mirror of
https://github.com/labmlai/annotated_deep_learning_paper_implementations.git
synced 2025-10-30 02:08:50 +08:00
266 lines
13 KiB
HTML
266 lines
13 KiB
HTML
<!DOCTYPE html>
|
|
<html>
|
|
<head>
|
|
<meta http-equiv="content-type" content="text/html;charset=utf-8"/>
|
|
<meta name="viewport" content="width=device-width, initial-scale=1.0"/>
|
|
<meta name="description" content="A simple PyTorch implementation/tutorial of Wasserstein Generative Adversarial Networks (WGAN) loss functions."/>
|
|
|
|
<meta name="twitter:card" content="summary"/>
|
|
<meta name="twitter:image:src" content="https://avatars1.githubusercontent.com/u/64068543?s=400&v=4"/>
|
|
<meta name="twitter:title" content="Wasserstein GAN (WGAN)"/>
|
|
<meta name="twitter:description" content="A simple PyTorch implementation/tutorial of Wasserstein Generative Adversarial Networks (WGAN) loss functions."/>
|
|
<meta name="twitter:site" content="@labmlai"/>
|
|
<meta name="twitter:creator" content="@labmlai"/>
|
|
|
|
<meta property="og:url" content="https://nn.labml.ai/gan/wasserstein/index.html"/>
|
|
<meta property="og:title" content="Wasserstein GAN (WGAN)"/>
|
|
<meta property="og:image" content="https://avatars1.githubusercontent.com/u/64068543?s=400&v=4"/>
|
|
<meta property="og:site_name" content="LabML Neural Networks"/>
|
|
<meta property="og:type" content="object"/>
|
|
<meta property="og:title" content="Wasserstein GAN (WGAN)"/>
|
|
<meta property="og:description" content="A simple PyTorch implementation/tutorial of Wasserstein Generative Adversarial Networks (WGAN) loss functions."/>
|
|
|
|
<title>Wasserstein GAN (WGAN)</title>
|
|
<link rel="shortcut icon" href="/icon.png"/>
|
|
<link rel="stylesheet" href="../../pylit.css">
|
|
<link rel="canonical" href="https://nn.labml.ai/gan/wasserstein/index.html"/>
|
|
<!-- Global site tag (gtag.js) - Google Analytics -->
|
|
<script async src="https://www.googletagmanager.com/gtag/js?id=G-4V3HC8HBLH"></script>
|
|
<script>
|
|
window.dataLayer = window.dataLayer || [];
|
|
|
|
function gtag() {
|
|
dataLayer.push(arguments);
|
|
}
|
|
|
|
gtag('js', new Date());
|
|
|
|
gtag('config', 'G-4V3HC8HBLH');
|
|
</script>
|
|
</head>
|
|
<body>
|
|
<div id='container'>
|
|
<div id="background"></div>
|
|
<div class='section'>
|
|
<div class='docs'>
|
|
<p>
|
|
<a class="parent" href="/">home</a>
|
|
<a class="parent" href="../index.html">gan</a>
|
|
<a class="parent" href="index.html">wasserstein</a>
|
|
</p>
|
|
<p>
|
|
|
|
<a href="https://github.com/lab-ml/labml_nn/tree/master/labml_nn/gan/wasserstein/__init__.py">
|
|
<img alt="Github"
|
|
src="https://img.shields.io/github/stars/lab-ml/nn?style=social"
|
|
style="max-width:100%;"/></a>
|
|
<a href="https://join.slack.com/t/labforml/shared_invite/zt-egj9zvq9-Dl3hhZqobexgT7aVKnD14g/"
|
|
rel="nofollow">
|
|
<img alt="Join Slact"
|
|
src="https://img.shields.io/badge/slack-chat-green.svg?logo=slack"
|
|
style="max-width:100%;"/></a>
|
|
<a href="https://twitter.com/labmlai"
|
|
rel="nofollow">
|
|
<img alt="Twitter"
|
|
src="https://img.shields.io/twitter/follow/labmlai?style=social"
|
|
style="max-width:100%;"/></a>
|
|
</p>
|
|
</div>
|
|
</div>
|
|
<div class='section' id='section-0'>
|
|
<div class='docs doc-strings'>
|
|
<div class='section-link'>
|
|
<a href='#section-0'>#</a>
|
|
</div>
|
|
<h1>Wasserstein GAN (WGAN)</h1>
|
|
<p>This is an implementation of
|
|
<a href="https://arxiv.org/abs/1701.07875">Wasserstein GAN</a>.</p>
|
|
<p>The original GAN loss is based on Jensen-Shannon (JS) divergence
|
|
between the real distribution $\mathbb{P}_r$ and generated distribution $\mathbb{P}_g$.
