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<time class="post-full-meta-date" datetime="15 December 2021">15 December 2021</time>
<span class="date-divider">/</span>
<a href='/tag/cs231n/'>CS231N</a>
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<h1 class="post-full-title">cs231n - Lecture 4. Neural Networks and Backpropagation</h1>
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<h3 id="image-features">Image Features</h3>
<ul>
<li>Problem: Linear Classifiers are not very powerful
<ul>
<li>Visual Viewpoint: Linear classifiers learn one template per class</li>
<li>Geometric Viewpoint: Linear classifiers can only draw linear decision boundaries</li>
</ul>
</li>
<li>
<p>Image Features: Motivation<br />
After applying feature transform, points can be separated by linear classifier<br />
$f(x,y) = (r(x,y), \theta(x,y))$</p>
</li>
<li>Image Features vs. ConvNets<br />
<img src="/assets/images/cs231n_lec4_0.png" alt="png" width="60%", height="60%" /></li>
</ul>
<h3 id="neural-networks">Neural Networks</h3>
<ul>
<li>
<p>Neural networks, also called <em>Fully connected networks</em>(FCN) or sometimes <em>multi-layer perceptrons</em>(MLP)<br />
(Before) Linear score function:<br />
\(\begin{align*}& f=Wx \\
& x\in\mathbb{R}^D, W\in\mathbb{R}^{C\times D}
\end{align*}\)<br />
$\rightarrow$ <em>2</em>-layer Neural Network:<br />
\(\begin{align*}& f=W_2 \mbox{max}(0,W_1 x) \\
& x\in\mathbb{R}^D, W_1\in\mathbb{R}^{H\times D},
W_2\in\mathbb{R}^{C\times H}
\end{align*}\)<br />
$\rightarrow$ or <em>3</em>-layer Neural Network:<br />
\(f=W_3\mbox{max}(0,W_2 \mbox{max}(0,W_1 x)) \\
\vdots\)<br />
(In practice we will usually add a learnable bias at each layer as well)</p>
</li>
<li>
<p>Neural networks: hierarchical computation<br />
<img src="/assets/images/cs231n_lec4_1.png" alt="png" width="50%", height="50%" /><br />
Learning 100s of templates instead of 10 and share templates between classes</p>
</li>
<li>
<p>Why is <em>max</em> operator important?<br />
The function $\mbox{max}(0,z)$ is called the activation function.<br />
Q: What if we try to build a neural network without one?<br />
A: We end up with a linear classifier again!<br />
$f=W_2 W_1 x, W_3=W_1 W_2, f = W_3 x$</p>
</li>
<li>
<p>Activation functions<br />
ReLU($\mbox{max}(0,z)$) is a good default choice for most problems<br />
Others: Sigmoid, tanh, Leaky ReLU, Maxout, ELU, etc.</p>
</li>
<li>
<p>Neural networks: Architectures<br />
Example feed-forward computation of a neural network<br />
<img src="/assets/images/cs231n_lec4_2.png" alt="png" width="50%", height="50%" /></p>
</li>
</ul>
<div class="language-python highlighter-rouge"><div class="highlight"><pre class="highlight"><code><span class="c1"># forward-pass of a 3-layer neural network:
</span><span class="n">f</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="mf">1.0</span><span class="o">/</span><span class="p">(</span><span class="mf">1.0</span> <span class="o">+</span> <span class="n">np</span><span class="p">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">x</span><span class="p">))</span> <span class="c1"># activation function (use sigmoid)
</span><span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="p">.</span><span class="n">random</span><span class="p">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span> <span class="c1"># random input vector of three numbers (3x1)
</span><span class="n">h1</span> <span class="o">=</span> <span class="n">f</span><span class="p">(</span><span class="n">np</span><span class="p">.</span><span class="n">dot</span><span class="p">(</span><span class="n">W1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span> <span class="o">+</span> <span class="n">b1</span><span class="p">)</span> <span class="c1"># calculate first hidden layer activations (4x1)
</span><span class="n">h2</span> <span class="o">=</span> <span class="n">f</span><span class="p">(</span><span class="n">np</span><span class="p">.</span><span class="n">dot</span><span class="p">(</span><span class="n">W2</span><span class="p">,</span> <span class="n">h1</span><span class="p">)</span> <span class="o">+</span> <span class="n">b2</span><span class="p">)</span> <span class="c1">#calculate second hidden layer activations (4x1)
</span><span class="n">out</span> <span class="o">=</span> <span class="n">np</span><span class="p">.</span><span class="n">dot</span><span class="p">(</span><span class="n">W3</span><span class="p">,</span> <span class="n">h2</span><span class="p">)</span> <span class="o">+</span> <span class="n">b3</span> <span class="c1">#output neuron (1x1)
</span></code></pre></div></div>
<p>Full implementation of training a <em>2</em>-layer Neural Network:</p>
<div class="language-python highlighter-rouge"><div class="highlight"><pre class="highlight"><code><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="n">np</span>
