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batch norm formulas
This commit is contained in:
Varuna Jayasiri
@ -3,12 +3,12 @@
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<meta http-equiv="content-type" content="text/html;charset=utf-8"/>
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<meta name="viewport" content="width=device-width, initial-scale=1.0"/>
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<meta name="description" content="A PyTorch implementations/tutorials of batch normalization."/>
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<meta name="description" content="A PyTorch implementation/tutorial of batch normalization."/>
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<meta name="twitter:card" content="summary"/>
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<meta name="twitter:image:src" content="https://avatars1.githubusercontent.com/u/64068543?s=400&v=4"/>
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<meta name="twitter:title" content="Batch Normalization"/>
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<meta name="twitter:description" content="A PyTorch implementations/tutorials of batch normalization."/>
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<meta name="twitter:description" content="A PyTorch implementation/tutorial of batch normalization."/>
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<meta name="twitter:site" content="@labmlai"/>
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<meta name="twitter:creator" content="@labmlai"/>
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@ -18,7 +18,7 @@
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<meta property="og:site_name" content="LabML Neural Networks"/>
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<meta property="og:type" content="object"/>
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<meta property="og:title" content="Batch Normalization"/>
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<meta property="og:description" content="A PyTorch implementations/tutorials of batch normalization."/>
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<meta property="og:description" content="A PyTorch implementation/tutorial of batch normalization."/>
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<title>Batch Normalization</title>
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<link rel="shortcut icon" href="/icon.png"/>
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@ -151,6 +151,25 @@ a CNN classifier that use batch normalization for MNIST dataset.</p>
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<a href='#section-1'>#</a>
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</div>
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<h2>Batch Normalization Layer</h2>
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<p>Batch normalization layer $\text{BN}$ normalizes the input $X$ as follows:</p>
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<p>When input $X \in \mathbb{R}^{B \times C \times H \times W}$ is a batch of image representations,
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where $B$ is the batch size, $C$ is the number of channels, $H$ is the height and $W$ is the width.
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<script type="math/tex; mode=display">\text{BN}(X) = \gamma
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\frac{X - \underset{B, H, W}{\mathbb{E}}[X]}{\sqrt{\underset{B, H, W}{Var}[X] + \epsilon}}
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+ \beta</script>
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</p>
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<p>When input $X \in \mathbb{R}^{B \times C}$ is a batch of vector embeddings,
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where $B$ is the batch size and $C$ is the number of features.
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<script type="math/tex; mode=display">\text{BN}(X) = \gamma
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\frac{X - \underset{B}{\mathbb{E}}[X]}{\sqrt{\underset{B}{Var}[X] + \epsilon}}
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+ \beta</script>
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</p>
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<p>When input $X \in \mathbb{R}^{B \times C \times L}$ is a batch of sequence embeddings,
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where $B$ is the batch size, $C$ is the number of features, and $L$ is the length of the sequence.
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<script type="math/tex; mode=display">\text{BN}(X) = \gamma
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\frac{X - \underset{B, L}{\mathbb{E}}[X]}{\sqrt{\underset{B, L}{Var}[X] + \epsilon}}
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+ \beta</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">102</span><span class="k">class</span> <span class="nc">BatchNorm</span><span class="p">(</span><span class="n">nn</span><span class="o">.</span><span class="n">Module</span><span class="p">):</span></pre></div>
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@ -171,9 +190,9 @@ a CNN classifier that use batch normalization for MNIST dataset.</p>
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<p>We’ve tried to use the same names for arguments as PyTorch <code>BatchNorm</code> implementation.</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">107</span> <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">channels</span><span class="p">:</span> <span class="nb">int</span><span class="p">,</span> <span class="o">*</span><span class="p">,</span>
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<span class="lineno">108</span> <span class="n">eps</span><span class="p">:</span> <span class="nb">float</span> <span class="o">=</span> <span class="mf">1e-5</span><span class="p">,</span> <span class="n">momentum</span><span class="p">:</span> <span class="nb">float</span> <span class="o">=</span> <span class="mf">0.1</span><span class="p">,</span>
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<span class="lineno">109</span> <span class="n">affine</span><span class="p">:</span> <span class="nb">bool</span> <span class="o">=</span> <span class="kc">True</span><span class="p">,</span> <span class="n">track_running_stats</span><span class="p">:</span> <span class="nb">bool</span> <span class="o">=</span> <span class="kc">True</span><span class="p">):</span></pre></div>
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<div class="highlight"><pre><span class="lineno">127</span> <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">channels</span><span class="p">:</span> <span class="nb">int</span><span class="p">,</span> <span class="o">*</span><span class="p">,</span>
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<span class="lineno">128</span> <span class="n">eps</span><span class="p">:</span> <span class="nb">float</span> <span class="o">=</span> <span class="mf">1e-5</span><span class="p">,</span> <span class="n">momentum</span><span class="p">:</span> <span class="nb">float</span> <span class="o">=</span> <span class="mf">0.1</span><span class="p">,</span>
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<span class="lineno">129</span> <span class="n">affine</span><span class="p">:</span> <span class="nb">bool</span> <span class="o">=</span> <span class="kc">True</span><span class="p">,</span> <span class="n">track_running_stats</span><span class="p">:</span> <span class="nb">bool</span> <span class="o">=</span> <span class="kc">True</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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@ -184,14 +203,14 @@ a CNN classifier that use batch normalization for MNIST dataset.</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">119</span> <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__init__</span><span class="p">()</span>
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<span class="lineno">120</span>
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<span class="lineno">121</span> <span class="bp">self</span><span class="o">.</span><span class="n">channels</span> <span class="o">=</span> <span class="n">channels</span>
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<span class="lineno">122</span>
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<span class="lineno">123</span> <span class="bp">self</span><span class="o">.</span><span class="n">eps</span> <span class="o">=</span> <span class="n">eps</span>
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<span class="lineno">124</span> <span class="bp">self</span><span class="o">.</span><span class="n">momentum</span> <span class="o">=</span> <span class="n">momentum</span>
