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<meta property="og:description" content="A simple PyTorch implementation/tutorial of Wasserstein Generative Adversarial Networks (WGAN) loss functions."/>
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<title>Wasserstein GAN (WGAN)</title>
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<a class="parent" href="/">home</a>
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<a class="parent" href="../index.html">gan</a>
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<a class="parent" href="index.html">wasserstein</a>
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<div class='section' id='section-0'>
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<a href='#section-0'>#</a>
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</div>
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<p>This is an implementation of
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<a href="https://arxiv.org/abs/1701.07875">Wasserstein GAN</a>.</p>
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<p>The original GAN loss is based on Jensen-Shannon (JS) divergence
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between the real distribution $\mathbb{P}_r$ and generated distribution $\mathbb{P}_g$.
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The Wasserstein GAN is based on Earth Mover distance between these distributions.</p>
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<p>
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<script type="math/tex; mode=display">
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W(\mathbb{P}_r, \mathbb{P}_g) =
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\underset{\gamma \in \Pi(\mathbb{P}_r, \mathbb{P}_g)} {\mathrm{inf}}
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\mathbb{E}_{(x,y) \sim \gamma}
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\Vert x - y \Vert
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</script>
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</p>
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<p>$\Pi(\mathbb{P}_r, \mathbb{P}_g)$ is the set of all joint distributions, whose
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marginal probabilities are $\gamma(x, y)$.</p>
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<p>$\mathbb{E}_{(x,y) \sim \gamma} \Vert x - y \Vert$ is the earth mover distance for
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a given joint distribution ($x$ and $y$ are probabilities).</p>
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<p>So $W(\mathbb{P}_r, \mathbb{P}g)$ is equal to the least earth mover distance for
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any joint distribution between the real distribution $\mathbb{P}_r$ and generated distribution $\mathbb{P}_g$.</p>
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<p>The paper shows that Jensen-Shannon (JS) divergence and other measures for difference between two probability
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distributions are not smooth. And therefore if we are doing a gradient descent on one of the probability
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distributions (parameterized) it will not converge.</p>
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<p>Based on Kantorovich-Rubinstein duality,
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<script type="math/tex; mode=display">
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W(\mathbb{P}_r, \mathbb{P}_g) =
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\underset{\Vert f \Vert_L \le 1} {\mathrm{sup}}
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\mathbb{E}_{x \sim \mathbb{P}_r} [f(x)]- \mathbb{E}_{x \sim \mathbb{P}_g} [f(x)]
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</script>
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</p>
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<p>where $\Vert f \Vert_L \le 1$ are all 1-Lipschitz functions.</p>
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<p>That is, it is equal to the greatest difference
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<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>
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among all 1-Lipschitz functions.</p>
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<p>For $K$-Lipschitz functions,
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<script type="math/tex; mode=display">
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W(\mathbb{P}_r, \mathbb{P}_g) =
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\underset{\Vert f \Vert_L \le K} {\mathrm{sup}}
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\mathbb{E}_{x \sim \mathbb{P}_r} \Bigg[\frac{1}{K} f(x) \Bigg]
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- \mathbb{E}_{x \sim \mathbb{P}_g} \Bigg[\frac{1}{K} f(x) \Bigg]
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</script>
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</p>
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<p>If all $K$-Lipschitz functions can be represented as $f_w$ where $f$ is parameterized by
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$w \in \mathcal{W}$,</p>
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<p>
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<script type="math/tex; mode=display">
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K \cdot W(\mathbb{P}_r, \mathbb{P}_g) =
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\max_{w \in \mathcal{W}}
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\mathbb{E}_{x \sim \mathbb{P}_r} [f_w(x)]- \mathbb{E}_{x \sim \mathbb{P}_g} [f_w(x)]
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</script>
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</p>
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<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
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distribution $z \sim p(z)$,</p>
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<p>
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<script type="math/tex; mode=display">
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K \ cdot W(\mathbb{P}_r, \mathbb{P}_\theta) =
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\max_{w \in \mathcal{W}}
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\mathbb{E}_{x \sim \mathbb{P}_r} [f_w(x)]- \mathbb{E}_{z \sim p(z)} [f_w(g_\theta(z))]
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</script>
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</p>
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<p>Now to converge $g_\theta$ with $\mathbb{P}_{r}$ we can gradient descent on $\theta$
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to minimize above formula.</p>
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<p>Similarly we can find $\max_{w \in \mathcal{W}}$ by ascending on $w$,
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while keeping $K$ bounded. <em>One way to keep $K$ bounded is to clip all weights in the neural
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network that defines $f$ clipped within a range.</em></p>
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<p>Here is the code to try this on a <a href="experiment.html">simple MNIST generation experiment</a>.</p>
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<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>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">85</span><span></span><span class="kn">import</span> <span class="nn">torch.utils.data</span>
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<span class="lineno">86</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>
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<span class="lineno">87</span>
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<span class="lineno">88</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>
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</div>
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</div>
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<div class='section' id='section-1'>
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<div class='docs doc-strings'>
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<div class='section-link'>
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<a href='#section-1'>#</a>
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</div>
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<h2>Discriminator Loss</h2>
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<p>We want to find $w$ to maximize
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<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>,
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so we minimize,
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<script type="math/tex; mode=display">-\frac{1}{m} \sum_{i=1}^m f_w \big(x^{(i)} \big) +
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\frac{1}{m} \sum_{i=1}^m f_w \big( g_\theta(z^{(i)}) \big)</script>
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</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">91</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>
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</div>
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</div>
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<div class='section' id='section-2'>
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<div class='docs doc-strings'>
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<div class='section-link'>
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<a href='#section-2'>#</a>
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</div>
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<ul>
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<li><code>f_real</code> is $f_w(x)$</li>
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<li><code>f_fake</code> is $f_w(g_\theta(z))$</li>
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</ul>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">102</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>
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</div>
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</div>
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<div class='section' id='section-3'>
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<div class='docs'>
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<div class='section-link'>
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<a href='#section-3'>#</a>
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</div>
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<p>We use ReLUs to clip the loss to keep $f \in [-1, +1]$ range.</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">109</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>
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</div>
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</div>
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<div class='section' id='section-4'>
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<div class='docs doc-strings'>
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<div class='section-link'>
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<a href='#section-4'>#</a>
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</div>
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<h2>Generator Loss</h2>
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<p>We want to find $\theta$ to minimize
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<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>
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The first component is independent of $\theta$,
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so we minimize,
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<script type="math/tex; mode=display">-\frac{1}{m} \sum_{i=1}^m f_w \big( g_\theta(z^{(i)}) \big)</script>
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</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">112</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>
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</div>
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</div>
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<div class='section' id='section-5'>
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<div class='docs doc-strings'>
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<div class='section-link'>
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<a href='#section-5'>#</a>
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</div>
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<ul>
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<li><code>f_fake</code> is $f_w(g_\theta(z))$</li>
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</ul>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">124</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>
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</div>
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</div>
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<div class='section' id='section-6'>
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<div class='docs'>
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<div class='section-link'>
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<a href='#section-6'>#</a>
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</div>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">128</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>
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