=& \\frac{1}{2} \\bigg( \\log4 + L(G, D^*) \\bigg)\n", + "\\end{align*}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "b636eff3", + "metadata": { + "editable": true + }, + "source": [ + "Thus," + ] + }, + { + "cell_type": "markdown", + "id": "c20b731a", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "L(G, D^*) = 2D_{JS}(p_{r} \\| p_g) - 2\\log2\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "ba5eb4c6", + "metadata": { + "editable": true + }, + "source": [ + "Essentially the loss function of GAN quantifies the similarity between\n", + "the generative data distribution $p_g$ and the real sample\n", + "distribution $p_{r}$ by JS divergence when the discriminator is\n", + "optimal. The best $G^*$ that replicates the real data distribution\n", + "leads to the minimum $L(G^*, D^*) = -2\\log2$ which is aligned with\n", + "equations above." + ] + }, + { + "cell_type": "markdown", + "id": "6ccd7370", "metadata": { "editable": true }, @@ -1513,7 +1849,7 @@ }, { "cell_type": "markdown", - "id": "c0522fb3", + "id": "7bf93525", "metadata": { "editable": true }, @@ -1527,7 +1863,7 @@ }, { "cell_type": "markdown", - "id": "fc2b9ad9", + "id": "e1d7f1d1", "metadata": { "editable": true }, @@ -1543,7 +1879,7 @@ }, { "cell_type": "markdown", - "id": "45c07637", + "id": "1c26f00a", "metadata": { "editable": true }, @@ -1554,7 +1890,7 @@ { "cell_type": "code", "execution_count": 1, - "id": "b4dc82a1", + "id": "3f9bdab9", "metadata": { "collapsed": false, "editable": true @@ -1579,7 +1915,7 @@ }, { "cell_type": "markdown", - "id": "f87b63c1", + "id": "280dbb97", "metadata": { "editable": true }, @@ -1590,7 +1926,7 @@ { "cell_type": "code", "execution_count": 2, - "id": "139f0467", + "id": "7e0df89a", "metadata": { "collapsed": false, "editable": true @@ -1639,7 +1975,7 @@ }, { "cell_type": "markdown", - "id": "2c1f7b6f", + "id": "3ec46b09", "metadata": { "editable": true }, @@ -1650,7 +1986,7 @@ { "cell_type": "code", "execution_count": 3, - "id": "ad486a71", + "id": "309e2e3a", "metadata": { "collapsed": false, "editable": true @@ -1679,7 +2015,7 @@ { "cell_type": "code", "execution_count": 4, - "id": "632212e4", + "id": "646796ec", "metadata": { "collapsed": false, "editable": true @@ -1696,7 +2032,7 @@ }, { "cell_type": "markdown", - "id": "e204ed36", + "id": "7a0473da", "metadata": { "editable": true }, @@ -1707,7 +2043,7 @@ { "cell_type": "code", "execution_count": 5, - "id": "0ca2ac08", + "id": "b1f5b069", "metadata": { "collapsed": false, "editable": true @@ -1735,7 +2071,7 @@ }, { "cell_type": "markdown", - "id": "1791b464", + "id": "332100d5", "metadata": { "editable": true }, @@ -1746,7 +2082,7 @@ { "cell_type": "code", "execution_count": 6, - "id": "386b381a", + "id": "4d6be600", "metadata": { "collapsed": false, "editable": true @@ -1787,7 +2123,7 @@ }, { "cell_type": "markdown", - "id": "f28eaf09", + "id": "2f8ab669", "metadata": { "editable": true }, @@ -1798,7 +2134,7 @@ { "cell_type": "code", "execution_count": 7, - "id": "352db36d", + "id": "c858d332", "metadata": { "collapsed": false, "editable": true @@ -1823,7 +2159,7 @@ }, { "cell_type": "markdown", - "id": "d02e0c9a", + "id": "6cb82728", "metadata": { "editable": true }, @@ -1834,7 +2170,7 @@ { "cell_type": "code", "execution_count": 8, - "id": "3c157e73", + "id": "ec774a9e", "metadata": { "collapsed": false, "editable": true @@ -1916,7 +2252,7 @@ { "cell_type": "code", "execution_count": 9, - "id": "639686f4", + "id": "67c4d3de", "metadata": { "collapsed": false, "editable": true @@ -1964,7 +2300,7 @@ }, { "cell_type": "markdown", - "id": "4a7df60c", + "id": "9156f147", "metadata": { "editable": true }, @@ -1975,7 +2311,7 @@ { "cell_type": "code", "execution_count": 