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<span class="lineno">125</span> <span class="bp">self</span><span class="o">.</span><span class="n">affine</span> <span class="o">=</span> <span class="n">affine</span>
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<span class="lineno">126</span> <span class="bp">self</span><span class="o">.</span><span class="n">track_running_stats</span> <span class="o">=</span> <span class="n">track_running_stats</span></pre></div>
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<div class="highlight"><pre><span class="lineno">139</span> <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__init__</span><span class="p">()</span>
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<span class="lineno">140</span>
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<span class="lineno">141</span> <span class="bp">self</span><span class="o">.</span><span class="n">channels</span> <span class="o">=</span> <span class="n">channels</span>
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<span class="lineno">142</span>
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<span class="lineno">143</span> <span class="bp">self</span><span class="o">.</span><span class="n">eps</span> <span class="o">=</span> <span class="n">eps</span>
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<span class="lineno">144</span> <span class="bp">self</span><span class="o">.</span><span class="n">momentum</span> <span class="o">=</span> <span class="n">momentum</span>
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<span class="lineno">145</span> <span class="bp">self</span><span class="o">.</span><span class="n">affine</span> <span class="o">=</span> <span class="n">affine</span>
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<span class="lineno">146</span> <span class="bp">self</span><span class="o">.</span><span class="n">track_running_stats</span> <span class="o">=</span> <span class="n">track_running_stats</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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@ -202,9 +221,9 @@ a CNN classifier that use batch normalization for MNIST dataset.</p>
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<p>Create parameters for $\gamma$ and $\beta$ for scale and shift</p>
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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">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">affine</span><span class="p">:</span>
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<span class="lineno">129</span> <span class="bp">self</span><span class="o">.</span><span class="n">scale</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">Parameter</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">ones</span><span class="p">(</span><span class="n">channels</span><span class="p">))</span>
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<span class="lineno">130</span> <span class="bp">self</span><span class="o">.</span><span class="n">shift</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">Parameter</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">channels</span><span class="p">))</span></pre></div>
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<div class="highlight"><pre><span class="lineno">148</span> <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">affine</span><span class="p">:</span>
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<span class="lineno">149</span> <span class="bp">self</span><span class="o">.</span><span class="n">scale</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">Parameter</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">ones</span><span class="p">(</span><span class="n">channels</span><span class="p">))</span>
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<span class="lineno">150</span> <span class="bp">self</span><span class="o">.</span><span class="n">shift</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">Parameter</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">channels</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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@ -216,9 +235,9 @@ a CNN classifier that use batch normalization for MNIST dataset.</p>
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mean $\mathbb{E}[x^{(k)}]$ and variance $Var[x^{(k)}]$</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">133</span> <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">track_running_stats</span><span class="p">:</span>
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<span class="lineno">134</span> <span class="bp">self</span><span class="o">.</span><span class="n">register_buffer</span><span class="p">(</span><span class="s1">'exp_mean'</span><span class="p">,</span> <span class="n">torch</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">channels</span><span class="p">))</span>
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<span class="lineno">135</span> <span class="bp">self</span><span class="o">.</span><span class="n">register_buffer</span><span class="p">(</span><span class="s1">'exp_var'</span><span class="p">,</span> <span class="n">torch</span><span class="o">.</span><span class="n">ones</span><span class="p">(</span><span class="n">channels</span><span class="p">))</span></pre></div>
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<div class="highlight"><pre><span class="lineno">153</span> <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">track_running_stats</span><span class="p">:</span>
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<span class="lineno">154</span> <span class="bp">self</span><span class="o">.</span><span class="n">register_buffer</span><span class="p">(</span><span class="s1">'exp_mean'</span><span class="p">,</span> <span class="n">torch</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">channels</span><span class="p">))</span>
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<span class="lineno">155</span> <span class="bp">self</span><span class="o">.</span><span class="n">register_buffer</span><span class="p">(</span><span class="s1">'exp_var'</span><span class="p">,</span> <span class="n">torch</span><span class="o">.</span><span class="n">ones</span><span class="p">(</span><span class="n">channels</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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@ -232,7 +251,7 @@ mean $\mathbb{E}[x^{(k)}]$ and variance $Var[x^{(k)}]$</p>
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<code>[batch_size, channels, height, width]</code></p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">137</span> <span class="k">def</span> <span class="nf">forward</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</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 class="highlight"><pre><span class="lineno">157</span> <span class="k">def</span> <span class="nf">forward</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</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-7'>
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@ -243,7 +262,7 @@ mean $\mathbb{E}[x^{(k)}]$ and variance $Var[x^{(k)}]$</p>
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<p>Keep the original shape</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">145</span> <span class="n">x_shape</span> <span class="o">=</span> <span class="n">x</span><span class="o">.</span><span class="n">shape</span></pre></div>
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<div class="highlight"><pre><span class="lineno">165</span> <span class="n">x_shape</span> <span class="o">=</span> <span class="n">x</span><span class="o">.</span><span class="n">shape</span></pre></div>
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</div>
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</div>
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<div class='section' id='section-8'>
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@ -254,7 +273,7 @@ mean $\mathbb{E}[x^{(k)}]$ and variance $Var[x^{(k)}]$</p>
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<p>Get the batch size</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">147</span> <span class="n">batch_size</span> <span class="o">=</span> <span class="n">x_shape</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span></pre></div>
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<div class="highlight"><pre><span class="lineno">167</span> <span class="n">batch_size</span> <span class="o">=</span> <span class="n">x_shape</span><span class="p">[</span><span class="mi">0</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-9'>
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@ -265,7 +284,7 @@ mean $\mathbb{E}[x^{(k)}]$ and variance $Var[x^{(k)}]$</p>