|
|
The Wasserstein GAN is based on Earth Mover distance between these distributions.</p>
|
|
<p>
|
|
<script type="math/tex; mode=display">
|
|
W(\mathbb{P}_r, \mathbb{P}_g) =
|
|
\underset{\gamma \in \Pi(\mathbb{P}_r, \mathbb{P}_g)} {\mathrm{inf}}
|
|
\mathbb{E}_{(x,y) \sim \gamma}
|
|
\Vert x - y \Vert
|
|
</script>
|
|
</p>
|
|
<p>$\Pi(\mathbb{P}_r, \mathbb{P}_g)$ is the set of all joint distributions, whose
|
|
marginal probabilities are $\gamma(x, y)$.</p>
|
|
<p>$\mathbb{E}_{(x,y) \sim \gamma} \Vert x - y \Vert$ is the earth mover distance for
|
|
a given joint distribution ($x$ and $y$ are probabilities).</p>
|
|
<p>So $W(\mathbb{P}_r, \mathbb{P}g)$ is equal to the least earth mover distance for
|
|
any joint distribution between the real distribution $\mathbb{P}_r$ and generated distribution $\mathbb{P}_g$.</p>
|
|
<p>The paper shows that Jensen-Shannon (JS) divergence and other measures for the difference between two probability
|
|
distributions are not smooth. And therefore if we are doing gradient descent on one of the probability
|
|
distributions (parameterized) it will not converge.</p>
|
|
<p>Based on Kantorovich-Rubinstein duality,
|
|
<script type="math/tex; mode=display">
|
|
W(\mathbb{P}_r, \mathbb{P}_g) =
|
|
\underset{\Vert f \Vert_L \le 1} {\mathrm{sup}}
|
|
\mathbb{E}_{x \sim \mathbb{P}_r} [f(x)]- \mathbb{E}_{x \sim \mathbb{P}_g} [f(x)]
|
|
</script>
|
|
</p>
|
|
<p>where $\Vert f \Vert_L \le 1$ are all 1-Lipschitz functions.</p>
|
|
<p>That is, it is equal to the greatest difference
|
|
<script type="math/tex; mode=display">\mathbb{E}_{x \sim \mathbb{P}_r} [f(x)] - \mathbb{E}_{x \sim \mathbb{P}_g} [f(x)]</script>
|
|
among all 1-Lipschitz functions.</p>
|
|
<p>For $K$-Lipschitz functions,
|
|
<script type="math/tex; mode=display">
|
|
W(\mathbb{P}_r, \mathbb{P}_g) =
|
|
\underset{\Vert f \Vert_L \le K} {\mathrm{sup}}
|
|
\mathbb{E}_{x \sim \mathbb{P}_r} \Bigg[\frac{1}{K} f(x) \Bigg]
|
|
- \mathbb{E}_{x \sim \mathbb{P}_g} \Bigg[\frac{1}{K} f(x) \Bigg]
|
|
</script>
|
|
</p>
|
|
<p>If all $K$-Lipschitz functions can be represented as $f_w$ where $f$ is parameterized by
|
|
$w \in \mathcal{W}$,</p>
|
|
<p>
|
|
<script type="math/tex; mode=display">
|
|
K \cdot W(\mathbb{P}_r, \mathbb{P}_g) =
|
|
\max_{w \in \mathcal{W}}
|
|
\mathbb{E}_{x \sim \mathbb{P}_r} [f_w(x)]- \mathbb{E}_{x \sim \mathbb{P}_g} [f_w(x)]
|
|
</script>
|
|
</p>
|
|
<p>If $(\mathbb{P}_{g})$ is represented by a generator <script type="math/tex; mode=display">g_\theta (z)</script> and $z$ is from a known
|
|
distribution $z \sim p(z)$,</p>
|
|
<p>
|
|
<script type="math/tex; mode=display">
|
|
K \ cdot W(\mathbb{P}_r, \mathbb{P}_\theta) =
|
|
\max_{w \in \mathcal{W}}
|
|
\mathbb{E}_{x \sim \mathbb{P}_r} [f_w(x)]- \mathbb{E}_{z \sim p(z)} [f_w(g_\theta(z))]
|
|
</script>
|
|
</p>
|
|
<p>Now to converge $g_\theta$ with $\mathbb{P}_{r}$ we can gradient descent on $\theta$
|
|
to minimize above formula.</p>
|
|
<p>Similarly we can find $\max_{w \in \mathcal{W}}$ by ascending on $w$,