10, - "id": "c575fd26", + "id": "42fef552", "metadata": { "collapsed": false, "editable": true @@ -2012,7 +2348,7 @@ { "cell_type": "code", "execution_count": 11, - "id": "15299086", + "id": "4514f6ff", "metadata": { "collapsed": false, "editable": true @@ -2043,7 +2379,7 @@ }, { "cell_type": "markdown", - "id": "f4b675b8", + "id": "3f004fd2", "metadata": { "editable": true }, @@ -2054,7 +2390,7 @@ { "cell_type": "code", "execution_count": 12, - "id": "4f8dd76d", + "id": "3dcb337e", "metadata": { "collapsed": false, "editable": true @@ -2109,7 +2445,7 @@ }, { "cell_type": "markdown", - "id": "604b1a6b", + "id": "57bbb2c1", "metadata": { "editable": true }, @@ -2126,7 +2462,7 @@ }, { "cell_type": "markdown", - "id": "69adfd20", + "id": "5dc8db83", "metadata": { "editable": true }, @@ -2151,7 +2487,7 @@ }, { "cell_type": "markdown", - "id": "db15b3f4", + "id": "61f738e9", "metadata": { "editable": true }, @@ -2169,7 +2505,7 @@ }, { "cell_type": "markdown", - "id": "8feb45ca", + "id": 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zUU0wBbJX>odANIDkRkkDVj<=a_-s8iQ6QaBxmQ#9TVUqNMD5o_!EZj{Gn@j|chWB+ z $(P_1, Q_1)$ match up. - \item Then move 2 shovelfuls from $P_2$ to $P_3$ => $(P_2, Q_2)$ match up. - \item Finally move 1 shovelfuls from $Q_3$ to $Q_4$ => $(P_3, Q_3)$ and $(P_4, Q_4)$ match up. -\end{itemize} - -\begin{figure}[!htb] - \centering - \includegraphics[width=\linewidth]{EM_distance_discrete.png} - \caption{Step-by-step plan of moving dirt between piles in $P$ and $Q$ to make them match.} - \label{fig:fig7} -\end{figure} - -If we label the cost to pay to make $P_i$ and $Q_i$ match as $\delta_i$, we would have $\delta_{i+1} = \delta_i + P_i - Q_i$ and in the example: - -\begin{align*} -\delta_0 &= 0\\ -\delta_1 &= 0 + 3 - 1 = 2\\ -\delta_2 &= 2 + 2 - 2 = 2\\ -\delta_3 &= 2 + 1 - 4 = -1\\ -\delta_4 &= -1 + 4 - 3 = 0 -\end{align*} - -Finally the Earth Mover's distance is $W = \sum \vert \delta_i \vert = 5$. - - -When dealing with the continuous probability domain, the distance formula becomes: - -\[ -W(p_r, p_g) = \inf_{\gamma \sim \Pi(p_r, p_g)} \mathbb{E}_{(x, y) \sim \gamma}[\| x-y \|] -\] - - -In the formula above, $\Pi(p_r, p_g)$ is the set of all possible joint probability distributions between $p_r$ and $p_g$. One joint distribution $\gamma \in \Pi(p_r, p_g)$ describes one dirt transport plan, same as the discrete example above, but in the continuous probability space. Precisely $\gamma(x, y)$ states the percentage of dirt should be transported from point $x$ to $y$ so as to make $x$ follows the same probability distribution of $y$. That's why the marginal distribution over $x$ adds up to $p_g$, $\sum_{x} \gamma(x, y) = p_g(y)$ (Once we finish moving the planned amount of dirt from every possible $x$ to the target $y$, we end up with exactly what $y$ has according to $p_g$.) and vice versa $\sum_{y} \gamma(x, y) = p_r(x)$. - -When treating $x$ as the starting point and $y$ as the destination, the total amount of dirt moved is $\gamma(x, y)$ and the traveling distance is $\| x-y \|$ and thus the cost is $\gamma(x, y) \cdot \| x-y \|$. The expected cost averaged across all the $(x,y)$ pairs can be easily computed as: - -\[ -\sum_{x, y} \gamma(x, y) \| x-y \| -= \mathbb{E}_{x, y \sim \gamma} \| x-y \| -\] - -Finally, we take the minimum one among the costs of all dirt moving solutions as the EM distance. In the definition of Wasserstein distance, the $\inf$ (infimum, also known as *greatest lower bound*) indicates that we are only interested in the smallest cost. - - -\subsection{Why Wasserstein is better than JS or KL Divergence?