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<p>Sanity check to make sure the number of features is same</p>
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</div>
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<div class='code'>
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<div class="highlight"><pre><span class="lineno">149</span> <span class="k">assert</span> <span class="bp">self</span><span class="o">.</span><span class="n">channels</span> <span class="o">==</span> <span class="n">x</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span></pre></div>
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<div class="highlight"><pre><span class="lineno">169</span> <span class="k">assert</span> <span class="bp">self</span><span class="o">.</span><span class="n">channels</span> <span class="o">==</span> <span class="n">x</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">1</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-10'>
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@ -276,7 +295,7 @@ mean $\mathbb{E}[x^{(k)}]$ and variance $Var[x^{(k)}]$</p>
|
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<p>Reshape into <code>[batch_size, channels, n]</code></p>
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</div>
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<div class='code'>
|
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<div class="highlight"><pre><span class="lineno">152</span> <span class="n">x</span> <span class="o">=</span> <span class="n">x</span><span class="o">.</span><span class="n">view</span><span class="p">(</span><span class="n">batch_size</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">channels</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">)</span></pre></div>
|
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<div class="highlight"><pre><span class="lineno">172</span> <span class="n">x</span> <span class="o">=</span> <span class="n">x</span><span class="o">.</span><span class="n">view</span><span class="p">(</span><span class="n">batch_size</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">channels</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</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-11'>
|
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@ -288,7 +307,7 @@ mean $\mathbb{E}[x^{(k)}]$ and variance $Var[x^{(k)}]$</p>
|
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if we are in training mode or if we have not tracked exponential moving averages</p>
|
||||
</div>
|
||||
<div class='code'>
|
||||
<div class="highlight"><pre><span class="lineno">156</span> <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">training</span> <span class="ow">or</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">track_running_stats</span><span class="p">:</span></pre></div>
|
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<div class="highlight"><pre><span class="lineno">176</span> <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">training</span> <span class="ow">or</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">track_running_stats</span><span class="p">:</span></pre></div>
|
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</div>
|
||||
</div>
|
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<div class='section' id='section-12'>
|
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@ -300,7 +319,7 @@ if we are in training mode or if we have not tracked exponential moving averages
|
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i.e. the means for each feature $\mathbb{E}[x^{(k)}]$</p>
|
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</div>
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<div class='code'>
|
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<div class="highlight"><pre><span class="lineno">159</span> <span class="n">mean</span> <span class="o">=</span> <span class="n">x</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">dim</span><span class="o">=</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">])</span></pre></div>
|
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<div class="highlight"><pre><span class="lineno">179</span> <span class="n">mean</span> <span class="o">=</span> <span class="n">x</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">dim</span><span class="o">=</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</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-13'>
|
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@ -312,7 +331,7 @@ i.e. the means for each feature $\mathbb{E}[x^{(k)}]$</p>
|
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i.e. the means for each feature $\mathbb{E}[(x^{(k)})^2]$</p>
|
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</div>
|
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<div class='code'>
|
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<div class="highlight"><pre><span class="lineno">162</span> <span class="n">mean_x2</span> <span class="o">=</span> <span class="p">(</span><span class="n">x</span> <span class="o">**</span> <span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">dim</span><span class="o">=</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">])</span></pre></div>
|
||||
<div class="highlight"><pre><span class="lineno">182</span> <span class="n">mean_x2</span> <span class="o">=</span> <span class="p">(</span><span class="n">x</span> <span class="o">**</span> <span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">dim</span><span class="o">=</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">])</span></pre></div>
|
||||
</div>
|
||||
</div>
|
||||
<div class='section' id='section-14'>
|
||||
@ -323,7 +342,7 @@ i.e. the means for each feature $\mathbb{E}[(x^{(k)})^2]$</p>
|
||||
<p>Variance for each feature $Var[x^{(k)}] = \mathbb{E}[(x^{(k)})^2] - \mathbb{E}[x^{(k)}]^2$</p>
|
||||
</div>
|
||||
<div class='code'>
|
||||
<div class="highlight"><pre><span class="lineno">164</span> <span class="n">var</span> <span class="o">=</span> <span class="n">mean_x2</span> <span class="o">-</span> <span class="n">mean</span> <span class="o">**</span> <span class="mi">2</span></pre></div>
|
||||
<div class="highlight"><pre><span class="lineno">184</span> <span class="n">var</span> <span class="o">=</span> <span class="n">mean_x2</span> <span class="o">-</span> <span class="n">mean</span> <span class="o">**</span> <span class="mi">2</span></pre></div>
|
||||
</div>
|
||||
</div>
|
||||
<div class='section' id='section-15'>
|
||||
@ -334,9 +353,9 @@ i.e. the means for each feature $\mathbb{E}[(x^{(k)})^2]$</p>
|
||||
<p>Update exponential moving averages</p>
|
||||
</div>
|
||||
<div class='code'>
|
||||
<div class="highlight"><pre><span class="lineno">167</span> <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">training</span> <span class="ow">and</span> <span class="bp">self</span><span class="o">.</span><span class="n">track_running_stats</span><span class="p">:</span>
|
||||
<span class="lineno">168</span> <span class="bp">self</span><span class="o">.</span><span class="n">exp_mean</span> <span class="o">=</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">momentum</span><span class="p">)</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">exp_mean</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">momentum</span> <span class="o">*</span> <span class="n">mean</span>
|
||||
<span class="lineno">169</span> <span class="bp">self</span><span class="o">.</span><span class="n">exp_var</span> <span class="o">=</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">momentum</span><span class="p">)</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">exp_var</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">momentum</span> <span class="o">*</span> <span class="n">var</span></pre></div>
|
||||
<div class="highlight"><pre><span class="lineno">187</span> <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">training</span> <span class="ow">and</span> <span class="bp">self</span><span class="o">.</span><span class="n">track_running_stats</span><span class="p">:</span>
|
||||
<span class="lineno">188</span> <span class="bp">self</span><span class="o">.</span><span class="n">exp_mean</span> <span class="o">=</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">momentum</span><span class="p">)</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">exp_mean</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">momentum</span> <span class="o">*</span> <span class="n">mean</span>
|
||||
<span class="lineno">189</span> <span class="bp">self</span><span class="o">.</span><span class="n">exp_var</span> <span class="o">=</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">momentum</span><span class="p">)</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">exp_var</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">momentum</span> <span class="o">*</span> <span class="n">var</span></pre></div>