|
|
while keeping $K$ bounded. <em>One way to keep $K$ bounded is to clip all weights in the neural
|
|
network that defines $f$ clipped within a range.</em></p>
|
|
<p>Here is the code to try this on a <a href="experiment.html">simple MNIST generation experiment</a>.</p>
|
|
<p><a href="https://colab.research.google.com/github/lab-ml/nn/blob/master/labml_nn/gan/wasserstein/experiment.ipynb"><img alt="Open In Colab" src="https://colab.research.google.com/assets/colab-badge.svg" /></a></p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">87</span><span></span><span class="kn">import</span> <span class="nn">torch.utils.data</span>
|
|
<span class="lineno">88</span><span class="kn">from</span> <span class="nn">torch.nn</span> <span class="kn">import</span> <span class="n">functional</span> <span class="k">as</span> <span class="n">F</span>
|
|
<span class="lineno">89</span>
|
|
<span class="lineno">90</span><span class="kn">from</span> <span class="nn">labml_helpers.module</span> <span class="kn">import</span> <span class="n">Module</span></pre></div>
|
|
</div>
|
|
</div>
|
|
<div class='section' id='section-1'>
|
|
<div class='docs doc-strings'>
|
|
<div class='section-link'>
|
|
<a href='#section-1'>#</a>
|
|
</div>
|
|
<h2>Discriminator Loss</h2>
|
|
<p>We want to find $w$ to maximize
|
|
<script type="math/tex; mode=display">\mathbb{E}_{x \sim \mathbb{P}_r} [f_w(x)]- \mathbb{E}_{z \sim p(z)} [f_w(g_\theta(z))]</script>,
|
|
so we minimize,
|
|
<script type="math/tex; mode=display">-\frac{1}{m} \sum_{i=1}^m f_w \big(x^{(i)} \big) +
|
|
\frac{1}{m} \sum_{i=1}^m f_w \big( g_\theta(z^{(i)}) \big)</script>
|
|
</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">93</span><span class="k">class</span> <span class="nc">DiscriminatorLoss</span><span class="p">(</span><span class="n">Module</span><span class="p">):</span></pre></div>
|
|
</div>
|
|
</div>
|
|
<div class='section' id='section-2'>
|
|
<div class='docs doc-strings'>
|
|
<div class='section-link'>
|
|
<a href='#section-2'>#</a>
|
|
</div>
|
|
<ul>
|
|
<li><code>f_real</code> is $f_w(x)$</li>
|
|
<li><code>f_fake</code> is $f_w(g_\theta(z))$</li>
|
|
</ul>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">104</span> <span class="k">def</span> <span class="fm">__call__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">f_real</span><span class="p">:</span> <span class="n">torch</span><span class="o">.</span><span class="n">Tensor</span><span class="p">,</span> <span class="n">f_fake</span><span class="p">:</span> <span class="n">torch</span><span class="o">.</span><span class="n">Tensor</span><span class="p">):</span></pre></div>
|
|
</div>
|
|
</div>
|
|
<div class='section' id='section-3'>
|
|
<div class='docs'>
|
|
<div class='section-link'>
|
|
<a href='#section-3'>#</a>
|
|
</div>
|
|