<span class="kn">from</span> <span class="nn">numpy.random</span> <span class="kn">import</span> <span class="n">randn</span>
<span class="n">N</span><span class="p">,</span> <span class="n">D_in</span><span class="p">,</span> <span class="n">H</span><span class="p">,</span> <span class="n">D_out</span> <span class="o">=</span> <span class="mi">64</span><span class="p">,</span> <span class="mi">1000</span><span class="p">,</span> <span class="mi">100</span><span class="p">,</span> <span class="mi">10</span> <span class="c1"># Define the network
</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">randn</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="n">D_in</span><span class="p">),</span> <span class="n">randn</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="n">D_out</span><span class="p">)</span>
<span class="n">w1</span><span class="p">,</span> <span class="n">w2</span> <span class="o">=</span> <span class="n">randn</span><span class="p">(</span><span class="n">D_in</span><span class="p">,</span> <span class="n">H</span><span class="p">),</span> <span class="n">randn</span><span class="p">(</span><span class="n">H</span><span class="p">,</span> <span class="n">D_out</span><span class="p">)</span>
<span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">2000</span><span class="p">):</span> <span class="c1"># Forward pass
</span> <span class="n">h</span> <span class="o">=</span> <span class="mi">1</span> <span class="o">/</span> <span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">np</span><span class="p">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">x</span><span class="p">.</span><span class="n">dot</span><span class="p">(</span><span class="n">w1</span><span class="p">)))</span>
<span class="n">y_pred</span> <span class="o">=</span> <span class="n">h</span><span class="p">.</span><span class="n">dot</span><span class="p">(</span><span class="n">w2</span><span class="p">)</span>
<span class="n">loss</span> <span class="o">=</span> <span class="n">np</span><span class="p">.</span><span class="n">square</span><span class="p">(</span><span class="n">y_pred</span> <span class="o">-</span> <span class="n">y</span><span class="p">).</span><span class="nb">sum</span><span class="p">()</span>
<span class="k">print</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">loss</span><span class="p">)</span>
<span class="n">grad_y_pred</span> <span class="o">=</span> <span class="mf">2.0</span> <span class="o">*</span> <span class="p">(</span><span class="n">y_pred</span> <span class="o">-</span> <span class="n">y</span><span class="p">)</span> <span class="c1"># Calculate the analytical gradients
</span> <span class="n">grad_w2</span> <span class="o">=</span> <span class="n">h</span><span class="p">.</span><span class="n">T</span><span class="p">.</span><span class="n">dot</span><span class="p">(</span><span class="n">grad_y_pred</span><span class="p">)</span>
<span class="n">grad_h</span> <span class="o">=</span> <span class="n">grad_y_pred</span><span class="p">.</span><span class="n">dot</span><span class="p">(</span><span class="n">w2</span><span class="p">.</span><span class="n">T</span><span class="p">)</span>
<span class="n">grad_w1</span> <span class="o">=</span> <span class="n">x</span><span class="p">.</span><span class="n">T</span><span class="p">.</span><span class="n">dot</span><span class="p">(</span><span class="n">grad_h</span> <span class="o">*</span> <span class="n">h</span> <span class="o">*</span> <span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">h</span><span class="p">))</span>
<span class="n">w1</span> <span class="o">-=</span> <span class="mf">1e-4</span> <span class="o">*</span> <span class="n">grad_w1</span> <span class="c1"># Gradient descent
</span> <span class="n">w2</span> <span class="o">-=</span> <span class="mf">1e-4</span> <span class="o">*</span> <span class="n">grad_w2</span>
</code></pre></div></div>
<ul>
<li>
<p>Plugging in neural networks with loss functions<br />
$s = f(x;W_1,W_2) = W_2\mbox{max}(0,W_1 x)$ Nonlinear score function<br />
$L_i = \sum_{j\ne y_i}\mbox{max}(0,s_j-s_{y_i}+1)$ SVM Loss on predictions<br />
$R(W)=\sum_k W_k^2$ Regularization<br />
$L=\frac{1}{N}\sum_{i=1}^N L_i + \lambda R(W_1) + \lambda R(W_2)$ Total loss: data loss + regularization</p>
</li>
<li>
<p>Problem: How to compute gradients?<br />
If we can compute partial derivaties, then we can learn $W_1$ and $W_2$.</p>
</li>
</ul>
<h3 id="backpropagation">Backpropagation</h3>
<ul>
<li>
<p>Chain rule:<br />
\(\begin{align*}
\frac{\partial f}{\partial y}=\frac{\partial f}{\partial q} \frac{\partial q}{\partial y} \mbox{Upstream gradient} \times \mbox{Local gradient}
\end{align*}\)</p>
</li>
<li>
<p>Patterns in gradient flow<br />
<img src="/assets/images/cs231n_lec4_3.png" alt="png" width="60%", height="60%" /></p>
</li>
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