|
||||
</div>
|
||||
</div>
|
||||
<div class='section' id='section-16'>
|
||||
@ -347,9 +366,9 @@ i.e. the means for each feature $\mathbb{E}[(x^{(k)})^2]$</p>
|
||||
<p>Use exponential moving averages as estimates</p>
|
||||
</div>
|
||||
<div class='code'>
|
||||
<div class="highlight"><pre><span class="lineno">171</span> <span class="k">else</span><span class="p">:</span>
|
||||
<span class="lineno">172</span> <span class="n">mean</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">exp_mean</span>
|
||||
<span class="lineno">173</span> <span class="n">var</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">exp_var</span></pre></div>
|
||||
<div class="highlight"><pre><span class="lineno">191</span> <span class="k">else</span><span class="p">:</span>
|
||||
<span class="lineno">192</span> <span class="n">mean</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">exp_mean</span>
|
||||
<span class="lineno">193</span> <span class="n">var</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">exp_var</span></pre></div>
|
||||
</div>
|
||||
</div>
|
||||
<div class='section' id='section-17'>
|
||||
@ -361,7 +380,7 @@ i.e. the means for each feature $\mathbb{E}[(x^{(k)})^2]$</p>
|
||||
</p>
|
||||
</div>
|
||||
<div class='code'>
|
||||
<div class="highlight"><pre><span class="lineno">176</span> <span class="n">x_norm</span> <span class="o">=</span> <span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="n">mean</span><span class="o">.</span><span class="n">view</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span> <span class="o">/</span> <span class="n">torch</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">var</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">eps</span><span class="p">)</span><span class="o">.</span><span class="n">view</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span></pre></div>
|
||||
<div class="highlight"><pre><span class="lineno">196</span> <span class="n">x_norm</span> <span class="o">=</span> <span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="n">mean</span><span class="o">.</span><span class="n">view</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span> <span class="o">/</span> <span class="n">torch</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">var</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">eps</span><span class="p">)</span><span class="o">.</span><span class="n">view</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span></pre></div>
|
||||
</div>
|
||||
</div>
|
||||
<div class='section' id='section-18'>
|
||||
@ -373,8 +392,8 @@ i.e. the means for each feature $\mathbb{E}[(x^{(k)})^2]$</p>
|
||||
</p>
|
||||
</div>
|
||||
<div class='code'>
|
||||
<div class="highlight"><pre><span class="lineno">178</span> <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">affine</span><span class="p">:</span>
|
||||
<span class="lineno">179</span> <span class="n">x_norm</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">scale</span><span class="o">.</span><span class="n">view</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span> <span class="o">*</span> <span class="n">x_norm</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">shift</span><span class="o">.</span><span class="n">view</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span></pre></div>
|
||||
<div class="highlight"><pre><span class="lineno">198</span> <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">affine</span><span class="p">:</span>
|
||||
<span class="lineno">199</span> <span class="n">x_norm</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">scale</span><span class="o">.</span><span class="n">view</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span> <span class="o">*</span> <span class="n">x_norm</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">shift</span><span class="o">.</span><span class="n">view</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span></pre></div>
|
||||
</div>
|
||||
</div>
|
||||
<div class='section' id='section-19'>
|
||||
@ -385,7 +404,7 @@ i.e. the means for each feature $\mathbb{E}[(x^{(k)})^2]$</p>
|
||||
<p>Reshape to original and return</p>
|
||||
</div>
|
||||
<div class='code'>
|
||||
<div class="highlight"><pre><span class="lineno">182</span> <span class="k">return</span> <span class="n">x_norm</span><span class="o">.</span><span class="n">view</span><span class="p">(</span><span class="n">x_shape</span><span class="p">)</span></pre></div>
|
||||
<div class="highlight"><pre><span class="lineno">202</span> <span class="k">return</span> <span class="n">x_norm</span><span class="o">.</span><span class="n">view</span><span class="p">(</span><span class="n">x_shape</span><span class="p">)</span></pre></div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
380
docs/normalization/layer_norm/index.html
Normal file
380
docs/normalization/layer_norm/index.html
Normal file
@ -0,0 +1,380 @@
|
||||
<!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 PyTorch implementation/tutorial of layer normalization."/>
|
||||
|
||||
<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="Layer Normalization"/>
|
||||
<meta name="twitter:description" content="A PyTorch implementation/tutorial of layer normalization."/>
|
||||
<meta name="twitter:site" content="@labmlai"/>
|
||||
<meta name="twitter:creator" content="@labmlai"/>
|
||||
|
||||
<meta property="og:url" content="https://nn.labml.ai/normalization/layer_norm/index.html"/>
|
||||
<meta property="og:title" content="Layer Normalization"/>
|
||||
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|
||||
<meta property="og:site_name" content="LabML Neural Networks"/>
|
||||
<meta property="og:type" content="object"/>
|
||||
<meta property="og:title" content="Layer Normalization"/>
|
||||
<meta property="og:description" content="A PyTorch implementation/tutorial of layer normalization."/>
|
||||
|
||||
<title>Layer Normalization</title>
|
||||
<link rel="shortcut icon" href="/icon.png"/>
|
||||
<link rel="stylesheet" href="../../pylit.css">
|
||||
<link rel="canonical" href="https://nn.labml.ai/normalization/layer_norm/index.html"/>
|
||||
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|
||||
<script async src="https://www.googletagmanager.com/gtag/js?id=G-4V3HC8HBLH"></script>
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|
||||
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||||
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|
||||
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||||
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|
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|
||||
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|
||||
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||||
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|
||||
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|
||||
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|
||||
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|
||||
<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">normalization</a>
|
||||
<a class="parent" href="index.html">layer_norm</a>
|
||||
</p>
|
||||
<p>
|
||||
|
||||
<a href="https://github.com/lab-ml/labml_nn/tree/master/labml_nn/normalization/layer_norm/__init__.py">
|
||||
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|
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|
||||
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|
||||
</div>
|
||||
</div>
|
||||
<div class='section' id='section-0'>
|
||||
<div class='docs doc-strings'>
|
||||
<div class='section-link'>
|
||||
<a href='#section-0'>#</a>
|
||||
</div>
|
||||
<h1>Layer Normalization</h1>
|
||||
<p>This is a <a href="https://pytorch.org">PyTorch</a> implementation of
|
||||
<a href="https://arxiv.org/abs/1607.06450">Layer Normalization</a>.</p>
|
||||
<h3>Limitations of <a href="../batch_norm/index.html">Batch Normalization</a></h3>
|
||||
<ul>
|
||||
<li>You need to maintain running means.</li>
|
||||
<li>Tricky for RNNs. Do you need different normalizations for each step?</li>
|
||||
<li>Doesn’t work with small batch sizes;
|
||||
large NLP models are usually trained with small batch sizes.</li>
|
||||
<li>Need to compute means and variances across devices in distributed training</li>
|
||||
</ul>
|
||||
<h2>Layer Normalization</h2>
|
||||
<p>Layer normalization is a simpler normalization method that works
|
||||
on a wider range of settings.
|
||||
Layer normalization transformers the inputs to have zero mean and unit variance
|
||||
across the features.
|
||||
*Note that batch normalization, fixes the zero mean and unit variance for each vector.