} - -Even when two distributions are located in lower dimensional manifolds without overlaps, Wasserstein distance can still provide a meaningful and smooth representation of the distance in-between. - -The WGAN paper exemplified the idea with a simple example. - -Suppose we have two probability distributions, $P$ and $Q$: - -\[ -\forall (x, y) \in P, x = 0 \text{ and } y \sim U(0, 1)\\ -\forall (x, y) \in Q, x = \theta, 0 \leq \theta \leq 1 \text{ and } y \sim U(0, 1)\\ -\] - - -\begin{figure}[!htb] - \centering - \includegraphics[width=0.7\linewidth]{wasserstein_simple_example.png} - \caption{There is no overlap between $P$ and $Q$ when $\theta \neq 0$.} - \label{fig:fig8} -\end{figure} - -When $\theta \neq 0$: - - -\begin{align*} -D_{KL}(P \| Q) &= \sum_{x=0, y \sim U(0, 1)} 1 \cdot \log\frac{1}{0} = +\infty \\ -D_{KL}(Q \| P) &= \sum_{x=\theta, y \sim U(0, 1)} 1 \cdot \log\frac{1}{0} = +\infty \\ -D_{JS}(P, Q) &= \frac{1}{2}(\sum_{x=0, y \sim U(0, 1)} 1 \cdot \log\frac{1}{1/2} + \sum_{x=0, y \sim U(0, 1)} 1 \cdot \log\frac{1}{1/2}) = \log 2\\ -W(P, Q) &= |\theta| -\end{align*} - -But when $\theta = 0$, two distributions are fully overlapped: - -\begin{align*} -D_{KL}(P \| Q) &= D_{KL}(Q \| P) = D_{JS}(P, Q) = 0\\ -W(P, Q) &= 0 = \lvert \theta \rvert -\end{align*} - - -$D_{KL}$ gives us infinity when two distributions are disjoint. The value of $D_{JS}$ has sudden jump, not differentiable at $\theta = 0$. Only Wasserstein metric provides a smooth measure, which is super helpful for a stable learning process using gradient descents. - - -\subsection{Use Wasserstein Distance as GAN Loss Function} - -It is intractable to exhaust all the possible joint distributions in $\Pi(p_r, p_g)$ to compute $\inf_{\gamma \sim \Pi(p_r, p_g)}$. Thus the authors proposed a smart transformation of the formula based on the Kantorovich-Rubinstein duality to: - -\[ -W(p_r, p_g) = \frac{1}{K} \sup_{\| f \|_L \leq K} \mathbb{E}_{x \sim p_r}[f(x)] - \mathbb{E}_{x \sim p_g}[f(x)] -\] - -where $\sup$ (supremum) is the opposite of $inf$ (infimum); we want to measure the least upper bound or, in even simpler words, the maximum value. - - -\subsubsection{Lipschitz Continuity} - -The function $f$ in the new form of Wasserstein metric is demanded to satisfy $\| f \|_L \leq K$, meaning it should be \textit{K-Lipschitz continuous}. - -A real-valued function $f: \mathbb{R} \rightarrow \mathbb{R}$ is called $K$-Lipschitz continuous if there exists a real constant $K \geq 0$ such that, for all $x_1, x_2 \in \mathbb{R}$, - -\[ -\lvert f(x_1) - f(x_2) \rvert \leq K \lvert x_1 - x_2 \rvert -\] - -Here $K$ is known as a Lipschitz constant for function $f(.)$. Functions that are everywhere continuously differentiable is Lipschitz continuous, because the derivative, estimated as $\frac{\lvert f(x_1) - f(x_2) \rvert}{\lvert x_1 - x_2 \rvert}$, has bounds. However, a Lipschitz continuous function may not be everywhere differentiable, such as $f(x) = \lvert x \rvert$. - -Explaining how the transformation happens on the Wasserstein distance formula is worthy of a long post by itself, so I skip the details here. If you are interested in how to compute Wasserstein metric using linear programming, or how to transfer Wasserstein metric into its dual form according to the Kantorovich-Rubinstein Duality, read this awesome \href{https://vincentherrmann.github.io/blog/wasserstein/}{post}. - - -\subsubsection{Wasserstein Loss Function} - -Suppose this function $f$ comes from a family of K-Lipschitz continuous functions, $\{ f_w \}_{w \in W}$, parameterized by $w$. In the modified Wasserstein-GAN, the "discriminator" model is used to learn $w$ to find a good $f_w$ and the loss function is configured as measuring the Wasserstein distance between $p_r$ and $p_g$. - -\[ -L(p_r, p_g) = W(p_r, p_g) = \max_{w \in W} \mathbb{E}_{x \sim p_r}[f_w(x)] - \mathbb{E}_{z \sim p_r(z)}[f_w(g_\theta(z))] -\] -Thus the "discriminator" is not a direct critic of telling the fake samples apart from the real ones anymore. Instead, it is trained to learn a $K$-Lipschitz continuous function to help compute Wasserstein distance. As the loss function decreases in the training, the Wasserstein distance gets smaller and the generator model's output grows closer to the real data distribution. - -One big problem is to maintain the $K$-Lipschitz continuity of $f_w$ during the training in order to make everything work out. The paper presents a simple but very practical trick: After every gradient update, clamp the weights $w$ to a small window, such as $[-0.01, 0.01]$, resulting in a compact parameter space $W$ and thus $f_w$ obtains its lower and upper bounds to preserve the Lipschitz continuity. - - -\begin{figure}[!htb] - \centering - \includegraphics[width=0.85\linewidth]{WGAN_algorithm.png} - \caption{Algorithm of Wasserstein generative adversarial network. (Image source:~\cite{wgan2017})} - \label{fig:fig9} -\end{figure} - - -Compared to the original GAN algorithm, the WGAN undertakes the following changes: -\begin{itemize} - \item After every gradient update on the critic function, clamp the weights to a small fixed range, $[-c, c]$. - \item Use a new loss function derived from the Wasserstein distance, no logarithm anymore. The "discriminator" model does not play as a direct critic but a helper for estimating the Wasserstein metric between real and generated data distribution. - \item Empirically the authors recommended RMSProp optimizer on the critic, rather than a momentum based optimizer such as Adam which could cause instability in the model training. I haven't seen clear theoretical explanation on this point through. -\end{itemize} - -Sadly, Wasserstein GAN is not perfect. Even the authors of the original WGAN paper mentioned that \textit{"Weight clipping is a clearly terrible way to enforce a Lipschitz constraint"}. WGAN still suffers from unstable training, slow convergence after weight clipping (when clipping window is too large), and vanishing gradients (when clipping window is too small). - -Some improvement, precisely replacing weight clipping with \textit{gradient penalty}, has been discussed in~\cite{wgan2017improve}. - - -\bibliographystyle{plain} -%\bibliography{../references} - -\begin{thebibliography}{1} - - \bibitem{gan2014} - Ian Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, - Sherjil Ozair, Aaron Courville, and Yoshua Bengio. - \newblock Generative adversarial nets. - \newblock In {\em NIPS}, pages 2672--2680. 2014. - - \bibitem{gan2015train} - Ferenc Huszár. - \newblock How (not) to train your generative model: Scheduled sampling, - likelihood, adversary? - \newblock {\em arXiv:1511.05101}, 2015. - - \bibitem{salimans2016nips} - Tim Salimans, Ian Goodfellow, Wojciech Zaremba, Vicki Cheung, Alec Radford, and - Xi~Chen. - \newblock Improved techniques for training gans. - \newblock {\em NIPS}, 2016. - - \bibitem{arjovsky2017} - Martin Arjovsky and Léon Bottou. - \newblock Towards principled methods for training generative adversarial - networks. - \newblock {\em ICML}, 2017. - - \bibitem{wgan2017} - Martin Arjovsky, Soumith Chintala, and Léon Bottou. - \newblock Wasserstein gan. - \newblock {\em arXiv:1701.07875}, 2017. - - \bibitem{wgan2017improve} - Ishaan Gulrajani, Faruk Ahmed, Martin Arjovsky, Vincent Dumoulin, and Aaron - Courville. - \newblock Improved training of wasserstein gans. - \newblock {\em arXiv:1704.00028}, 2017. - -\end{thebibliography} - - -\end{document}