<p>We use ReLUs to clip the loss to keep $f \in [-1, +1]$ range.</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">111</span> <span class="k">return</span> <span class="n">F</span><span class="o">.</span><span class="n">relu</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">f_real</span><span class="p">)</span><span class="o">.</span><span class="n">mean</span><span class="p">(),</span> <span class="n">F</span><span class="o">.</span><span class="n">relu</span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">f_fake</span><span class="p">)</span><span class="o">.</span><span class="n">mean</span><span class="p">()</span></pre></div>
|
|
</div>
|
|
</div>
|
|
<div class='section' id='section-4'>
|
|
<div class='docs doc-strings'>
|
|
<div class='section-link'>
|
|
<a href='#section-4'>#</a>
|
|
</div>
|
|
<h2>Generator Loss</h2>
|
|
<p>We want to find $\theta$ to minimize
|
|
<script type="math/tex; mode=display">\mathbb{E}_{x \sim \mathbb{P}_r} [f_w(x)]- \mathbb{E}_{z \sim p(z)} [f_w(g_\theta(z))]</script>
|
|
The first component is independent of $\theta$,
|
|
so we minimize,
|
|
<script type="math/tex; mode=display">-\frac{1}{m} \sum_{i=1}^m f_w \big( g_\theta(z^{(i)}) \big)</script>
|
|
</p>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">114</span><span class="k">class</span> <span class="nc">GeneratorLoss</span><span class="p">(</span><span class="n">Module</span><span class="p">):</span></pre></div>
|
|
</div>
|
|
</div>
|
|
<div class='section' id='section-5'>
|
|
<div class='docs doc-strings'>
|
|
<div class='section-link'>
|
|
<a href='#section-5'>#</a>
|
|
</div>
|
|
<ul>
|
|
<li><code>f_fake</code> is $f_w(g_\theta(z))$</li>
|
|
</ul>
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">126</span> <span class="k">def</span> <span class="fm">__call__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">f_fake</span><span class="p">:</span> <span class="n">torch</span><span class="o">.</span><span class="n">Tensor</span><span class="p">):</span></pre></div>
|
|
</div>
|
|
</div>
|
|
<div class='section' id='section-6'>
|
|
<div class='docs'>
|
|
<div class='section-link'>
|
|
<a href='#section-6'>#</a>
|
|
</div>
|
|
|
|
</div>
|
|
<div class='code'>
|
|
<div class="highlight"><pre><span class="lineno">130</span> <span class="k">return</span> <span class="o">-</span><span class="n">f_fake</span><span class="o">.</span><span class="n">mean</span><span class="p">()</span></pre></div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
</div>
|
|
<script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.4/MathJax.js?config=TeX-AMS_HTML">
|
|
</script>
|
|
<!-- MathJax configuration -->
|
|
<script type="text/x-mathjax-config">
|
|
MathJax.Hub.Config({
|
|
tex2jax: {
|
|
inlineMath: [ ['$','$'] ],
|
|
displayMath: [ ['$$','$$'] ],
|
|
processEscapes: true,
|
|
processEnvironments: true
|
|
},
|
|
// Center justify equations in code and markdown cells. Elsewhere
|
|
// we use CSS to left justify single line equations in code cells.
|
|
displayAlign: 'center',
|
|
"HTML-CSS": { fonts: ["TeX"] }
|
|
});
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
</script>
|
|
</body>
|
|
</html> |