|
||||
Layer normalization does it for each batch across all elements.</p>
|
||||
<p>Layer normalization is generally used for NLP tasks.</p>
|
||||
<p>Here’s <a href="mnist.html">the training code</a> and a notebook for training
|
||||
a CNN classifier that use batch normalization for MNIST dataset.</p>
|
||||
<p><a href="https://colab.research.google.com/github/lab-ml/nn/blob/master/labml_nn/normalization/batch_norm/mnist.ipynb"><img alt="Open In Colab" src="https://colab.research.google.com/assets/colab-badge.svg" /></a>
|
||||
<a href="https://web.lab-ml.com/run?uuid=011254fe647011ebbb8e0242ac1c0002"><img alt="View Run" src="https://img.shields.io/badge/labml-experiment-brightgreen" /></a></p>
|
||||
</div>
|
||||
<div class='code'>
|
||||
<div class="highlight"><pre><span class="lineno">39</span><span></span><span class="kn">import</span> <span class="nn">torch</span>
|
||||
<span class="lineno">40</span><span class="kn">from</span> <span class="nn">torch</span> <span class="kn">import</span> <span class="n">nn</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>Batch Normalization Layer</h2>
|
||||
</div>
|
||||
<div class='code'>
|
||||
<div class="highlight"><pre><span class="lineno">43</span><span class="k">class</span> <span class="nc">BatchNorm</span><span class="p">(</span><span class="n">nn</span><span class="o">.</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>channels</code> is the number of features in the input</li>
|
||||
<li><code>eps</code> is $\epsilon$, used in $\sqrt{Var[x^{(k)}] + \epsilon}$ for numerical stability</li>
|
||||
<li><code>momentum</code> is the momentum in taking the exponential moving average</li>
|
||||
<li><code>affine</code> is whether to scale and shift the normalized value</li>
|
||||
<li><code>track_running_stats</code> is whether to calculate the moving averages or mean and variance</li>
|
||||
</ul>
|
||||
<p>We’ve tried to use the same names for arguments as PyTorch <code>BatchNorm</code> implementation.</p>
|
||||
</div>
|
||||
<div class='code'>
|
||||
<div class="highlight"><pre><span class="lineno">48</span> <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">channels</span><span class="p">:</span> <span class="nb">int</span><span class="p">,</span> <span class="o">*</span><span class="p">,</span>
|
||||
<span class="lineno">49</span> <span class="n">eps</span><span class="p">:</span> <span class="nb">float</span> <span class="o">=</span> <span class="mf">1e-5</span><span class="p">,</span> <span class="n">momentum</span><span class="p">:</span> <span class="nb">float</span> <span class="o">=</span> <span class="mf">0.1</span><span class="p">,</span>
|
||||
<span class="lineno">50</span> <span class="n">affine</span><span class="p">:</span> <span class="nb">bool</span> <span class="o">=</span> <span class="kc">True</span><span class="p">,</span> <span class="n">track_running_stats</span><span class="p">:</span> <span class="nb">bool</span> <span class="o">=</span> <span class="kc">True</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>
|
||||
|
||||
</div>
|
||||
<div class='code'>
|
||||
<div class="highlight"><pre><span class="lineno">60</span> <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__init__</span><span class="p">()</span>
|
||||
<span class="lineno">61</span>
|
||||
<span class="lineno">62</span> <span class="bp">self</span><span class="o">.</span><span class="n">channels</span> <span class="o">=</span> <span class="n">channels</span>
|
||||
<span class="lineno">63</span>
|
||||
<span class="lineno">64</span> <span class="bp">self</span><span class="o">.</span><span class="n">eps</span> <span class="o">=</span> <span class="n">eps</span>
|
||||
<span class="lineno">65</span> <span class="bp">self</span><span class="o">.</span><span class="n">momentum</span> <span class="o">=</span> <span class="n">momentum</span>
|
||||
<span class="lineno">66</span> <span class="bp">self</span><span class="o">.</span><span class="n">affine</span> <span class="o">=</span> <span class="n">affine</span>
|
||||
<span class="lineno">67</span> <span class="bp">self</span><span class="o">.</span><span class="n">track_running_stats</span> <span class="o">=</span> <span class="n">track_running_stats</span></pre></div>
|
||||
</div>
|
||||
</div>
|
||||
<div class='section' id='section-4'>
|
||||
<div class='docs'>
|
||||
<div class='section-link'>
|
||||
<a href='#section-4'>#</a>
|
||||
</div>
|
||||
<p>Create parameters for $\gamma$ and $\beta$ for scale and shift</p>
|
||||
</div>
|
||||
<div class='code'>
|
||||
<div class="highlight"><pre><span class="lineno">69</span> <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">affine</span><span class="p">:</span>
|
||||
<span class="lineno">70</span> <span class="bp">self</span><span class="o">.</span><span class="n">scale</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">Parameter</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">ones</span><span class="p">(</span><span class="n">channels</span><span class="p">))</span>
|
||||
<span class="lineno">71</span> <span class="bp">self</span><span class="o">.</span><span class="n">shift</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">Parameter</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">channels</span><span class="p">))</span></pre></div>
|
||||
</div>
|
||||
</div>
|
||||
<div class='section' id='section-5'>
|
||||
<div class='docs'>
|
||||
<div class='section-link'>
|
||||
<a href='#section-5'>#</a>
|
||||
</div>
|
||||
<p>Create buffers to store exponential moving averages of
|
||||
mean $\mathbb{E}[x^{(k)}]$ and variance $Var[x^{(k)}]$</p>
|
||||
</div>
|
||||
<div class='code'>
|
||||
<div class="highlight"><pre><span class="lineno">74</span> <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">track_running_stats</span><span class="p">:</span>
|
||||
<span class="lineno">75</span> <span class="bp">self</span><span class="o">.</span><span class="n">register_buffer</span><span class="p">(</span><span class="s1">'exp_mean'</span><span class="p">,</span> <span class="n">torch</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">channels</span><span class="p">))</span>
|
||||
<span class="lineno">76</span> <span class="bp">self</span><span class="o">.</span><span class="n">register_buffer</span><span class="p">(</span><span class="s1">'exp_var'</span><span class="p">,</span> <span class="n">torch</span><span class="o">.</span><span class="n">ones</span><span class="p">(</span><span class="n">channels</span><span class="p">))</span></pre></div>
|
||||
</div>
|
||||
</div>
|
||||
<div class='section' id='section-6'>
|
||||
<div class='docs doc-strings'>
|
||||
<div class='section-link'>
|
||||
<a href='#section-6'>#</a>
|
||||
</div>
|
||||
<p><code>x</code> is a tensor of shape <code>[batch_size, channels, *]</code>.
|
||||
<code>*</code> could be any (even *) dimensions.
|
||||
For example, in an image (2D) convolution this will be
|
||||
<code>[batch_size, channels, height, width]</code></p>
|
||||
</div>
|
||||
<div class='code'>
|
||||
<div class="highlight"><pre><span class="lineno">78</span> <span class="k">def</span> <span class="nf">forward</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</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-7'>
|
||||
<div class='docs'>
|
||||
<div class='section-link'>
|
||||
<a href='#section-7'>#</a>
|
||||
</div>
|
||||
<p>Keep the original shape</p>
|
||||
</div>
|
||||
<div class='code'>
|
||||
<div class="highlight"><pre><span class="lineno">86</span> <span class="n">x_shape</span> <span class="o">=</span> <span class="n">x</span><span class="o">.</span><span class="n">shape</span></pre></div>
|
||||
</div>
|
||||
</div>
|
||||
<div class='section' id='section-8'>
|
||||
<div class='docs'>
|
||||
<div class='section-link'>
|
||||
<a href='#section-8'>#</a>
|
||||
</div>
|
||||
<p>Get the batch size</p>
|
||||
</div>
|
||||
<div class='code'>
|
||||
<div class="highlight"><pre><span class="lineno">88</span> <span class="n">batch_size</span> <span class="o">=</span> <span class="n">x_shape</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span></pre></div>
|
||||
</div>
|
||||
</div>
|
||||
<div class='section' id='section-9'>
|
||||
<div class='docs'>
|
||||
<div class='section-link'>
|
||||
<a href='#section-9'>#</a>
|
||||
</div>
|
||||
<p>Sanity check to make sure the number of features is same</p>
|
||||
</div>
|
||||
<div class='code'>
|
||||
<div class="highlight"><pre><span class="lineno">90</span> <span class="k">assert</span> <span class="bp">self</span><span class="o">.</span><span class="n">channels</span> <span class="o">==</span> <span class="n">x</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span></pre></div>
|
||||
</div>
|
||||
</div>
|
||||
<div class='section' id='section-10'>
|
||||
<div class='docs'>
|
||||
<div class='section-link'>
|
||||
<a href='#section-10'>#</a>
|
||||
</div>
|
||||
<p>Reshape into <code>[batch_size, channels, n]</code></p>
|
||||
</div>
|
||||
<div class='code'>
|
||||
<div class="highlight"><pre><span class="lineno">93</span> <span class="n">x</span> <span class="o">=</span> <span class="n">x</span><span class="o">.</span><span class="n">view</span><span class="p">(</span><span class="n">batch_size</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">channels</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">)</span></pre></div>
|
||||
</div>
|
||||
</div>
|
||||
<div class='section' id='section-11'>
|
||||
<div class='docs'>
|
||||
<div class='section-link'>
|
||||
<a href='#section-11'>#</a>
|
||||
</div>
|
||||
<p>We will calculate the mini-batch mean and variance
|
||||
if we are in training mode or if we have not tracked exponential moving averages</p>
|
||||
</div>
|
||||
<div class='code'>
|
||||
<div class="highlight"><pre><span class="lineno">97</span> <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">training</span> <span class="ow">or</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">track_running_stats</span><span class="p">:</span></pre></div>
|
||||
</div>
|
||||
</div>
|
||||
<div class='section' id='section-12'>
|
||||
<div class='docs'>
|
||||
<div class='section-link'>
|
||||
<a href='#section-12'>#</a>
|
||||
</div>
|
||||
<p>Calculate the mean across first and last dimension;
|
||||
i.e. the means for each feature $\mathbb{E}[x^{(k)}]$</p>
|
||||
</div>
|
||||
<div class='code'>
|
||||
<div class="highlight"><pre><span class="lineno">100</span> <span class="n">mean</span> <span class="o">=</span> <span class="n">x</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">dim</span><span class="o">=</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">])</span></pre></div>
|
||||
</div>
|
||||
</div>
|
||||
<div class='section' id='section-13'>
|
||||
<div class='docs'>
|
||||
<div class='section-link'>
|
||||
<a href='#section-13'>#</a>
|
||||
</div>
|
||||
<p>Calculate the squared mean across first and last dimension;
|
||||
i.e. the means for each feature $\mathbb{E}[(x^{(k)})^2]$</p>
|
||||
</div>
|
||||
<div class='code'>
|
||||
<div class="highlight"><pre><span class="lineno">103</span> <span class="n">mean_x2</span> <span class="o">=</span> <span class="p">(</span><span class="n">x</span> <span class="o">**</span> <span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">dim</span><span class="o">=</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">])</span></pre></div>
|
||||
</div>
|
||||
</div>
|
||||
<div class='section' id='section-14'>
|
||||
<div class='docs'>
|
||||
<div class='section-link'>
|
||||
<a href='#section-14'>#</a>
|
||||
</div>
|
||||
<p>Variance for each feature $Var[x^{(k)}] = \mathbb{E}[(x^{(k)})^2] - \mathbb{E}[x^{(k)}]^2$</p>
|
||||
</div>
|
||||
<div class='code'>
|
||||
<div class="highlight"><pre><span class="lineno">105</span> <span class="n">var</span> <span class="o">=</span> <span class="n">mean_x2</span> <span class="o">-</span> <span class="n">mean</span> <span class="o">**</span> <span class="mi">2</span></pre></div>
|
||||
</div>
|
||||
</div>
|
||||
<div class='section' id='section-15'>
|
||||
<div class='docs'>
|
||||
<div class='section-link'>
|
||||
<a href='#section-15'>#</a>
|
||||
</div>
|
||||
<p>Update exponential moving averages</p>
|
||||
</div>
|
||||
<div class='code'>
|
||||
<div class="highlight"><pre><span class="lineno">108</span> <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">training</span> <span class="ow">and</span> <span class="bp">self</span><span class="o">.</span><span class="n">track_running_stats</span><span class="p">:</span>
|
||||
<span class="lineno">109</span> <span class="bp">self</span><span class="o">.</span><span class="n">exp_mean</span> <span class="o">=</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">momentum</span><span class="p">)</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">exp_mean</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">momentum</span> <span class="o">*</span> <span class="n">mean</span>
|
||||
<span class="lineno">110</span> <span class="bp">self</span><span class="o">.</span><span class="n">exp_var</span> <span class="o">=</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">momentum</span><span class="p">)</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">exp_var</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">momentum</span> <span class="o">*</span> <span class="n">var</span></pre></div>
|
||||
</div>
|
||||
</div>
|
||||
<div class='section' id='section-16'>
|
||||
<div class='docs'>
|
||||
<div class='section-link'>
|
||||
<a href='#section-16'>#</a>
|
||||
</div>
|
||||
<p>Use exponential moving averages as estimates</p>
|
||||
</div>
|
||||
<div class='code'>
|
||||
<div class="highlight"><pre><span class="lineno">112</span> <span class="k">else</span><span class="p">:</span>
|
||||
<span class="lineno">113</span> <span class="n">mean</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">exp_mean</span>
|
||||
<span class="lineno">114</span> <span class="n">var</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">exp_var</span></pre></div>
|
||||
</div>
|
||||
</div>
|
||||
<div class='section' id='section-17'>
|
||||
<div class='docs'>
|
||||
<div class='section-link'>
|
||||
<a href='#section-17'>#</a>
|
||||
</div>
|
||||
<p>Normalize <script type="math/tex; mode=display">\hat{x}^{(k)} = \frac{x^{(k)} - \mathbb{E}[x^{(k)}]}{\sqrt{Var[x^{(k)}] + \epsilon}}</script>
|
||||
</p>
|
||||
</div>
|
||||
<div class='code'>
|
||||
<div class="highlight"><pre><span class="lineno">117</span> <span class="n">x_norm</span> <span class="o">=</span> <span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="n">mean</span><span class="o">.</span><span class="n">view</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span> <span class="o">/</span> <span class="n">torch</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">var</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">eps</span><span class="p">)</span><span class="o">.</span><span class="n">view</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span></pre></div>
|
||||
</div>
|
||||
</div>
|
||||
<div class='section' id='section-18'>
|
||||
<div class='docs'>
|
||||
<div class='section-link'>
|
||||
<a href='#section-18'>#</a>
|
||||
</div>
|
||||
<p>Scale and shift <script type="math/tex; mode=display">y^{(k)} =\gamma^{(k)} \hat{x}^{(k)} + \beta^{(k)}</script>
|
||||
</p>
|
||||
</div>
|
||||
<div class='code'>
|
||||
<div class="highlight"><pre><span class="lineno">119</span> <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">affine</span><span class="p">:</span>
|
||||
<span class="lineno">120</span> <span class="n">x_norm</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">scale</span><span class="o">.</span><span class="n">view</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span> <span class="o">*</span> <span class="n">x_norm</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">shift</span><span class="o">.</span><span class="n">view</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span></pre></div>
|
||||
</div>
|
||||
</div>
|
||||
<div class='section' id='section-19'>
|
||||
<div class='docs'>
|
||||
<div class='section-link'>
|
||||
<a href='#section-19'>#</a>
|
||||
</div>
|
||||
<p>Reshape to original and return</p>
|
||||
</div>
|
||||
<div class='code'>
|
||||
<div class="highlight"><pre><span class="lineno">123</span> <span class="k">return</span> <span class="n">x_norm</span><span class="o">.</span><span class="n">view</span><span class="p">(</span><span class="n">x_shape</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>
|
||||
@ -99,7 +99,7 @@
|
||||
|
||||
<url>
|
||||
<loc>https://nn.labml.ai/normalization/batch_norm/index.html</loc>
|
||||
<lastmod>2021-02-01T16:30:00+00:00</lastmod>
|
||||
<lastmod>2021-02-02T16:30:00+00:00</lastmod>
|
||||
<priority>1.00</priority>
|
||||
</url>
|
||||
|
||||
|
||||
@ -2,7 +2,7 @@
|
||||
---
|
||||
title: Batch Normalization
|
||||
summary: >
|
||||
A PyTorch implementations/tutorials of batch normalization.
|
||||
A PyTorch implementation/tutorial of batch normalization.
|
||||
---
|
||||
|
||||
# Batch Normalization
|
||||
@ -100,8 +100,28 @@ from torch import nn
|
||||
|
||||
|
||||
class BatchNorm(nn.Module):
|
||||
"""
|
||||
r"""
|
||||
## Batch Normalization Layer
|
||||
|
||||
Batch normalization layer $\text{BN}$ normalizes the input $X$ as follows:
|
||||
|
||||
When input $X \in \mathbb{R}^{B \times C \times H \times W}$ is a batch of image representations,
|
||||
where $B$ is the batch size, $C$ is the number of channels, $H$ is the height and $W$ is the width.
|
||||
$$\text{BN}(X) = \gamma
|
||||
\frac{X - \underset{B, H, W}{\mathbb{E}}[X]}{\sqrt{\underset{B, H, W}{Var}[X] + \epsilon}}
|
||||
+ \beta$$
|
||||
|
||||
When input $X \in \mathbb{R}^{B \times C}$ is a batch of vector embeddings,
|
||||
where $B$ is the batch size and $C$ is the number of features.
|
||||
$$\text{BN}(X) = \gamma
|
||||
\frac{X - \underset{B}{\mathbb{E}}[X]}{\sqrt{\underset{B}{Var}[X] + \epsilon}}
|
||||
+ \beta$$
|
||||
|
||||
When input $X \in \mathbb{R}^{B \times C \times L}$ is a batch of sequence embeddings,
|
||||
where $B$ is the batch size, $C$ is the number of features, and $L$ is the length of the sequence.
|
||||
$$\text{BN}(X) = \gamma
|
||||
\frac{X - \underset{B, L}{\mathbb{E}}[X]}{\sqrt{\underset{B, L}{Var}[X] + \epsilon}}
|
||||
+ \beta$$
|
||||
"""
|
||||
|
||||
def __init__(self, channels: int, *,
|
||||
|
||||
123
labml_nn/normalization/layer_norm/__init__.py
Normal file
123
labml_nn/normalization/layer_norm/__init__.py
Normal file
@ -0,0 +1,123 @@
|
||||
"""
|
||||
---
|
||||
title: Layer Normalization
|
||||
summary: >
|
||||
A PyTorch implementation/tutorial of layer normalization.
|
||||
---
|
||||
|
||||
# Layer Normalization
|
||||
|
||||
This is a [PyTorch](https://pytorch.org) implementation of
|
||||
[Layer Normalization](https://arxiv.org/abs/1607.06450).
|
||||
|
||||
### Limitations of [Batch Normalization](../batch_norm/index.html)
|
||||
|
||||
* You need to maintain running means.
|
||||
* Tricky for RNNs. Do you need different normalizations for each step?
|
||||
* Doesn't work with small batch sizes;
|
||||
large NLP models are usually trained with small batch sizes.
|
||||
* Need to compute means and variances across devices in distributed training
|
||||
|
||||
## Layer Normalization
|
||||
|
||||
Layer normalization is a simpler normalization method that works
|
||||
on a wider range of settings.
|
||||
Layer normalization transformers the inputs to have zero mean and unit variance
|
||||
across the features.
|
||||
*Note that batch normalization, fixes the zero mean and unit variance for each vector.
|
||||
Layer normalization does it for each batch across all elements.
|
||||
|
||||
Layer normalization is generally used for NLP tasks.
|
||||
|
||||
Here's [the training code](mnist.html) and a notebook for training
|
||||
a CNN classifier that use batch normalization for MNIST dataset.
|
||||
|
||||
[](https://colab.research.google.com/github/lab-ml/nn/blob/master/labml_nn/normalization/batch_norm/mnist.ipynb)
|
||||
[](https://web.lab-ml.com/run?uuid=011254fe647011ebbb8e0242ac1c0002)
|
||||
"""
|
||||
|
||||
import torch
|
||||
from torch import nn
|
||||
|
||||
|
||||
class BatchNorm(nn.Module):
|
||||
"""
|
||||
## Batch Normalization Layer
|
||||
"""
|
||||
|
||||
def __init__(self, channels: int, *,
|
||||
eps: float = 1e-5, momentum: float = 0.1,
|
||||
affine: bool = True, track_running_stats: bool = True):
|
||||
"""
|
||||
* `channels` is the number of features in the input
|
||||
* `eps` is $\epsilon$, used in $\sqrt{Var[x^{(k)}] + \epsilon}$ for numerical stability
|
||||
* `momentum` is the momentum in taking the exponential moving average
|
||||
* `affine` is whether to scale and shift the normalized value
|
||||
* `track_running_stats` is whether to calculate the moving averages or mean and variance
|
||||
|
||||
We've tried to use the same names for arguments as PyTorch `BatchNorm` implementation.
|
||||
"""
|
||||
super().__init__()
|
||||
|
||||
self.channels = channels
|
||||
|
||||
self.eps = eps
|
||||
self.momentum = momentum
|
||||
self.affine = affine
|
||||
self.track_running_stats = track_running_stats
|
||||
# Create parameters for $\gamma$ and $\beta$ for scale and shift
|
||||
if self.affine:
|
||||
self.scale = nn.Parameter(torch.ones(channels))
|
||||
self.shift = nn.Parameter(torch.zeros(channels))
|
||||
# Create buffers to store exponential moving averages of
|
||||
# mean $\mathbb{E}[x^{(k)}]$ and variance $Var[x^{(k)}]$
|
||||
if self.track_running_stats:
|
||||
self.register_buffer('exp_mean', torch.zeros(channels))
|
||||
self.register_buffer('exp_var', torch.ones(channels))
|
||||
|
||||
def forward(self, x: torch.Tensor):
|
||||
"""
|
||||
`x` is a tensor of shape `[batch_size, channels, *]`.
|
||||
`*` could be any (even *) dimensions.
|
||||
For example, in an image (2D) convolution this will be
|
||||
`[batch_size, channels, height, width]`
|
||||
"""
|
||||
# Keep the original shape
|
||||
x_shape = x.shape
|
||||
# Get the batch size
|
||||
batch_size = x_shape[0]
|
||||
# Sanity check to make sure the number of features is same
|
||||
assert self.channels == x.shape[1]
|
||||
|
||||
# Reshape into `[batch_size, channels, n]`
|
||||
x = x.view(batch_size, self.channels, -1)
|
||||
|
||||
# We will calculate the mini-batch mean and variance
|
||||
# if we are in training mode or if we have not tracked exponential moving averages
|
||||
if self.training or not self.track_running_stats:
|
||||
# Calculate the mean across first and last dimension;
|
||||
# i.e. the means for each feature $\mathbb{E}[x^{(k)}]$
|
||||
mean = x.mean(dim=[0, 2])
|
||||
# Calculate the squared mean across first and last dimension;
|
||||
# i.e. the means for each feature $\mathbb{E}[(x^{(k)})^2]$
|
||||
mean_x2 = (x ** 2).mean(dim=[0, 2])
|
||||
# Variance for each feature $Var[x^{(k)}] = \mathbb{E}[(x^{(k)})^2] - \mathbb{E}[x^{(k)}]^2$
|
||||
var = mean_x2 - mean ** 2
|
||||
|
||||
# Update exponential moving averages
|
||||
if self.training and self.track_running_stats:
|
||||
self.exp_mean = (1 - self.momentum) * self.exp_mean + self.momentum * mean
|
||||
self.exp_var = (1 - self.momentum) * self.exp_var + self.momentum * var
|
||||
# Use exponential moving averages as estimates
|
||||
else:
|
||||
mean = self.exp_mean
|
||||
var = self.exp_var
|
||||
|
||||
# Normalize $$\hat{x}^{(k)} = \frac{x^{(k)} - \mathbb{E}[x^{(k)}]}{\sqrt{Var[x^{(k)}] + \epsilon}}$$
|
||||
x_norm = (x - mean.view(1, -1, 1)) / torch.sqrt(var + self.eps).view(1, -1, 1)
|
||||
# Scale and shift $$y^{(k)} =\gamma^{(k)} \hat{x}^{(k)} + \beta^{(k)}$$
|
||||
if self.affine:
|
||||
x_norm = self.scale.view(1, -1, 1) * x_norm + self.shift.view(1, -1, 1)
|
||||
|
||||
# Reshape to original and return
|
||||
return x_norm.view(x_shape)
|
||||
Reference in New Issue
Block a user